text:
\section{Introduction}

Gromov introduced the simplicial volume $\| M\| $ of closed manifolds in the beginning of the 80's and established the following general proportionality principle for closed manifolds.

\begin{thm}[Gromov-Thurston \cite{Gromov,Th78}] Let $(M,g)$ be a closed Riemannian manifold. Then there exists a constant $c(\widetilde{M},\widetilde{g})\in \mathbb{R}_{>0}\cup \{+\infty \}$ depending only on the universal cover $(\widetilde{M},\widetilde{g})$ such that
$$ \| M\| =\frac{\mathrm{Vol}(M,g)}{c(\widetilde{M},\widetilde{g})}.$$
\end{thm}

This fundamental principle, which gives an alternative for the Hirzebruch proportionality principle in odd dimension, allows, whenever $\| M\|\neq 0$, to consider the Riemannian volume of manifolds with a fixed universal cover as a topological invariant.

For symmetric spaces of noncompact type, the proportionality constant is easily seen to be equal to the sup norm $\|  \omega_{\widetilde{M}}\|_\infty \in \mathbb{R}_{>0} $ of the volume form  $\omega_{\widetilde{M}} \in H^n_c(\mathrm{Isom}(\widetilde{M}))$ (see \cite{Bucher}). The fact that in the compact case, the simplicial volume of locally symmetric spaces of non compact type is strictly positive  was established by Lafont and Schmidt (see also \cite{Sa82} and \cite{Bu07b} for the case of locally $\mathrm{SL}(3,\mathbb{R})/\mathrm{SO}(3)$ factors) answering a conjecture of Gromov. As a consequence the proportionality constant $\|  \omega_{\widetilde{M}}\|_\infty$ is finite for symmetric spaces $\widetilde{M}$ of noncompact type.

For open manifolds, the natural generalization of the simplicial volume, is the $\ell^1$-norm of the locally finite fundamental class of $M$, which we denote by $\| M\|_\mathrm{lf}$ (see Section \ref{section: Locally finite and relative homology} below for the definition). It is however in general not well behaved as far as the proportionality principle is concerned. Indeed, Gromov showed \cite{Gromov} that $\| M_1\times M_2 \times M_3 \|_\mathrm{lf}=0$ for any open manifolds $M_1,M_2,M_3$. L\"oh and Sauer generalized this vanishing result for symmetric spaces by showing that $\| M \|_\mathrm{lf}=0$ for any ${\mathbb Q}$-rank $\geq 3$ locally symmetric space \cite{LoehSauerDegreeThms}. On the positive side, Gromov and Thurston noted that the proportionality principle holds for the locally finite simplicial volume of complete hyperbolic manifolds of finite volume. This result was extended by L\"oh and Sauer to symmetric spaces with amenable cusp fundamental group \cite{LoehSauerHilbert}, which includes all  $\mathbb{R}$-rank $1$ symmetric spaces and  covers some $\mathbb{Q}$-rank $1$ symmetric spaces, such as Hilbert modular varieties. A further generalization to Riemannian manifolds with amenable boundary has just been provided by the third author and Kuessner \cite{Kim-Kuessner}.

L\"oh and Sauer's proof is based on a discrete approximation of smearing in measure homology and on Gromov's equivalence theorem \cite[page 57]{Gromov}, which has just recently admitted a straightforward (and complete) proof \cite{BBFIPP}. The  proof in \cite{Kim-Kuessner}  builds on Gromov's theory of multicomplexes \cite[Section 3]{Gromov} and the duality between $\ell^1$-homology and bounded cohomology. We propose an alternative complete self contained proof based on standard techniques from bounded cohomology. We recall the precise statement of L\"oh and Sauer's result which involves also the relative simplicial volume whose definition is given in Section \ref{section: Locally finite and relative homology}:

\begin{thm}[\cite{LoehSauerHilbert}] \label{thm:PropPrincLocFin}Let $V$ be a compact manifold with boundary $\partial V$ such that its interior $M=\mathrm{Int}(V)$ is a
 complete finite volume locally symmetric space. If the fundamental group of any connected component of $\partial V$ is amenable, then
$$\| V,\partial V \|=\| M\|_\mathrm{lf}=\frac{\mathrm{Vol}(M)}{\| \omega_{\widetilde{M}}\| _\infty}.$$
\end{thm}

\begin{co}[\cite{LoehSauerHilbert}] Let $M$ be as in the theorem. Then
$$\| M\|_\mathrm{lf}>0.$$
\end{co}

Other important consequences include the existence of degree theorems or positivity of the minimal volume (see \cite{LoehSauerHilbert}).

Note that in the case of a hyperbolic manifold $V$ of dimension $\geq 3$ with geodesic boundary $V$, Jungreis \cite{Jungreis} and Kuessner \cite{Kuessner} showed the strict inequality $\| V,\partial V \| <\frac{ \mathrm{Vol}(V)}{ \| \omega_{\mathbb{H}^n}\|_\infty}$. (See also \cite{FrigerioPagliantini} for a proof that this strict inequality is optimal.)

The proportionality principle in Theorem \ref{thm:PropPrincLocFin} is established only for the case that a compact manifold $V$ has amenable ends.
The following theorem implies that the proportionality principle may hold even though the fundamental group of a connected component of $\partial V$ is not amenable.

\begin{thm}\label{thm:1.4} Let $M$ be a $\mathbb{Q}$-rank $1$ locally symmetric $n$-space covered by a product of ${\mathbb R}$-rank $1$ symmetric spaces. Let $H$ be  the isometry group of the universal cover $\widetilde{M}$ of $M$. Suppose that the comparison map $c : H^n_{c,b}(H,{\mathbb R}_\varepsilon) \rightarrow H^n_c(H,{\mathbb R}_\varepsilon)$ is an isomorphism. Then
$$\| M\|_\mathrm{lf}=\frac{\mathrm{Vol}(M)}{\| \omega_{\widetilde{M}}\| _\infty}.$$
\end{thm}

The cohomology groups will be introduced in Section \ref{section: Cont coho}. Note that the comparison map is known to be surjective in top degree, while its injectivity is only conjecturally true. When $\widetilde{M}$ is the real hyperbolic $n$-space ($n\geq 2$), the injectivity was established in \cite{BBI}. Since the proof of the validity of the K\"unneth formula passes to bounded cohomology, the following corollary is immediate:

\begin{co}\label{co: hyp prod}Let $M$ be a $\mathbb{Q}$-rank $1$ locally symmetric $n$-space covered by a product of real hyperbolic spaces $\mathbb{H}^{n_1}\times ... \times\mathbb{H}^{n_k}$ with $n_1,...,n_k\geq 2$.  Then
$$\| M\|_\mathrm{lf}=\frac{\mathrm{Vol}(M)}{\| \omega_{\mathbb{H}^{n_1}\times ... \times \mathbb{H}^{n_k}}\| _\infty}.$$
\end{co}

Theorem \ref{thm:1.4} and Corollary \ref{co: hyp prod} in particular hold for reducible $\mathbb{Q}$-rank $1$ locally symmetric spaces. Note that a reducible $\mathbb{Q}$-rank $1$ locally symmetric space is a product of an irreducible $\mathbb{Q}$-rank $1$ locally symmetric space and closed locally symmetric spaces. Hence, it is easy to see that each component of its ends does not have amenable fundamental group.
To our knowledge, this gives the first class of examples of $\mathbb{Q}$-rank $1$ locally symmetric spaces for which the proportionality principle is known without the assumption that the fundamental group of each component of their ends is amenable.

\subsection*{On the proofs} The idea of the proof of Theorem \ref{thm:PropPrincLocFin} is as follows. Let $\omega_M\in H^n_{cpct}(M)$ be the top dimensional compact support cohomology class which equals to the volume of $M$ when evaluated on the locally finite fundamental class $[M]_\mathrm{lf}\in H_n^\mathrm{lf}(M)$ (see Sections \ref{section: Locally finite and relative homology} and \ref{section: Cohomology with compact support} for more details). Two seminorms $\| \omega_M \|_\infty$ and $\| \omega_M \|^\infty$ are defined on $H^n_{cpct}(M)$ and it is a simple consequence of the Hahn-Banach theorem (see \cite{Gromov}, \cite{Loeh} or Theorems \ref{thm:dualityFORopen} and \ref{thm:dualityFORrelative} here) that
$$ \| M\|_\mathrm{lf}=\frac{\mathrm{Vol}(M)}{\| \omega_{{M}}\| ^\infty} \ \ \mathrm{and} \ \  \| V,\partial V \|=\frac{\mathrm{Vol}(M)}{\| \omega_{{M}}\| _\infty}.$$

To prove Theorem \ref{thm:PropPrincLocFin} we hence just need to show that
$$\| \omega_{{M}}\| _\infty = \| \omega_{{M}}\| ^\infty = \| \omega_{\widetilde{M}}\| _\infty,$$
as stated in Proposition \ref{mainProp}. In case $M$ is closed, the compact support cohomology of $M$ is nothing else than the usual singular cohomology, the norms $\| \omega_{{M}}\| _\infty $ and $ \| \omega_{{M}}\| ^\infty $ are easily seen to be equal and the equality  $\| \omega_{{M}}\| _\infty = \| \omega_{\widetilde{M}}\| _\infty$ follows from the fact that the top dimensional isomorphism
$$H^n_c(\mathrm{Isom}(\widetilde{M}))\rightarrow H^n(\pi_1(M))\cong H^n(M)$$
is isometric (see \cite{Bucher} for more details on the compact case).

If $M$ is noncompact, then $H^n(\pi_1(M))=H^n(M)=0$ and the above approach fails. We choose to pass instead to bounded cohomology. We introduce transfer maps
\[
\begin{CD}
  H^n_{cpct, b}(M)    @>trans_b>>          H^n_{c,b}(G,{\mathbb R}) \\
                      @VV c V                             @VV c V \\
  H^n_{cpct}(M) @>trans>>  H^n_{c}(G,{\mathbb R})    \end{CD} \]
directly on the compact support cohomology in such a way that the unbounded transfer map agrees with the de Rham transfer (see Section \ref{Section: Transfer}) and hence sends the volume class $\omega_M$ to $\omega_{\widetilde{M}}$. Our proof amounts to showing that the transfer map
\[
\begin{CD}
  trans:H^n_{cpct}(M) @> >>  H^n_{c}(G,{\mathbb R})    \end{CD} \]
  is an isometric isomorphism in top dimension.

Transfer maps in cohomology are the dual analogue of smearing in homology, but the cohomology approach has the striking advantage that it does not rely on the technical and difficult fact that measure homology is isometrically isomorphic to $\ell^1$-homology. Instead, all our proofs are straightforward and self contained.

The proof of Theorem \ref{thm:1.4} is a development of ideas of \cite{Kim-Kim}, where the positivity of the simplicial volume for ${\mathbb Q}$-rank $1$ lattices lying in the product of ${\mathbb R}$-rank $1$ simple groups is proven. The injectivity of the comparison map allows to conclude that $\| \omega_{{M}}\| ^\infty \leq \| \omega_{\widetilde{M}}\| _\infty$ (compare with the inequality $\| \omega_{{M}}\| ^\infty \leq \|\Theta \| _\infty<+\infty$, where $\Theta$ is one particular bounded cocycle representing $ \omega_{\widetilde{M}}$ established in \cite{Kim-Kim} for the mere positivity of the simplicial volume), and the other inequality is proven in Section \ref{sec:5.3} for the proof of Theorem \ref{thm:PropPrincLocFin}.

\subsection*{Structure of the paper} We recall some basics of locally finite and relative homology in Section \ref{section: Locally finite and relative homology}, cohomology with compact support in Section \ref{section: Cohomology with compact support}  and continuous group cohomology in Section \ref{section: Cont coho}. We introduce our new transfer maps in Section \ref{Section: Transfer} and establish there a few simple properties, in particular the first inequality of Proposition \ref{mainProp}. In Section \ref{Section: coning}, we prove the second inequality of Proposition \ref{mainProp}  by exhibiting a norm nonincreasing cohomology map $f^*:H^*_b(\pi_1(M))\rightarrow H^*_{cpct,b}(M)$ induced by a cusp map. This map should be of independent interest. Finally, the proof of Theorem \ref{thm:1.4}  is given is section \ref{section: product of rank 1}

\section{Locally finite and relative homology}\label{section: Locally finite and relative homology}

Let $M$ be a manifold.
Borel and Moore \cite{Borel-Moore} introduce a homology theory for locally compact spaces, called the Borel-Moore homology. There are several ways to describe this homology theory and we will define it here as the homology of the locally finite chain complex of $M$ and call this the locally finite homology of $M$.

Let $R$ be a field of characteristic zero.
The locally finite chain complex $C^\mathrm{lf}_*(M,R)$ of $M$ with coefficients in $R$ is defined by the chain complex of infinite singular chains $c=\sum_{i=0}^\infty a_i \sigma_i$ where $\sigma_i$ is a singular simplex, $a_i \in R$ and the sum is locally finite in the following sense: Any compact subset of $M$ intersects the image of only finitely many singular simplices occurring in $c$.
The usual boundary map $\partial$ on the singular chains is well-defined on $C^\mathrm{lf}_*(M,R)$. Then,
the locally finite homology $H^\mathrm{lf}_*(M,R)$ with coefficients in $R$ is defined by
$$H^\text{lf}_* (M,R)=H_*(C^\mathrm{lf}_*(M,R), \partial).$$

For compact manifolds, the locally finite homology coincides with the usual singular homology, but
it gives useful homology groups for noncompact manifolds. An essential  advantage of locally finite homology as opposed to singular homology is the existence of a fundamental class of any oriented manifold. Indeed, the $n$-th singular homology of a noncompact manifold vanishes, and hence cannot contain any fundamental class. The existence of a well-defined fundamental class $[M]\in H^\mathrm{lf}_n(M,\mathbb{R})$ for any oriented $n$-dimensional manifold is established via the Poincar\'e duality
$$H^\mathrm{lf}_i(M,R) \cong H_{cpct}^{n-i}(M,R)$$
with cohomology with compact support, a cohomology theory which will be briefly recalled in the next section. The duality implies that $H^\mathrm{lf}_n(M,\mathbb{R}) \cong H_{cpct}^{0}(M,\mathbb{R})\cong \mathbb{R}$ and the existence of a canonical fundamental class $[M]\in H^\mathrm{lf}_n(M,\mathbb{R})$. We refer the reader to \cite{Borel-Moore} and \cite{Bredon} for more detailed explanations about locally finite homology.

The simplicial $\ell^1$-seminorm in $C^\mathrm{lf}_*(M,\mathbb{R})$ is defined by setting $\| c\|_1 =\sum_{i=0}^\infty |a_i|$ for a chain $c=\sum_{i=0}^\infty a_i \sigma_i$ in $C^\mathrm{lf}_*(M,\mathbb{R})$. This norm gives rise to a seminorm on the locally finite homology $H^\mathrm{lf}_*(M,\mathbb{R})$ as follows:
$$ \| \alpha \|_1 = \inf_{z} \|z \|_1,$$
where the infimum is taken over all locally finite cycles representing $\alpha \in H^\mathrm{lf}_*(M,\mathbb{R})$. For an oriented $n$-manifold $M$, the simplicial volume $\| M \|_\mathrm{lf}$ of $M$ is defined as the seminorm of its fundamental class $[M]\in H^\mathrm{lf}_n(M,\mathbb{R})$.

Now, let $V$ be a compact $n$-manifold with boundary $\partial V$. For a relative chain $c$ in the relative singular chain complex $C_*(V,\partial V,\mathbb{R})$, the simplicial $\ell^1$-norm is given by the infimum of the $\ell^1$-norms of its representatives. This norm induces a seminorm on the relative singular homology $H_*(V,\partial V,\mathbb{R})$ as in the case of locally finite homology. Then, the relative simplicial volume $\| V,\partial V \|$ is defined as the seminorm of its relative fundamental class $[V,\partial V] \in H_*(V,\partial V,\mathbb{R})$.

Let $M$ be the interior of $V$. Suppose that $M$ is an oriented manifold. We can obtain the inequality $\| V,\partial V \| \leq \| M\|_\mathrm{lf}$ as follows: Consider a compact submanifold $V_0$ of $M$ by removing a collared neighborhood of $\partial V$. It is clear that $V_0$ is homeomorphic to $V$. Let $c=\sum_{i=0}^\infty a_i \sigma_i$ be any locally finite cycle representing the fundamental class $[M] \in H^\mathrm{lf}_n(M,\mathbb{R})$. It is a standard fact that $c|_{V_0}=\sum_{im \sigma_i \cap V_0 \neq \emptyset} a_i \sigma_i$
represents the relative fundamental class $[V, V-V_0] \in H_n(V,V-V_0,\mathbb{R})$.
Because $V-V_0$ is the collared neighborhood of $\partial V$, one can think of $c|_{V_0}$ as a cycle representing
$[V,\partial V] \in H_n(V,\partial V)$. Clearly, we have an inequality
$$ \| c|_{V_0} \|_1 \leq \|c\|_1. $$
Taking the infimum over all locally finite cycles $c$ representing $[M]$, we have
$$ \|V,\partial V \| =\inf_{c'} \|c' \|_1 \leq  \inf_{c} \| c|_{V_0} \|_1 \leq \inf_{c} \|c\|_1 = \|M\|_\mathrm{lf},$$
where $c'$ runs over all relative cycles representing $[V,\partial V]$.

\section{Cohomology with compact support}\label{section: Cohomology with compact support}
\subsection{Singular cohomology}\label{section:Singular cohomology}
Let $M$ be an oriented $n$-manifold.
A singular cochain $f : C_q(M,{\mathbb R}) \rightarrow {\mathbb R}$ is said to be a cochain with compact support if there exists a compact subset $K$ such that $f(\sigma)=0$ whenever $\mathrm{Im}(\sigma)\cap K=\emptyset $.
The cohomology with compact supports of $M$, denoted by $H^*_{cpct}(M,{\mathbb R})$, is defined as the cohomology of the subcocomplex $C^*_{cpct}(M,{\mathbb R})$ of cochains with compact support in $C^*(M,{\mathbb R})$ with the usual coboundary operator $\delta$ on the singular cochain complex. For more details, see \cite[Chapter 3]{Ha02}.

Consider the sup norm with respect to singular simplices, that is, for a cochain $f\in C^q_{cpct}(M,{\mathbb R})$, set
$$\| f\|_\infty = \sup_{\sigma} |f(\sigma)|,$$
where $\sigma :\Delta^q \rightarrow M $ runs over all singular $q$-simplices.

Noting that $\| \delta(f)\|_\infty \leq (q+2)\cdot \| f\|_\infty$, we can consider the subcocomplex of bounded cochains with compact support
$$C^q_{cpct,b}(M,{\mathbb R})=\{f\in C^q_{cpct}(M,{\mathbb R})\mid \| f\|_\infty <+\infty \}$$
which gives rise to the bounded cohomology with compact support, denoted by $H^*_{cpct,b}(M,{\mathbb R})$. The inclusion of cocomplexes induces a cohomology homomorphism
$$c:H^*_{cpct,b}(M,{\mathbb R})\longrightarrow H^*_{cpct}(M,{\mathbb R})$$
which is traditionally named {\it comparison map}.

The sup norm induces a seminorm $\| \cdot \|_\infty$ on cohomology, both in the bounded and unbounded case.
For $\beta \in H^q_{cpct}(M,{\mathbb R})$, its seminorm is defined by
$$ \|\beta\|_\infty = \inf\{\|b\|_\infty \mid b\in C^q_{cpct}(M,{\mathbb R}), \  \delta b=0, \ [b]=\beta\}$$
where $[b]$ as usual denotes the cohomology class represented by the cocycle $b$.
A seminorm $\| \cdot \|_\infty$ is defined on $H^q_{cpct,b}(M,{\mathbb R})$ as above.
Note that
$$ \|\beta\|_\infty=\inf\{\|\beta_b\|_\infty \mid \beta_b\in H^q_{cpct,b}(M,{\mathbb R}), \ c(\beta_b)=\beta\}.$$

\subsection{De Rham cohomology}\label{derham}

Let $M$ be a smooth manifold. Let $\Omega_{cpct}^q(M)$ denote the set of all differential $q$--forms on $M$ with compact support. The de Rham cohomology of $M$ with compact support, denoted by $H^*_{dR,cpct}(M,\mathbb{R})$, is defined as the cohomology of the cocomplex $\Omega_{cpct}^*(M)$ with the standard exterior derivative. By \cite{deRham}, the cohomology $H^*_{cpct}(M,{\mathbb R})$ with compact support can be computed as $H^*_{dR,cpct}(M,\mathbb{R})$.

It is a standard fact that there is no difference between singular and smooth singular cohomology (with compact support).  Hence, we will use smooth singular cohomology with compact support to describe the de Rham isomorphism between de Rham cohomology and singular cohomology with compact support.
To denote smooth singular cohomology with compact support, we use the same notations as in Section \ref{section:Singular cohomology}.
For $\omega \in \Omega_{cpct}^q(M)$ and a smooth $q$-simplex $\sigma : \Delta^q \rightarrow M$, define a map $$I(\omega)(\sigma)=\int_\sigma \omega := \int_{\Delta^q} \sigma^* \omega,$$
and extend this linearly on $C_q(M,{\mathbb R})$. Then, it can be easily seen that $I : \Omega_{cpct}^*(M) \rightarrow C_{cpct}^*(M,{\mathbb R})$ is well-defined and a cochain map. Furthermore, this cochain map induces an isomorphism
$I : H^*_{dR,cpct}(M,{\mathbb R}) \rightarrow H^*_{cpct}(M,\mathbb{R})$ in cohomology.
This is known as the de Rham theorem in the case that $M$ is a closed manifold. We refer the reader to \cite[Chapter IV]{Demaily} or \cite[Chapter 16]{Lee} for further details.

Now, suppose that $M$ is a $\mathbb{Q}$-rank $1$ locally symmetric space.
Note that the geodesic straightening map $str : C^\mathrm{lf}_*(M,\mathbb{R}) \rightarrow C^\mathrm{lf}_*(M,\mathbb{R})$ is well-defined and moreover, chain homotopic to the identity \cite{Kim-Kim}.
Hence, it can be seen that if $f$ is a $q$-cochain with compact support, $f\circ str : C_q(M,{\mathbb R}) \rightarrow {\mathbb R}$ is also a cochain with compact support.
This allows us to have a map $C^*_{cpct}(M,{\mathbb R}) \rightarrow C^*_{cpct}(M,{\mathbb R})$ defined by $f \mapsto f\circ str$. Furthermore, this map induces the identity map in cohomology since the geodesic straightening map $str$ is chain homotopic to the identity. Thus, the de Rham isomorphism $I : H^*_{dR,cpct}(M,{\mathbb R}) \rightarrow H^*_{cpct}(M,\mathbb{R})$ can also be induced, at cochain level, by the map
$$I_{str}(\omega)(\sigma)=I(\omega)(str(\sigma))=\int_{str(\sigma)} \omega.$$
Observe that for arbitrary manifolds with a given straightening, the map $I_{str}$ is not well defined on the compact support cohomology, as the straightening will not map locally finite chains to locally finite chains in general.

\subsection{Dual norms}

There is a natural pairing
$$\langle \ , \ \rangle:H^q_{cpct}(M,{\mathbb R})\otimes H^\mathrm{lf}_q(M,\mathbb{R})\longrightarrow {\mathbb R}$$
 between cohomology with compact support and locally finite homology defined for $\beta=[b]\in H^q_{cpct}(M,{\mathbb R})$ and $\alpha=[z]\in H_q^\mathrm{lf}(M,\mathbb{R})$ as
 $$\langle \beta , \alpha \rangle=\langle [b],[z]\rangle:= b(z).$$

It is straightforward to check that this definition is independent of the choices of cocycles and cycles representing $\beta$ and $\alpha$ respectively. It is now easy to show that
\begin{equation}  | \langle \beta , \alpha \rangle|\leq \| \beta\|_\infty \cdot \| \alpha \|_1,\label{equ:dualityInequalityFORnorm_}\end{equation}
but equality does not hold in general even for $\alpha=[M]$. Examples are given below. In order to get a duality of norms in top dimension, Gromov introduced \cite[page 17]{Gromov} a different norm on cohomology with compact
support as follows : Let $\mathcal{S}_q$ be the set of singular $q$-simplices $\Delta^q \rightarrow M$.
A subset $\Phi$ of $\mathcal{S}_q$ is said to be locally finite if any compact subset of $M$ intersects the image of only finitely many elements of $\Phi$.
Let $\mathcal{S}_q^\mathrm{lf}$ denote the set of all locally finite subsets of $\mathcal{S}_q$.

For a $q$-cochain $b$ with compact support in $C^q_{cpct}(M,{\mathbb R})$, define a seminorm $\| b \|_\Phi$ by setting $$\| b \|_\Phi = \sup_{\sigma \in \Phi} |b(\sigma)|,$$
for each $\Phi \in \mathcal{S}^\mathrm{lf}_q$. Subsequently, obtain a seminorm $\| \cdot \|_\Phi$ on $H^q_{cpct}(M,{\mathbb R})$ and take the supremum of these seminorms over all locally finite subsets of $\mathcal{S}_q$.
Then, we have a new seminorm $\| \cdot \|^\infty$ on $H^q_{cpct}(M,{\mathbb R})$ defined by
$$\|\beta \| ^\infty = \sup_{\Phi \in \mathcal{S}^\mathrm{lf}_q} \| \beta \|_\Phi=\sup_{\Phi \in \mathcal{S}^\mathrm{lf}_q}\inf_{\beta=[b]}\sup_{\sigma\in \Phi}|b(\sigma)|.$$

Observe that for any $\Phi \in \mathcal{S}^\mathrm{lf}_q$, any $\sigma\in \Phi$ and $b\in C^q_{cpct}(M,{\mathbb R})$, we have
$$ \sup_{\sigma\in\Phi}|b(\sigma)|\leq \|b\|_\infty.$$
Hence it follows that
\begin{equation}
\left\Vert
\beta\right\Vert ^{\infty}\leq\left\Vert \beta\right\Vert
_{\infty}.\label{equ:norm^LEQnorm_}
\end{equation}

In general the above inequality is not an equality. Take $V=[0,1]$
and $M=(0,1)$ with a standard Euclidean metric. Let $\omega_M\in
H^1_{cpct}(M)$ be the unique cohomology class with $\langle
\omega_M, [M]\rangle=1$. It is not difficult to show that
$\|M \|_\mathrm{lf}=\infty$, whereas $\| V,\partial V \| =1<\infty$. In view of
Theorems \ref{thm:dualityFORopen} and
\ref{thm:dualityFORrelative}, $||\omega_M||^\infty=0$ and
$||\omega_M||_\infty>0$ for $M=(0,1)$.

\begin{lemma} \label{Lemma:dualityInequalityFORnorm^} Let $\beta \in H^q_{cpct}(M,{\mathbb R})$ and $\alpha \in H_q^\mathrm{lf}(M,\mathbb{R})$. Then
 $$| \langle \beta , \alpha \rangle |\leq \|\beta\|^\infty \cdot \|\alpha \|_1. $$
\end{lemma}

\begin{proof}
Let $f$ be a representative of $\beta$ and $z=\sum_{\sigma \in \Phi} a_\sigma \sigma$ be a representative of $\alpha$. Then,
$$ | \langle \beta , \alpha \rangle |
= | \langle f , z \rangle |
= \Big|\sum_{\sigma \in \Phi} a_\sigma \cdot f(\sigma) \Big|
\leq \| f \|_\Phi \cdot \sum_{\sigma \in \Phi} | a_\sigma | = \| f \|_\Phi \cdot \| z \|_1.$$
Taking the infimum over all representatives of $\beta$ and then taking the supremum over all $\Phi$,
$$ | \langle \beta , \alpha \rangle |
\leq \| \beta \|_\Phi \cdot \| z \|_1
\leq \| \beta \|^\infty \cdot \| z \|_1.$$
The desired inequality follows by taking the infimum over all representatives of $\alpha$.
\end{proof}

Note that in view of (\ref{equ:norm^LEQnorm_}) the inequality (\ref{equ:dualityInequalityFORnorm_}) immediately follows from the lemma. Gromov \cite{Gromov} showed that the simplicial volume of a non-compact manifold can be computed in terms of this norm $\| \cdot \|^\infty$ of the dual class
in cohomology with compact support as follows:

\begin{thm}\label{thm:dualityFORopen} Let $M$ be an oriented $n$-manifold, and let $0\neq \beta\in H^n_{cpct}(M,{\mathbb R})$. Then,
 $$\frac{| \langle \beta , [M] \rangle |}{\|\beta\|^\infty}= \|M\|_\mathrm{lf}. $$
\end{thm}

The original sup norm $\| \beta \|_\infty$ is still of interest, as it turns out to be dual to the $\ell^1$-norm of manifolds $V$ with boundary $\partial V$, in the case where $M$ is the interior of $V$.

\begin{thm}\label{thm:dualityFORrelative} Let $V$ be a compact, oriented $n$-manifold with boundary $\partial V$,
and let $M$ denote its interior. Let $0\neq \beta\in
H^n_{cpct}(M,{\mathbb R})$. Then,
 $$\frac{| \langle \beta , [M] \rangle |}{\|\beta\|_\infty}= \|V,\partial V \|. $$
 \end{thm}

For detailed proofs of Theorem \ref{thm:dualityFORopen} and \ref{thm:dualityFORrelative},
we refer the reader to \cite{Loeh}.
Observe that Theorems \ref{thm:dualityFORopen} and \ref{thm:dualityFORrelative}, together with the simple inequality (\ref{equ:norm^LEQnorm_}) immediately reproves the inequality
 \[ \left\Vert V,\partial V\right\Vert \leq\left\Vert
M\right\Vert_\mathrm{lf} ,\]
when $M$ is the interior of $V$.

Let $\omega_M\in H^n_{cpct}(M,{\mathbb R})$ denote the unique cohomology class defined by $\langle \omega_M,[M]\rangle=\mathrm{Vol}(M)$. In view of Theorems \ref{thm:dualityFORopen} and \ref{thm:dualityFORrelative}, our proportionality principles in Theorem \ref{thm:PropPrincLocFin}  will immediately follow from:
\begin{Prop}\label{mainProp}There is an equality of norms
$$ \| \omega_M \|_\infty = \| \omega_M \|^\infty = \| \omega_{\widetilde{M}}\|_\infty.$$
\end{Prop}
Given the inequality $ \| \omega_M \|_\infty \geq  \| \omega_M \|^\infty $ established above (\ref{equ:norm^LEQnorm_}), to prove the proposition, and hence Theorem  \ref{thm:PropPrincLocFin} it suffices to prove the inequalities
\begin{itemize}
\item $  \| \omega_M \|^\infty \geq \| \omega_{\widetilde{M}}\|_\infty$; this will be proven at the end of Section \ref{Section: Transfer},
\item $ \| \omega_M \|_\infty \leq \| \omega_{\widetilde{M}}\|_\infty$; this will be proven at the end of Section \ref{Section: coning}.
\end{itemize}

\section{Continuous cohomology}\label{section: Cont coho}

Let $G$ be a topological group. Recall that the continuous cohomology $H^*_c(G,\mathbb{R})$ of $G$ (with trivial $\mathbb{R}$ coefficients) is the cohomology of the cocomplex
$$C^{q+1}(G,\mathbb{R})=\{ f: G^{q+1}\rightarrow \mathbb{R} \mid f \mathrm{ \ continuous \,and \,  } G\mathrm{-invariant} \},$$
endowed with its homogeneous coboundary operator $\delta:C^{q+1}(G,\mathbb{R})\rightarrow C^{q+2}(G,\mathbb{R})$. The subcocomplex of bounded functions
$$C^{q+1}_b(G,\mathbb{R})=\{ f\in C^{q+1}(G,\mathbb{R}) \mid \| f \|_\infty = \sup_{g_0,...,g_q\in G} |f(g_0,...,g_q)|<+\infty   \}$$
leads to the continuous bounded cohomology $H^q_{c,b}(G,\mathbb{R})$ of $G$. The inclusion of cocomplexes $C^{q+1}_b(G,\mathbb{R})\subset C^{q+1}(G,\mathbb{R})$ induces a cohomology map $H^q_{c,b}(G,\mathbb{R})\rightarrow H^q_c(G,\mathbb{R})$. The sup norm defines a cohomology seminorm as
$$\| \beta \|_\infty = \inf \{ \| f\|_\infty \mid [f]=\beta, \ f\in C^{q+1}_b(G,\mathbb{R})\},$$
for $\beta $ in $H^q_c(G,\mathbb{R})$ or $H^q_{c,b}(G,\mathbb{R})$. Note that in the case where $\beta\in H^q_c(G,\mathbb{R})$ cannot be represented by a bounded cocycle, the infimum over the empty set is defined as $\| \beta \|_\infty = +\infty $.

If the group $G$ is endowed with the discrete topology, then the continuity condition is void and we remove the subscript ``c" from the notation. Note that one then recovers the Eilenberg-MacLane group cohomology.

Many different cocomplexes can be used to compute these cohomology groups. An important example is the following: Let $K$ be a maximal compact subgroup of $G$. The cohomology of the cocomplex of $G$-invariant continuous functions on the product of $q+1$ copies of $G/K$,
$$C((G/K)^{q+1},\mathbb{R})^G=\{ f: (G/K)^{q+1}\rightarrow \mathbb{R} \mid f \mathrm{ \ continuous \, and \,  } G \mathrm{-invariant} \},$$
endowed with its homogeneous coboundary operator is isomorphic to the continuous cohomology of $G$ \cite[Ch. III, Prop. 2.3]{Guichardet}. Similarly, the subcocomplex of bounded functions on $(G/K)^{q+1}$,
$$C_b((G/K)^{q+1},\mathbb{R})^G=\{ f\in C((G/K)^{q+1},\mathbb{R})^G  \mid \| f \|_\infty <+\infty  \},$$
computes the continuous bounded cohomology of $G$ \cite[Cor. 7.4.10]{Monod}. Furthermore, the sup norm on these cocomplexes induces a seminorm on cohomology which agrees with the above defined seminorm \cite[Cor. 7.4.10]{Monod}.

For Theorem \ref{thm:1.4} we will need also nontrivial coefficients, namely we will consider actions of $G$ on ${\mathbb R}$ by multiplication by $\pm 1$ given by a homomorphism $G\rightarrow \mathbb Z /2\mathbb Z$ (see the beginning of Section \ref{section: product of rank 1}). The corresponding cohomology groups are obtained by replacing $G$-invariant by $G$-equivariant cochains in the various cochain complexes.

Let now $G$ be a Lie group and $\Gamma <G$ a lattice. The inclusion $\Gamma <G$ induces restriction maps $C^{q+1}(G,\mathbb{R})\rightarrow C^{q+1}(\Gamma,\mathbb{R})$ and $C((G/K)^{q+1},\mathbb{R})^G\subset C((G/K)^{q+1},\mathbb{R})^\Gamma$ which in turn induces cohomology maps
$$H^*_{c}(G,\mathbb{R})\rightarrow H^*(\Gamma,\mathbb{R}) \mathrm{ \ and \ } H^*_{c,b}(G,\mathbb{R})\rightarrow H^*_b(\Gamma,\mathbb{R}).$$
If $\Gamma$ is cocompact, both maps admit a norm decreasing left inverse, while if $\Gamma$ is not cocompact, only the bounded cohomology map admits a norm decreasing left inverse. The left inverse is given by a transfer map
$$trans_\Gamma:C((G/K)^{q+1},\mathbb{R})^\Gamma \rightarrow C((G/K)^{q+1},\mathbb{R})^G $$
which is, for $c\in  C((G/K)^{q+1},\mathbb{R})^\Gamma $ and $g_0K,...,g_qK\in G/K$ defined as follows:
$$  trans_\Gamma(c)(g_0K,...,g_qK)=\int_{g\in D } c(gg_0K,...,gg_qK)d\mu(g),$$
where $D$ is any fundamental domain for $\Gamma \setminus G$ and the Haar measure $\mu$ is normalized so that $\mu(D)=1$. Note that $ trans_\Gamma(c)$ is finite if either $c$ is a bounded cochain or if the lattice $\Gamma <G$ is cocompact. As the transfer map commutes with the coboundary operator, it induces in these cases cohomology maps which are indeed left inverse to the restriction map. Finally observe that the transfer map does not increase seminorms.

\section{Transfer maps}\label{Section: Transfer}

\subsection{Transfer on de Rham cohomology}\label{deRham}

It is easy to define a transfer map $H^*_{cpct}(M)\rightarrow H^*_c(G)$ through the de Rham cohomology with compact support and the Van Est isomorphism. Indeed, at the cochain level, one defines
$$trans_{dR}:\Omega^q(G/K)_{cpct}^\Gamma \longrightarrow \Omega^q(G/K)^G$$
by sending the differential $q$-form $\alpha\in \Omega^q(G/K)_{cpct}^\Gamma$ to the form $\int_{g\in D} g^*\alpha d\mu(g)$, or more precisely to the form defined, for $x\in G/K$ and $V_1,...,V_q\in T_x(G/K)$, by
$$trans_{dR}(\alpha)_x(V_1,...,V_q)=\int_{g\in D} \alpha_{gx}(g_*(V_1),...,g_*(V_q)) d\mu(g).$$
As above, $D$ is any fundamental domain for $\Gamma\setminus G$ and the Haar measure $\mu$ is normalized so that the measure of $D$ is $1$. It is easy to check that this definition is independent of the fundamental domain $D$ and that the resulting differential form $trans_{dR}(\alpha)$ is $G$-invariant. Furthermore, the transfer map clearly commutes with the differential operator, and hence induces a cohomology map
$$trans_{dR}:H^*_{dR,cpct}(M)\longrightarrow  H^*(\Omega^*(G/K)^G)\cong \Omega^*(G/K)^G\cong H^*_c(G).$$
Naturally, we now want to understand the transfer map on singular cohomology and in particular on bounded singular cohomology, or in other words, to find transfer maps so that the following diagram commutes,
\begin{equation}
\begin{CD}  \label{equ: CD with trans}
  H^q_{cpct, b}(M)    @>trans_b>>          H^q_{c,b}(G,{\mathbb R}) \\
                      @VV c V                             @VV c V \\
  H^q_{cpct}(M) @>trans>>  H^q_{c}(G,{\mathbb R})   \\
     @AA \cong A                             @AA \cong A \\
     H^q_{dR,cpct}(M) @>trans_{dR}>>  \Omega^q(G/K)^G.
   \end{CD}
   \end{equation}

This is the purpose of the next subsection. Note that the existence of these transfer maps would be easy to show for arbitrary manifolds, given that the singular cohomology with compact support can be computed on measurable cochains. We prefer to restrict to $\mathbb{Q}$-rank $1$ manifolds where the proof is straightforward.

\subsection{Transfer on singular (bounded) cohomology}\label{subsec: def of p}

As usual, when defining a transfer map, we would like to integrate the evaluation of a cochain $c\in C^q_{cpct}(M)$ on translates $g\cdot \sigma$ of a singular simplex $\sigma$ over a fundamental domain $g\in D$ for $\Gamma\setminus G$. This is only possible if the cochain presents certain regularity properties. However, at this point, the cochain $c$ is completely arbitrary. We will thus start by replacing $c$ by a better behaved cochain.

The $\mathbb{Q}$-rank $1$ locally symmetric space $M$ admits the following description \cite{Leuzinger}. There is an exhaustion function $h:M\rightarrow [0,+ \infty )$ such that for any $s\geq 0$,
$$M=M(s)\cup \amalg_{i=1}^k E_i(s),$$
where the sublevel set $M(s)=\{x\in M\mid h(x)\leq s \}$ is a compact submanifold and $E_1(s),...,E_k(s)$ are the disjoint cusp ends of $M\setminus M(s)$. Note that furthermore, each cusp end $E_i(s)$ is geodesically convex for any $s\geq 0$, i.e.  for any two points $x,y\in E_i(s)$, the unique geodesic between $x$ and $y$ is contained in $E_i(s)$. Also, $\pi^{-1}(E_i(s))$ is a disjoint union of horoballs.

Choose $b_0\in M(0)$ and, for every $1\leq i\leq k$ and $j \in \mathbb{N} $, points $b_{ij}\in E_i(j-1)\setminus E_i(j)$. Define a measurable map $\overline{p}:M\rightarrow
\{b_0\}\cup \{b_{ij}\}_{\{1\leq i \leq k,  j \in \mathbb{N} \}}$ by  $\overline{p}(M(0))=b_0$ and $\overline p(E_i(j-1)\setminus E_i(j))={b_{ij}}$
for every $i$ and $j\in \mathbb{N}$. Lift $\overline p$ to a $\Gamma$-equivariant measurable map
$$p:\widetilde{M}\longrightarrow \pi^{-1}(\{b_0\}\cup \{b_{ij}\}_{\{1\leq i \leq k, j\in \mathbb{N} \}} )\subset \widetilde{M}$$
with the property that the image by $p$ of a horoball of $\pi^{-1}(E_i(j))$ remains in the given horoball,
where $\pi$ is the  projection $\pi:\widetilde{M}\rightarrow M$.

The pullback map $p^*: C^q_{cpct}(M,\mathbb{R})\rightarrow C^q_{cpct}(M,\mathbb{R})$ is defined by sending a cochain $c\in  C^q_{cpct}(M,\mathbb{R})$ to the cochain
$$p^*(c)(\sigma)=c(\pi_* str(p(\sigma_0),...,p(\sigma_q))),$$
where $\sigma_0,...,\sigma_q\in \widetilde{M}$ are the vertices of a lift of $\sigma$ to $\widetilde{M}$. Note that $p^*(c)$ indeed has compact support. To see that, let $C$ be the compact support of $c$. There exists $s$ such that $C\subset M(s)$. Let $\sigma:\Delta^q\rightarrow M$ be a singular simplex whose image does not intersect $M(s)$, say $\mathrm{Im}(\sigma)\subset E_i(s)$. Any lift of $\sigma$ will be contained in a single horoball in the preimage $\pi^{-1}(E_i(s))$. The same is true for the vertices of this lift, and hence for the straightened simplex $str(p(\sigma_0),...,p(\sigma_q))$. It follows that the image of $\pi_* str(p(\sigma_0),...,p(\sigma_q))$ is contained in $E_i(s)$ so that the evaluation of $c$ on it indeed vanishes. It is obvious that $p^*$ commutes with the coboundary, and it is easy to check that it is homotopic to the identity. Thus it induces the identity on cohomology. Clearly, the cochain map restricts to bounded cochains and also induces the identity on the bounded cohomology group. Note that it is easy to show that these maps, at cohomology level, are independent of the chosen points $b_0$ and $b_{ij}$, though we are not going to need this fact.

The image of $p^*$ is contained in a subcocomplex which we denote by $C^q_{cpct,meas}(M,\mathbb{R})\subset C^q_{cpct}(M,\mathbb{R})$ and $C^q_{cpct,meas,b}(M,\mathbb{R})\subset $ $C^q_{cpct,b}(M,\mathbb{R})$ in the bounded case, of cochains whose evaluation on singular simplices $\sigma$ is given by evaluation on  the vertices of a lift on a $\Gamma$-equivariant measurable function $f:\widetilde{M}^{q+1}\rightarrow \mathbb{R}$.

A transfer map
$$trans:C^q_{cpct,meas}(M,\mathbb{R})\longrightarrow C(\widetilde{M}^{q+1},\mathbb{R})^G$$
is defined by sending a cochain $c$ to
$$trans(c)(x_0,...,x_q)=\int_{g\in D} p^*(c)(gx_0,...,gx_q)d\mu(g),$$
where $D$ is a fundamental domain for  $\Gamma\setminus G$ normalized to have measure $1$ and $x_0,...,x_q$ are points in $\widetilde{M}$.
To see that the integral is finite, let $d=\mathrm{max}_i d(x_0,x_i)$ and denote by $M(s)_d$ the closure of the $d$-neighborhood of $M(s)$.
Here, we choose a positive integer $s$ with $C\subset M(s)$. Since $M(s)_d$ is compact, so is
$$D_0=\{ g\in D \subset G \mid \pi(gx_0)\in M(s)_d\}.$$
We claim that $p^*(c)(gx_0,...,gx_q)=0$ for $g\in D\setminus D_0$. Indeed, since $gx_0\in \pi^{-1}(M\setminus M(s)_d)$, it belongs to a horoball of $\pi^{-1}(E_i(s))$ for some cusp $E_i$ and furthermore a ball of radius $d$ centered at $gx_0$ is also contained in the same horoball. In particular, $gx_1,...,gx_q$ and the straightened simplex $str(p(gx_0),...,p(gx_q))$ all belong to the same horoball and hence project to $E_i(s)$, on which $c$ vanishes. Note that for  $g\in D_0$, the evaluation $p^*(c)(gx_0,...,gx_q)$ takes only finitely many values (depending on $x_0,...,x_q$). It is further easy to see that the integral is independent of the choice of fundamental domain $D$, that $trans(c)$ is $G$-invariant and that the transfer map commutes with coboundaries. It thus induces a cohomology map and we will denote by
$$trans:H^q_{cpct}(M)\longrightarrow H^q_c(G).$$
 We denote by $trans_b$ the map on
bounded cohomology. Since $trans_b$ is, at cochain level, the
restriction of $trans$, the commutativity of the diagram
\[
\begin{CD}
  H^q_{cpct, b}(M)    @>trans_b>>          H^n_{c,b}(G,{\mathbb R}) \\
                      @VV c V                             @VV c V \\
  H^q_{cpct}(M) @>trans>>  H^n_{c}(G,{\mathbb R})    \end{CD} \]
is obvious. Let us finally show that the full diagram (\ref{equ: CD with trans}) also commutes.

\begin{Prop} \label{prop: trans(omega)=omega}The diagram (\ref{equ: CD with trans}) commutes.
\end{Prop}

Note that in particular it follows that $trans(\omega_M)=\omega_{\widetilde{M}}$. Indeed, $\omega_M$ is represented
by $\rho \cdot (\Gamma\backslash \omega_{\widetilde M})$ where $\rho$ is a
compactly supported smooth function on $M$ so that $\int_M \rho \cdot
(\Gamma\backslash \omega_{\widetilde M})=vol(M)$. From section
\ref{deRham}, the image $trans(\omega_M)$ is independent of the
choice of $\rho$. Choose an exhausting sequence $(K_n)_{n\in \mathbb{N}}$ of compact connected subsets of $M$ with nonempty interior satisfying $\lim_{n\rightarrow \infty} \mathrm{Vol}(K_n)=\mathrm{Vol}(M)$. Then, there exist $\epsilon_n>0$ and a smooth function $\rho_n : M \rightarrow [0,1+\epsilon_n]$ supported on the $\epsilon_n$-neighborhood of $K_n$ such that $\rho_n |_{K_n} = 1+\epsilon_n$ and $\int_M \rho_n \cdot (\Gamma\backslash \omega_{\widetilde M})=vol(M)$. It is easy to see that the maps $\rho_n$ converges uniformly on any compact subset of $M$ to the constant function $c=1$ on $M$. Considering this sequence $(\rho_n)_{n\in \mathbb{N}}$, one can conclude that for any $\rho$, the average of a compactly supported differential form $\rho \cdot (\Gamma\backslash \omega_{\widetilde M})$ over the fundamental domain is $\omega_{\widetilde M}$.

\begin{proof} It remains to show that the lower diagram commutes. Note that the vertical isomorphisms are induced, at cochain level, by the map
$$\Phi: \Omega^q(G/K)^G \longrightarrow C((G/K)^{q+1},{\mathbb R})^G$$
sending the differential form $\alpha$ to the cochain $\Phi(\alpha)$ mapping a $(q+1)$-tuple of points $(x_0,...,x_q)\in (G/K)^{q+1}$ to
$$\int_{ str(x_0,...,x_q)}\alpha.$$
This map is clearly $G$ (and hence $\Gamma$) - equivariant and
restricts to the subcomplexes of compact support. Furthermore, for
the left vertical arrow, $\Phi$ should further be precomposed with
the map sending a singular simplex in $\widetilde{M}$ to its
vertices. Here we use again the ${\mathbb Q}$-rank oneness to restrict to the subcomplexes of compact support, see section \ref{derham}. To prove the proposition, we need to show that $trans\circ
\Phi$ and $\Phi\circ trans_{dR}$ differ by a coboundary. One checks
easily that
\begin{eqnarray*}
 trans\circ \Phi(\alpha)(x_0,...,x_q)&=& \int_{g\in \Gamma\setminus G}\int_{str(p(gx_0),...,p(gx_q))} \alpha d\mu(g)
\end{eqnarray*}
while
\begin{eqnarray*}
 \Phi\circ trans_{dR}(\alpha)(x_0,...,x_q)&=& \int_{g\in \Gamma\setminus G}\int_{str(gx_0,...,gx_q))} \alpha d\mu(g) .
\end{eqnarray*}
It is immediate that, since $d\alpha=0$, the coboundary of the $G$-invariant cochain
$$(x_0,...,x_{q-1})\longmapsto \sum_{i=0}^{q-1} (-1)^i \int_{g\in \Gamma\setminus G}\int_{str(gx_0,...,gx_i,p(gx_i),...,p(gx_q))} \alpha d\mu(g)$$
is equal to the difference of the two given cocycles, which finishes
the proof of the proposition.
\end{proof}

\subsection{Norms and first inequality of Proposition \ref{mainProp}}\label{sec:5.3}

The inequality $ \| \omega_M \|^\infty \geq \| \omega_{\widetilde{M}} \|_\infty$ of Proposition \ref{mainProp} is an immediate consequence of the following lemma:

\begin{lemma} For every $\beta\in H^q_{cpct}(M)$, one has
$$ \| trans(\beta) \|_\infty \leq \| \beta \|^\infty.$$
\end{lemma}

\begin{proof}
Let $\Lambda<G$ be a cocompact torsion free lattice. Because of the existence of a transfer map, the restriction map $i^*:H^*_{c,b}(G)\rightarrow H^*_b(\Lambda)\cong H^*_b(\Lambda \setminus \widetilde{M})$ is isometric. Thus, for every $\alpha\in H^*_{c,b}(G)$, we have the equality $\|\alpha \|_\infty=\|i^*(\alpha)\|_\infty $. Furthermore since $\Lambda \setminus \widetilde{M}$ is compact, the upper and lower infinity norms agree $\|i^*(\alpha)\|_\infty=\|i^*(\alpha)\|^\infty$. As on a compact manifold a locally finite set of singular simplices is actually finite, it immediately follows that the norm of $\alpha \in H^*_{c,b}(G)$ can be computed as
$$\| \alpha \|_\infty = \sup_{\Phi \mathrm{\ finite}} \inf_{[a]=\alpha} \sup_{\sigma\in \Phi}|a(\sigma_0,...,\sigma_q)|,$$
where here and in the sequel, $\sigma_0,...,\sigma_q$ are the vertices of the singular simplex $\sigma$. Applying this equality to $\alpha=trans(\beta)$ and restricting the infimum to the cocycles of the form $a=trans(b)$, where $b\in C^n_{cpct}(M)$ is a compact support cocycle representing $\beta$, we obtain the inequality
$$\| trans(\beta)\|_\infty \leq  \sup_{\Phi \mathrm{\ finite}} \inf_{[b]=\beta} \sup_{\sigma\in \Phi}|trans(b)(\sigma)|.$$
Let $\Phi_p$ denote the following set of singular simplices,
$$\Phi_p=\{ \pi_* str(p(g\sigma_0),...,p(g\sigma_q)) \mid \sigma\in \Phi, \ g\in D\},$$
where $D$ is a fundamental domain for $\Gamma\setminus G$. Note that as $\Phi$ is finite $\Phi_p$ is locally finite. Since by definition, $trans(b)$ is an average of evaluations of $b$ on singular simplices in $\Phi_p$, it is immediate that
$$\sup_{\sigma\in \Phi} | trans(b)(\sigma_0,...,\sigma_q)|\leq \sup_{\sigma\in \Phi_p} |b(\sigma)|.$$
It follows that
$$\| trans(\beta)\|_\infty \leq \sup_{\Phi \mathrm{\ finite}} \inf_{[b]=\beta} \sup_{\sigma\in \Phi_p} |b(\sigma)| = \sup_{\Phi_p } \inf_{[b]=\beta} \sup_{\sigma\in \Phi_p} |b(\sigma)|,$$
where the first sup of the latter expression is taken over all families of simplices of the form $\Phi_p$, for a finite family $\Phi$. Finally, as $\Phi_p$ is locally finite we obtain
$$\| trans(\beta)\|_\infty\leq \sup_{\Psi \in \mathcal{S}^\mathrm{lf}_q} \inf_{[b]=\beta} \sup_{\sigma\in \Psi} |b(\sigma)|=\|\beta\|^\infty,$$
which finishes the proof of the lemma.
\end{proof}

\section{Cusp map and relation to bounded cohomology}\label{Section: coning}

\subsection{Cusp map}
To prove the theorem, we will need the existence of a
$\Gamma$-equivariant cusp map, which we establish in this section. We first review the reduction theory for arithmetic lattices. The main reference is \cite{Borel-Ji}.

Let $\mathbf{G}$ be a connected, semisimple algebraic group
defined over ${\mathbb Q}$ with trivial center. Let $X=G/K$ be the
associated symmetric space of noncompact type with
$G=\mathbf{G}({\mathbb R})^0$ and a maximal compact subgroup $K$ of $G$. A
closed subgroup $\mathbf{P}$ of $\mathbf{G}$ defined over ${\mathbb Q}$ is
called \emph{rational parabolic subgroup} if $\mathbf{P}$ contains
a maximal, connected, solvable subgroup of $\mathbf{G}$. For any
rational parabolic subgroup $\mathbf{P}$ of $\mathbf{G}$, we
obtain the \emph{rational Langlands decomposition} of
$P=\mathbf{P}({\mathbb R})$:
$$P = N_{\mathbf{P}} \times A_\mathbf{P} \times M_\mathbf{P},$$
where $N_\mathbf{P}$ is the real locus of the unipotent radical $\mathbf{N}_\mathbf{P}$ of $\mathbf{P}$,
$A_\mathbf{P}$ is a stable lift of the identity component of the real locus of the the maximal ${\mathbb Q}$-torus in the Levi quotient $\mathbf{P} / \mathbf{N}_\mathbf{P}$ and $M_\mathbf{P}$ is a stable lift of the real locus of the complement of the maximal ${\mathbb Q}$-torus in $\mathbf{P} / \mathbf{N}_\mathbf{P}$.

Write $X_\mathbf{P}=M_\mathbf{P}/K \cap M_\mathbf{P}$. Let us denote by $\tau :  M_\mathbf{P} \rightarrow
X_\mathbf{P}$ the canonical projection. After fixing a basepoint $x_o \in X$, we have an analytic diffeomorphism
$$\mu : N_{\mathbf{P}} \times A_\mathbf{P} \times X_\mathbf{P} \rightarrow X, \ (n, a, \tau(m)) \rightarrow nam \cdot x_0,$$
This is called the \emph{rational horocyclic decomposition} of $X$.

Let $\Gamma$ be an arithmetic lattice in $G=\mathbf{G}({\mathbb R})$ with ${\mathbb Q}$-$\mathrm{rank}(\mathbf{G})=1$. It is a well known fact due to A. Borel and Harish-Chandra that there are only finitely many $\Gamma$-conjugacy classes of proper rational parabolic subgroups. Furthermore, all rational proper parabolic subgroups are minimal and these subgroups are conjugate under $\mathbf{G}({\mathbb Q})$ (see \cite[Theorem 11.4]{Borel}). Note that the set of simple positive ${\mathbb Q}$-roots contains only a single element and $\mathrm{dim} A_\mathbf{P}=1$ for any proper rational parabolic subgroup $\mathbf{P}$ of $\mathbf{G}$.

The locally symmetric space $M=\Gamma \backslash X$ has finitely many cusps and each cusp corresponds to a $\Gamma$-conjugacy class of a minimal rational parabolic subgroup. Let $\mathbf{P}_1,\ldots,\mathbf{P}_k$ denote representatives of the $\Gamma$-conjugacy classes. For each minimal rational parabolic subgroup $\mathbf{P}_i$, set $$V_{\mathbf{P}_i}(t) = \mu (N_{\mathbf{P}_i} \times A_{\mathbf{P}_i,t} \times X_{\mathbf{P}_i})$$ where $A_{\mathbf{P}_i,t}=\{ a\in A_{\mathbf{P}_i} : \alpha_i(\log a) > t\}$. Here $\alpha_i$ is the unique positive simple ${\mathbb Q}$-root corresponding to $\mathbf{P}_i$. Write $\Gamma_\mathbf{P_i}=\Gamma \cap \mathbf{P_i}$ for $i=1,\ldots,k$. There exists $t_0>0$ such that for all $t>t_0$, $M$ admits the following disjoint decomposition:
$$M=\Omega_t \cup  \amalg_{i=1}^k \Gamma_{\mathbf{P}_i} \backslash V_{\mathbf{P}_i}(t),$$
where $\Omega_t$ is a compact submanifold with boundary.
Furthermore, for $t>t_0$ and each $i$, we have
$$\Gamma_{\mathbf{P}_i} \backslash V_{\mathbf{P}_i}(t) \cong \Gamma_{\mathbf{P}_i} \backslash N_{\mathbf{P}_i} \times A_{\mathbf{P}_i,t} \times X_{\mathbf{P}_i} = \Gamma_{\mathbf{P}_i} \backslash ( N_{\mathbf{P}_i} \times X_{\mathbf{P}_i}) \times A_{\mathbf{P}_i,t}.$$
Geometrically, each $V_{\mathbf{P}_i}(t)$ is a horoball in $X$. In fact, the disjoint decomposition $M=\Omega_t \cup  \amalg_{i=1}^k \Gamma_{\mathbf{P}_i} \backslash V_{\mathbf{P}_i}(t)$ gives the same disjoint decomposition $M=M(s) \cup  \amalg_{i=1}^k E_i(s)$ for some $s>0$  in Section \ref{subsec: def of p}.

\begin{Prop}  \label{prop: coning map} Let $M$ be a ${\mathbb Q}$-rank $1$ locally
symmetric space and let $\Gamma=\pi_1(M)$ be its fundamental group.
Let $G$ denote the identity component of the isometry group of the universal cover
$\widetilde{M}$ of $M$ and let $P$ be a minimal rational parabolic subgroup of $G$. Then
there exists a $\Gamma$-equivariant measurable cusp map
$$ f:\widetilde{M} \longrightarrow G/P.$$
such that $f$ is constant on each horoball in $\widetilde{M}$ corresponding to a cusp of $M$.
\end{Prop}

\begin{proof}
Fix a $t>t_0$. Decompose $M=\Omega_t \cup \amalg_{i=1}^k \Gamma_{\mathbf{P}_i} \backslash V_{\mathbf{P}_i}(t)$ as before. We set $\mathbf{P}=\mathbf{P}_1$ and $P=\mathbf{P}({\mathbb R})^0$.
Since all minimal rational parabolic subgroups are conjugate under $\mathbf{G}({\mathbb Q})$, there exists $g_i \in G$ with $\mathbf{P}_i=g_i \mathbf{P} g_i^{-1}$ for $i=1,\ldots,k$. Let $F_{\mathbf{P}_i}(t)$ be a fundmental domain for the action of $\Gamma_{\mathbf{P}_i}$ on $V_{\mathbf{P}_i}(t)$ and $F_t$ be a fundamental domain for the action of $\Gamma$ on $\pi^{-1}(\Omega_t)$ where $\pi : \widetilde{M} \rightarrow M$ is the universal covering map of $M$.
Map $F_t$ to $P$ and $F_{\mathbf{P}_i}(t)$ to $g_iP$. Extending this map $\Gamma$-equivariantly, we get a $\Gamma$-equivariant measurable map $f : \widetilde{M} \rightarrow G/P$. Then it is easy to see that $f(x)=g_iP$ for all $x\in V_{\mathbf{P}_i}(t)$. This completes the proof.
\end{proof}

\subsection{The induced bounded cohomology map}

\begin{thm} \label{thm: induced from coning} Let $M$ be a ${\mathbb Q}$-rank $1$ locally symmetric space and let $\Gamma=\pi_1(M)$ be its fundamental group. Suppose that the fundamental group of any cusp of $M$ is amenable. Then the $\Gamma$-equivariant cusp map from Proposition \ref{prop: coning map}
$$  f:\widetilde{M} \longrightarrow G/P$$
induces for $q\geq 1$ a cohomology map
$$ f^*: H^q_{b}(\Gamma)\longrightarrow H^q_{cpct, b}(M)$$
such that
$$ trans_b \circ f^*=trans_\Gamma$$
and
$$ \| f^*(\alpha)\|_\infty \leq \| \alpha \|_\infty$$
for any $\alpha \in H^q_{b}(\Gamma)$.
\end{thm}

\begin{proof} It is shown in \cite[Section 6]{BucherMonod} that the bounded cohomology group $H^*_b(\Gamma)$ can be computed isometrically on the cochain complex of alternating $\Gamma$-invariant bounded functions $\ell^\infty_\mathrm{alt}((G/P)^{q+1},\mathbb{R})^\Gamma$ endowed with its homogeneous coboundary operator. We will use this cochain complex to induce the cohomology map $f^*:H^*_b(\Gamma)\rightarrow H^*_{cpct,b}(M)$ in degree $\geq 1$. Let thus $  f:\widetilde{M} \longrightarrow G/P$ be the cusp map. It
induces in degree $\geq 1$ a cochain map
$$f^*:\ell_{alt}^\infty((G/P)^{q+1},\mathbb{R})^\Gamma  \longrightarrow C^q_{cpct,b}(\widetilde{M})^\Gamma \cong C^q_{cpct,b}(M)$$
by sending a  bounded, $\Gamma$-invariant alternating cochain $c:(G/P)^{q+1}\rightarrow \mathbb{R}$ to the cochain $f^*(c)$ defined by mapping a singular simplex $\sigma:\Delta^q\rightarrow \widetilde{M}$ to $c(f(x_0),...,f(x_q))$, where $x_0,...,x_q\in \widetilde{M}$ are the vertices  of $\sigma$. The cochain $f^*(c)$ is clearly $\Gamma$-equivariant and bounded. Furthermore, it has compact support contained in $M(0)$ since if the image of $\pi(\sigma)$ does not intersect $M(0)$, then it is contained in a cusp $E_i$, so that $\sigma$ is contained in one horoball, and its vertices $x_0,...,x_q$ all project to the same boundary point $f(x_0)=...=f(x_q)\in G/P$. Since the cochain $c$ is alternating, $f^*(c)(\sigma)=0$. It is for this last assertion that we use the fact that we are in degree $\geq 1$.

It is immediate that $f^*$ commutes with the coboundary operator $\delta$ and hence induces a cohomology map
$$ f^*: H^*_{b}(\Gamma)\longrightarrow H^*_{cpct, b}(M).$$
Since the cochain map $f^*$ does not augment norm, the last assertion of the theorem is obvious. It remains to see that $ trans \circ f^*=trans_\Gamma$.

Consider the following diagram, where the cochain map $p^*$ is as defined in the beginning of Subsection \ref{subsec: def of p}:
\begin{equation*}
\xymatrix{ \ell_{alt}^\infty((G/P)^{q+1},\mathbb{R})^\Gamma  \ar@/^2pc/[rr]^{f^*} \ar[r]^{\ \ \ \ \ \  f^*  }&C^q_{cpct,b}(M) \ar[r]^-{p^* }  \ar@<-.7ex>[rd]_{trans } &  C^q_b(G/K)^\Gamma \ar[d]^{trans_\Gamma} \\
 & & C^q_b(G/K)^G.}
\end{equation*}
Observe that the composition $p^*\circ f^*$ is equal to $f^*$. Since the cochain map $f^*: \ell_{alt}^\infty((G/P)^{*+1})^\Gamma\rightarrow C^q_b(G/K)^\Gamma $ extends the identity $\mathbb{R}\rightarrow \mathbb{R}$ in degree $-1$ it induces the identity on cohomology. Thus, on cohomology, the
diagram becomes
\begin{equation*}
\xymatrix{H^q_b(\Gamma) \ar@/^2pc/[rr]^{f^*=Id} \ar[r]^{f^*  }&H^q_{cpct,b}(M) \ar[r]^{p^* \ \ \ }  \ar@<-.7ex>[rd]_{trans } &  H^q_b(\Gamma) \ar[d]^{trans_\Gamma} \\
 & & H^q_b(G/K)^G.}
\end{equation*}
It is now clear that $trans_\Gamma= trans_b \circ f^*$, which finishes the proof of the theorem. \end{proof}

\subsection{Proof of the second inequality of Proposition \ref{mainProp}}
We can now prove that $ ||\omega_M||_\infty \leq ||\omega_{\widetilde{M}}||_\infty$. Consider the following diagram:

\begin{equation*}
\xymatrix{ H^n_{c,b}(G,{\mathbb R})  \ar@<+.7ex>@{^{(}->}[r]^{i^*} \ar[d]^c & H^n_{b}(\Gamma,{\mathbb R}) \ar@<+.7ex>@{->>}[l]^{trans_\Gamma}\ar[r]^{f^* \ \ \ }  & H^n_{cpct, b}(M,{\mathbb R})  \ar@/_2pc/[ll]^{trans_b} \ar[d]^c \\
H^n_{c}(G,{\mathbb R})  & & \ar[ll]_{trans} H^n_{cpct}(M,{\mathbb R}). }
\end{equation*}
Using that $trans$ commutes with the comparison map $c$, that $trans_b \circ f^*=trans_\Gamma$ (Theorem \ref{thm: induced from coning}), and that $trans_\Gamma \circ i^*$ is the identity on $H^n_{c,b}(G,{\mathbb R})$, we show that
$$ trans \circ c \circ f^* \circ i^* = c. \label{equ: comm diag}$$
Indeed,
\begin{eqnarray*}
\underbrace{trans \circ c}_{=c\circ trans_b} \circ f^*\circ i^* &=& c\circ {\underbrace{trans_b \circ f^* }_{=trans_\Gamma}}\circ i^*\\
&=& c\circ \underbrace{trans_\Gamma \circ i^*}_{=\mathrm{Id}} =c.
\end{eqnarray*}
It remains to establish the desired inequality:
\begin{eqnarray*}
\| \omega_M\|_\infty&=& \mathrm{inf}\{ \| \alpha \|_\infty\mid \alpha\in H^n_{cpct, b}(M,{\mathbb R}),\  c(\alpha)=\omega_M \}\\
&\leq & \mathrm{inf}\{ \| \alpha \|_\infty\mid \alpha\in \mathrm{Im}(f^* \circ i^*),\  trans\circ c(\alpha)=trans(\omega_M)=\omega_{\widetilde{M}} \}\\
&=&\mathrm{inf}\{ \underbrace{\| (f^* \circ i^*)(\beta) \|_\infty}_{\leq \|\beta \|_\infty} \mid \beta\in H^n_{c,b}(G,{\mathbb R}),\  \underbrace{trans\circ c\circ f^* \circ i^*}_{=c}(\beta)=\omega_{\widetilde{M}} \}\\
&=& \| \omega_{\widetilde{M}} \|_\infty,
\end{eqnarray*}
where we have used the facts that the map $trans:H^n_{cpct}(M,{\mathbb R})\rightarrow H^n_c(G,{\mathbb R})$ is injective (even bijective since $trans(\omega_M)=\omega_{\widetilde{M}}$) in top degree and that $f^*$ and $i^*$ do not increase norms. This finishes the proof of the second inequality of Proposition \ref{mainProp}.

\section{The product of ${\mathbb R}$-rank $1$ symmetric spaces}\label{section: product of rank 1}

In this section, we establish the proportionality principle for the simplicial volume of certain $\mathbb{Q}$-rank $1$ locally symmetric spaces covered by a product of ${\mathbb R}$-rank $1$ symmetric spaces without any assumption on their ends.

Let $H$ be the full isometry group of $\widetilde{M}$. An action of $H$ on ${\mathbb R}$ is defined as follows: An element $h$ of $H$ acts by multiplication by $+1$ (resp. $-1$) if $h$ preserves (resp. reverses) the orientation in $\widetilde{M}$. We denote by ${\mathbb R}_\varepsilon$ the Banach space ${\mathbb R}$ endowed with the action of $H$. The continuous $H^n_{c}(H,{\mathbb R}_\varepsilon)$ and continuous, bounded $H^n_{c,b}(H,{\mathbb R}_\varepsilon)$ cohomology of $H$ with coefficients in ${\mathbb R}_\varepsilon$ is defined by replacing the $H$-invariant cochains by $H$-equivariant ones in the corresponding definitions.

\begin{proof}[Proof of Theorem \ref{thm:1.4}]
The simplicial volume $\| M \|_\mathrm{lf}$ of $M$ can be reformulated in terms of $\ell^1$-homology as follows:
\begin{eqnarray}\label{eqn:4} \| M \|_\mathrm{lf} =\inf_{\alpha \in [M]^{\ell^1}} \sup \left\{ \frac{1}{\| \varphi \|_\infty} \ \bigg| \ \varphi \in H^n_b(M) \text{ and } \langle \varphi, \alpha \rangle =1 \right\}, \end{eqnarray}
where $[M]^{\ell^1}$ is the set of all $\ell^1$-homology classes that are represented by at least one locally finite cycle with finite $\ell^1$-norm. For a proof, see \cite{Loeh}. For the reader's convenience, we sketch a proof that the Kronecker product $\langle \cdot, \cdot \rangle : H^*_b(M) \otimes H^{\ell^1}_*(M) \rightarrow {\mathbb R}$ is well defined. In fact, this follows from a well-known fact in functional analysis that $C_b^*(M)$ is the topological dual space of the $\ell^1$-completion $C^{\ell^1}_*(M)$ of $C_*(M)$. In other words, a linear functional on $C^{\ell^1}_*(M)$ is continous if and only if it is bounded. Let $f : C_*(M) \rightarrow {\mathbb R}$ be a bounded cochain in $C_b^*(M)$. Denote by $f' :C^{\ell^1}_*(M) \rightarrow {\mathbb R}$ the unique continuous extension of $f$. One can define a map $\langle \cdot,\cdot \rangle : C_b^*(M) \otimes C^{\ell^1}_*(M) \rightarrow {\mathbb R}$ via the evaluation map between $C^{\ell^1}_*(M)$ and its dual cochain complex by
$$\langle f, \cdot \rangle =f'(\cdot),$$
Noting that $\delta f (c) = f(\partial c)$ for $c \in C_{*+1}(M)$, it is easy to see that $(\delta f)'(c')=f'(\partial c')$ for $c' \in C^{\ell^1}_{*+1}(M)$. Hence, we have
$$\langle \delta f, c' \rangle = (\delta f)'(c')= f'(\partial c') = \langle f, \partial c' \rangle.$$
This implies that the Kronecker product between $H^*_b(M)$ and $H^{\ell^1}_*(M)$ is well defined.

Define a $H$-invariant cocycle $\Theta: \widetilde{M}^{n+1} \rightarrow {\mathbb R}_\varepsilon$ by
$$\Theta(x_0,\ldots,x_n) = \int_{[x_0,\ldots,x_n]} \omega_{\widetilde{M}}$$
where $[x_0,\ldots,x_n]$ is the geodesic simplex in $\widetilde{M}$ with an ordered vertex set $\{x_0,\ldots,x_n\}$ and $\omega_{\widetilde{M}}$ is the $H$-invariant volume form on $\widetilde{M}$. Note that the cocycle $\Theta$ is bounded because the volume of top dimensional geodesic simplices in a product of ${\mathbb R}$ rank $1$ symmetric spaces is uniformly bounded from above.
Hence, $\Theta$ determines a continuous cohomology class $[\Theta]^H \in H^n_c(H,{\mathbb R}_\varepsilon)$ and a continuous bounded cohomology class $[\Theta]^H_b$ in $H^n_{c,b}(H,{\mathbb R}_\varepsilon)$ with $c([\Theta]^H_b)=[\Theta]^H$.

Let $G$ be the connected component of the identity of $H$. Note that $G$ is a finite index subgroup of $H$.
Hence, the inclusion $G \subset H$ induces isometric embeddings $res^H : H^*_c(H,{\mathbb R}_\varepsilon) \rightarrow H^*_c(G,{\mathbb R})$ and $res^H_b : H^*_{c,b}(H,{\mathbb R}_\varepsilon) \rightarrow H^*_{c,b}(G,{\mathbb R})$ which are realized by the canonical inclusions of cocomplexes $C^*_c(\widetilde{M},{\mathbb R}_\varepsilon)^H \subset C^*_c(\widetilde{M},{\mathbb R})^G$ and $C^*_{c,b}(\widetilde{M},{\mathbb R}_\varepsilon)^H \subset C^*_{c,b}(\widetilde{M},{\mathbb R})^G$. Similarly, since $\Gamma$ is a lattice in $G$, the inclusion $\Gamma \subset G$ induces an isometric embedding $res^G_b : H^*_{c,b}(G,{\mathbb R}) \rightarrow H^*_b(\Gamma,{\mathbb R})$ where $\Gamma$ is the fundamental group of $M$.

It is clear that $res^H([\Theta]^H)=\omega_{\widetilde{M}}$. Since the comparison map is an isomorphism and $res^H$, $res^H_b$ and $res^G_b$ are isometric embeddings,
$$\| (res^G_b \circ res^H_b )([\Theta]^H_b) \|_\infty =\| [\Theta]^H_b \|_\infty = \| [\Theta]^H \|_\infty = \|\omega_{\widetilde{M}} \|_\infty.$$

Clearly, the cocycle $\Theta$ represents $(res^G_b \circ res^H_b )([\Theta]^H_b) \in H^n_b(M,{\mathbb R})$.
Let $\alpha \in [M]^{\ell^1}$. Then there is a locally finite fundamental cycle $c$ with $\| c\|_1 <\infty$ such that $c$ represents $\alpha$ in the $\ell^1$-homology of $M$. For $\mathbb{Q}$-rank $1$ locally symmetric spaces, the geodesic straightening map on the locally finite chain complex is well defined and chain homotopic to the identity \cite{Kim-Kim}. Hence, the geodesic straightening $str(c)$ of $c$ is again a locally finite fundamental cycle with $\|str(c) \|_1 \leq \|c \|_1 < \infty$ and we have $$str(c)-c=\partial H_n^\mathrm{lf}(c) + H_{n-1}^\mathrm{lf} \partial (c) = \partial H_n^\mathrm{lf}(c)$$
where $H_*^\mathrm{lf} : C_*^\mathrm{lf}(M,{\mathbb R}) \rightarrow  C_{*+1}^\mathrm{lf}(M,{\mathbb R})$ is a chain homotopy between the geodesic straightening map and the identity that is constructed by the canonical straight line homotopy. It is easy to see that $$\| H_n^\mathrm{lf}(c) \|_1 \leq (n+1) \|c\|_1 <\infty $$
and thus, $str(c)$ and $c$ represent the same $\ell^1$-homology class $\alpha$.
Furthermore, note that
\begin{eqnarray}\label{eqn:6} \langle \Theta, str(c) \rangle = \mathrm{Vol}(M).\end{eqnarray}
We refer the reader to \cite{Kim-Kim} for more details.

Equation (\ref{eqn:6}) implies that for any $\alpha \in [M]^{\ell^1}$,
$$\left\langle \frac{(res^G_b \circ res^H_b )([\Theta]^H_b)}{\mathrm{Vol}(M)}, \alpha \right\rangle = \frac{\langle \Theta, str(c) \rangle}{\mathrm{Vol}(M)}=1.$$
Hence, in the view of Equation (\ref{eqn:4}), we have
$$  \| M \|_\mathrm{lf} = \frac{\mathrm{Vol}(M)}{\| \omega_M \|^\infty} \geq \frac{\mathrm{Vol}(M)}{\| (res^G_b \circ res^H_b )([\Theta]^H_b) \|_\infty}=\frac
{\mathrm{Vol}(M)}{\| \omega_{\widetilde{M}} \|_\infty}.$$
Thus, we obtain an inequality $ \| \omega_M \|^\infty \leq \| \omega_{\widetilde{M}} \|_\infty$.
From the opposite inequality in Section \ref{sec:5.3}, we finally have
$$ \| \omega_M \|^\infty = \| \omega_{\widetilde{M}} \|_\infty,$$
which completes the proof.
\end{proof}

\begin{remark} Theorem \ref{thm:1.4} holds for reducible $\mathbb{Q}$-rank $1$ locally symmetric spaces.
Note that the fundamental groups of the ends of reducible $\mathbb{Q}$-rank $1$ locally symmetric spaces are not amenable.
Hence, this strongly supports the existence of the proportionality principle for the simplicial volume of arbitrary $\mathbb{Q}$-rank $1$ locally symmetric spaces. \end{remark}

redpajama_set_name: RedPajamaArXiv


text:
### Android setup

#### Link
By using `react-native link` ( __recommended__ ) [Next Step](facebook)

#### Manual

2. Open `android/settings.gradle` file and add following
```
...
include ':react-native-social-auth'
project(':react-native-social-auth').projectDir = new File(rootProject.projectDir, '../node_modules/react-native-social-auth/android')
```
3. Open `android/app/build.gradle` file and add following
```
...
dependencies {
    ...
    compile project(':react-native-social-auth')
}
```

4. Register React Package

  ```
  ...
  import com.xxsnakerxx.socialauth.SocialAuthPackage; // import

  public class MainActivity extends ReactActivity {
  ...
    /**
     * A list of packages used by the app. If the app uses additional views
     * or modules besides the default ones, add more packages here.
     */
    @Override
    protected List<ReactPackage> getPackages() {
        return Arrays.<ReactPackage>asList(
            new MainReactPackage(),
            new SocialAuthPackage() // <-- Add this line
        );
    }
  ...
  }
  ```

### Facebook

5. Follow these instructions [setup appId](https://developers.facebook.com/docs/android/getting-started/#app_id) and [setup login](https://developers.facebook.com/docs/android/getting-started/#login_share)
6. Create and setup Key Hash [Facebook docs](https://developers.facebook.com/docs/android/getting-started/#create_hash)

### Twitter

#### Not supported :-(

redpajama_set_name: RedPajamaGithub


text:
// Copyright 2000-2021 JetBrains s.r.o. and contributors. Use of this source code is governed by the Apache 2.0 license.
package com.intellij.openapi.fileEditor.impl;

import com.google.common.annotations.VisibleForTesting;
import com.intellij.ide.ui.UISettings;
import com.intellij.ide.ui.UISettingsListener;
import com.intellij.openapi.Disposable;
import com.intellij.openapi.application.ApplicationManager;
import com.intellij.openapi.components.*;
import com.intellij.openapi.diagnostic.Logger;
import com.intellij.openapi.extensions.ExtensionPointListener;
import com.intellij.openapi.extensions.PluginDescriptor;
import com.intellij.openapi.fileEditor.*;
import com.intellij.openapi.fileEditor.ex.FileEditorManagerEx;
import com.intellij.openapi.fileEditor.ex.FileEditorWithProvider;
import com.intellij.openapi.progress.ProcessCanceledException;
import com.intellij.openapi.project.Project;
import com.intellij.openapi.startup.StartupActivity;
import com.intellij.openapi.util.Pair;
import com.intellij.openapi.vfs.VfsUtilCore;
import com.intellij.openapi.vfs.VirtualFile;
import com.intellij.openapi.vfs.VirtualFileManager;
import com.intellij.psi.PsiDocumentManager;
import com.intellij.util.ArrayUtilRt;
import com.intellij.util.messages.SimpleMessageBusConnection;
import org.jdom.Element;
import org.jetbrains.annotations.ApiStatus;
import org.jetbrains.annotations.NotNull;
import org.jetbrains.annotations.Nullable;

import java.util.*;

@Service(Service.Level.PROJECT)
@State(name = "editorHistoryManager", storages = @Storage(StoragePathMacros.PRODUCT_WORKSPACE_FILE))
public final class EditorHistoryManager implements PersistentStateComponent<Element>, Disposable {
  private static final Logger LOG = Logger.getInstance(EditorHistoryManager.class);

  private final Project myProject;

  public static EditorHistoryManager getInstance(@NotNull Project project){
    return project.getService(EditorHistoryManager.class);
  }

  /**
   * State corresponding to the most recent file is the last
   */
  private final List<HistoryEntry> myEntriesList = new ArrayList<>();

  EditorHistoryManager(@NotNull Project project) {
    myProject = project;

    SimpleMessageBusConnection connection = project.getMessageBus().simpleConnect();
    connection.subscribe(UISettingsListener.TOPIC, uiSettings -> trimToSize());
    connection.subscribe(FileEditorManagerListener.Before.FILE_EDITOR_MANAGER, new FileEditorManagerListener.Before() {
      @Override
      public void beforeFileClosed(@NotNull FileEditorManager source, @NotNull VirtualFile file) {
        updateHistoryEntry(file, false);
      }
    });
    connection.subscribe(FileEditorManagerListener.FILE_EDITOR_MANAGER, new MyEditorManagerListener());

    FileEditorProvider.EP_FILE_EDITOR_PROVIDER.addExtensionPointListener(new ExtensionPointListener<>() {
      @Override
      public void extensionRemoved(@NotNull FileEditorProvider provider, @NotNull PluginDescriptor pluginDescriptor) {
        myEntriesList.forEach(e -> e.onProviderRemoval(provider));
      }
    }, this);
  }

  static final class EditorHistoryManagerStartUpActivity implements StartupActivity.DumbAware {
    @Override
    public void runActivity(@NotNull Project project) {
      getInstance(project);
    }
  }

  private synchronized void removeEntry(@NotNull HistoryEntry entry) {
    if (myEntriesList.remove(entry)) {
      entry.destroy();
    }
  }

  private synchronized void moveOnTop(@NotNull HistoryEntry entry) {
    myEntriesList.remove(entry);
    myEntriesList.add(entry);
  }

  @ApiStatus.Internal
  @ApiStatus.Experimental
  public interface IncludeInEditorHistoryFile {}

  /**
   * Makes file most recent one
   */
  private void fileOpenedImpl(@NotNull VirtualFile file, @Nullable FileEditor fallbackEditor, @Nullable FileEditorProvider fallbackProvider) {
    ApplicationManager.getApplication().assertIsDispatchThread();
    // don't add files that cannot be found via VFM (light & etc.)
    if (!(file instanceof IncludeInEditorHistoryFile) &&
        VirtualFileManager.getInstance().findFileByUrl(file.getUrl()) == null) {
      return;
    }

    FileEditorManagerEx editorManager = FileEditorManagerEx.getInstanceEx(myProject);

    Pair<FileEditor[], FileEditorProvider[]> editorsWithProviders = editorManager.getEditorsWithProviders(file);
    FileEditor[] editors = editorsWithProviders.getFirst();
    FileEditorProvider[] oldProviders = editorsWithProviders.getSecond();
    LOG.assertTrue(editors.length == oldProviders.length, "Different number of editors and providers");
    if (editors.length <= 0 && fallbackEditor != null && fallbackProvider != null) {
      editors = new FileEditor[] { fallbackEditor };
      oldProviders = new FileEditorProvider[] { fallbackProvider };
    }
    if (editors.length <= 0) {
      // fileOpened notification is asynchronous, file could have been closed by now due to some reason
      return;
    }
    FileEditor selectedEditor = editorManager.getSelectedEditor(file);
    if (selectedEditor == null) {
      selectedEditor = fallbackEditor;
    }
    LOG.assertTrue(selectedEditor != null);
    int selectedProviderIndex = ArrayUtilRt.find(editors, selectedEditor);
    LOG.assertTrue(selectedProviderIndex != -1, "Can't find " + selectedEditor + " among " + Arrays.asList(editors));

    HistoryEntry entry = getEntry(file);
    if (entry != null) {
      moveOnTop(entry);
    }
    else {
      FileEditorState[] states = new FileEditorState[editors.length];
      FileEditorProvider[] providers = new FileEditorProvider[editors.length];
      for (int i = states.length - 1; i >= 0; i--) {
        FileEditorProvider provider = oldProviders[i];
        LOG.assertTrue(provider != null);
        providers[i] = provider;
        FileEditor editor = editors[i];
        if (editor.isValid()) {
          states[i] = editor.getState(FileEditorStateLevel.FULL);
        }
      }
      //noinspection SynchronizeOnThis
      synchronized (this) {
        myEntriesList.add(HistoryEntry.createHeavy(myProject, file, providers, states, providers[selectedProviderIndex]));
      }
      trimToSize();
    }
  }

  @SuppressWarnings("WeakerAccess")
  public void updateHistoryEntry(@NotNull VirtualFile file, boolean changeEntryOrderOnly) {
    updateHistoryEntry(file, null, null, changeEntryOrderOnly);
  }

  private void updateHistoryEntry(@NotNull VirtualFile file,
                                  @Nullable FileEditor fileEditor,
                                  @Nullable FileEditorProvider fileEditorProvider,
                                  boolean changeEntryOrderOnly) {
    FileEditorManagerEx editorManager = FileEditorManagerEx.getInstanceEx(myProject);
    FileEditor[] editors;
    FileEditorProvider[] providers;
    if (fileEditor == null || fileEditorProvider == null) {
      Pair<FileEditor[], FileEditorProvider[]> editorsWithProviders = editorManager.getEditorsWithProviders(file);
      editors = editorsWithProviders.getFirst();
      providers = editorsWithProviders.getSecond();
    }
    else {
      editors = new FileEditor[] {fileEditor};
      providers = new FileEditorProvider[] {fileEditorProvider};
    }

    if (editors.length == 0) {
      // obviously not opened in any editor at the moment,
      // makes no sense to put the file in the history
      return;
    }
    HistoryEntry entry = getEntry(file);
    if (entry == null) {
      // Size of entry list can be less than number of opened editors (some entries can be removed)
      if (file.isValid()) {
        // the file could have been deleted, so the isValid() check is essential
        fileOpenedImpl(file, fileEditor, fileEditorProvider);
      }
      return;
    }

    if (!changeEntryOrderOnly) { // update entry state
      //LOG.assertTrue(editors.length > 0);
      for (int i = editors.length - 1; i >= 0; i--) {
        FileEditor           editor = editors   [i];
        FileEditorProvider provider = providers [i];
        if (provider == null) continue; // can happen if fileEditorProvider is null
        if (!editor.isValid()) {
          // this can happen for example if file extension was changed
          // and this method was called during corresponding myEditor close up
          continue;
        }

        FileEditorState oldState = entry.getState(provider);
        FileEditorState newState = editor.getState(FileEditorStateLevel.FULL);
        if (!newState.equals(oldState)) {
          entry.putState(provider, newState);
        }
      }
    }

    FileEditorWithProvider selectedEditorWithProvider = editorManager.getSelectedEditorWithProvider(file);
    if (selectedEditorWithProvider != null) {
      //LOG.assertTrue(selectedEditorWithProvider != null);
      entry.setSelectedProvider(selectedEditorWithProvider.getProvider());
      LOG.assertTrue(entry.getSelectedProvider() != null);

      if (changeEntryOrderOnly) {
        moveOnTop(entry);
      }
    }
  }

  /**
   * @return array of valid files that are in the history, oldest first.
   */
  public synchronized VirtualFile @NotNull [] getFiles() {
    List<VirtualFile> result = new ArrayList<>(myEntriesList.size());
    for (HistoryEntry entry : myEntriesList) {
      VirtualFile file = entry.getFile();
      if (file != null) result.add(file);
    }
    return VfsUtilCore.toVirtualFileArray(result);
  }

  /**
   * For internal or test-only usage.
   */
  @VisibleForTesting
  public synchronized void removeAllFiles() {
    for (HistoryEntry entry : myEntriesList) {
      entry.destroy();
    }
    myEntriesList.clear();
  }

  /**
   * @return a set of valid files that are in the history, oldest first.
   */
  @NotNull
  public synchronized List<VirtualFile> getFileList() {
    List<VirtualFile> result = new ArrayList<>();
    for (HistoryEntry entry : myEntriesList) {
      VirtualFile file = entry.getFile();
      if (file != null) {
        result.add(file);
      }
    }
    return result;
  }

  public synchronized boolean hasBeenOpen(@NotNull VirtualFile f) {
    for (HistoryEntry each : myEntriesList) {
      if (f.equals(each.getFile())) {
        return true;
      }
    }
    return false;
  }

  /**
   * Removes specified {@code file} from history. The method does
   * nothing if {@code file} is not in the history.
   *
   * @exception IllegalArgumentException if {@code file}
   * is {@code null}
   */
  public synchronized void removeFile(@NotNull VirtualFile file){
    HistoryEntry entry = getEntry(file);
    if(entry != null){
      removeEntry(entry);
    }
  }

  public FileEditorState getState(@NotNull VirtualFile file, @NotNull FileEditorProvider provider) {
    HistoryEntry entry = getEntry(file);
    return entry != null ? entry.getState(provider) : null;
  }

  /**
   * @return may be null
   */
  FileEditorProvider getSelectedProvider(@NotNull VirtualFile file) {
    HistoryEntry entry = getEntry(file);
    return entry != null ? entry.getSelectedProvider() : null;
  }

  private synchronized HistoryEntry getEntry(@NotNull VirtualFile file) {
    for (int i = myEntriesList.size() - 1; i >= 0; i--) {
      HistoryEntry entry = myEntriesList.get(i);
      VirtualFile entryFile = entry.getFile();
      if (file.equals(entryFile)) {
        return entry;
      }
    }
    return null;
  }

  /**
   * If total number of files in history more then {@code UISettings.RECENT_FILES_LIMIT}
   * then removes the oldest ones to fit the history to new size.
   */
  private synchronized void trimToSize() {
    int limit = UISettings.getInstance().getRecentFilesLimit() + 1;
    while (myEntriesList.size() > limit) {
      HistoryEntry removed = myEntriesList.remove(0);
      removed.destroy();
    }
  }

  @Override
  public synchronized void loadState(@NotNull Element state) {
    // each HistoryEntry contains myDisposable that must be disposed to dispose corresponding virtual file pointer
    removeAllFiles();

    // backward compatibility - previously entry maybe duplicated
    Map<String, Element> fileToElement = new LinkedHashMap<>();
    for (Element e : state.getChildren(HistoryEntry.TAG)) {
      String file = e.getAttributeValue(HistoryEntry.FILE_ATTR);
      fileToElement.remove(file);
      // last is the winner
      fileToElement.put(file, e);
    }

    for (Element e : fileToElement.values()) {
      try {
        myEntriesList.add(HistoryEntry.createHeavy(myProject, e));
      }
      catch (ProcessCanceledException ignored) {
      }
      catch (Exception anyException) {
        LOG.error(anyException);
      }
    }
  }

  @Override
  public synchronized Element getState() {
    Element element = new Element("state");
    // update history before saving
    VirtualFile[] openFiles = FileEditorManager.getInstance(myProject).getOpenFiles();
    for (int i = openFiles.length - 1; i >= 0; i--) {
      VirtualFile file = openFiles[i];
      // we have to update only files that are in history
      if (getEntry(file) != null) {
        updateHistoryEntry(file, false);
      }
    }

    for (HistoryEntry entry : myEntriesList) {
      entry.writeExternal(element, myProject);
    }
    return element;
  }

  @Override
  public synchronized void dispose() {
    removeAllFiles();
  }

  /**
   * Updates history
   */
  private final class MyEditorManagerListener implements FileEditorManagerListener {
    @Override
    public void fileOpened(@NotNull FileEditorManager source, @NotNull VirtualFile file){
      fileOpenedImpl(file, null, null);
    }

    @Override
    public void selectionChanged(@NotNull FileEditorManagerEvent event){
      // updateHistoryEntry does commitDocument which is 1) very expensive and 2) cannot be performed from within PSI change listener
      // so defer updating history entry until documents committed to improve responsiveness
      PsiDocumentManager.getInstance(myProject).performWhenAllCommitted(() -> {
        FileEditor newEditor = event.getNewEditor();
        if(newEditor != null && !newEditor.isValid())
          return;

        VirtualFile oldFile = event.getOldFile();
        if (oldFile != null) {
          updateHistoryEntry(oldFile, event.getOldEditor(), event.getOldProvider(), false);
        }
        VirtualFile newFile = event.getNewFile();
        if (newFile != null) {
          updateHistoryEntry(newFile, true);
        }
      });
    }
  }
}


Working with Amazon Web Services (AWS) technologies for the first time can be confusing sometimes. The most commonly perplexed situations in an AWS environment (apart from the billing structure) are the different AWS terminologies. Things like EC2, S3, EIP etc. can be overwhelming.
There are immense features offered by AWS to discuss. Although, this article covers a few basic services which helps to understand and get started with AWS.
Disclaimer: AWS has a pay-as-you-go billing model. Readers are advised to go through the AWS billing/ pricing documentation and FAQ's before making a decision.
AWS Management Console is a web-based interface for managing most of the AWS environment technologies.
Elastic Cloud Compute (EC2) is a virtual machine that is created in the AWS Management Console. Each virtual machine created is called an "instance".
Amazon Elastic Block Store (EBS) are virtual disk drives which are attached to EC2 instances. Each EBS created is called a "volume".
The EBS volumes can be "attached" or "detached" to EC2 instances as required. The size of a particular EBS volume can be specified at the time of its creation along with the required file system and partitions.
Elastic IP Address (EIP) is a static IP specifically to associate with an EC2 instance. When an EC2 instance is created a dynamic IP address is assigned to it by default. The dynamic IP address changes if a reboot is initiated on the instance. Hence to counter this situation an EIP is associated with an instance to keep it static.
Simple Storage Service (S3) is a cloud based storage service.
Static assets like images, cascading style sheet (CSS) files etc. used in web applications can be stored on S3. The S3 can be used to store data backups as well.
CloudFront is a content delivery system offered by AWS. CloudFront can be connected with S3 so that the static files are delivered in an efficient way.
CloudFront integrates with other Amazon web services to provide an easy method to distribute content.
The AWS environment technologies explained in this article are to provide a basic understanding to get started with Amazon AWS.
We will try to bring in more parts and series of AWS related articles and tutorials.
Till that time, stay tuned and go ahead explore more and get to know more about AWS.


Keep up-to-date with what is happening in your area by text, phone call or email.
In The Know is the free messaging service brought to you by Lancashire Police, supported by Lancashire Fire and Rescue Service and Lancashire County Council, where you can sign up to receive information about what is happening close to where you live or work, either by phone call, text or email.
You will receive regular crime updates, information on road closures, information about fires and other emergencies and learn about what your local neighbourhood policing team is doing, as well as being able to register for job alerts.
To find out more please visit: www.stayintheknow.co.uk.


Host Monster has been a leading hosting provider and offers hosting solutions to businesses and personal websites previously the year 1996. They have enough keep incredible affordability in terms of cost in row hosting solutions coupled along with mood retain and hosting equipment.
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All photos taken in 1961, sorted by posting date, most recent first.
Added to archive April 21, 2019 by Ken Patton.
Run by at Bridge 176.
Added to archive May 17, 2018 by Ken Patton.
Added to archive May 12, 2018 by Ken Patton.


This Aaron Hendra Is My Home Boy Funny Trending Country Music T Shirt is one of our favorites, it wont be around forever so order yours here today! The graphic is printed on a quality, preshrunk cotton t shirt you will love, satisfaction guaranteed. It would make a great addition to your wardrobe, or buy it as a gift for friends and family.
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text:
New blood at county health
Posted By Achille Ngoma on Wed, Jul 18, 2012 at 11:05 AM
El Paso County Public Health
With El Paso County Public Health for 17 years, Law has held leadership roles in the areas of strategic planning.
El Paso County Public Health has promoted Jill Law to the position of county public health director. She had been the interim public health director since March 2012, when executive director Kandi Buckland retired.
"Jill's challenge is to keep current level of services and funding," explains Susan Wheelan, public health communication director.
Letson has dedicated his career to public health, pediatrics and fighting infectious diseases.
The challenge will be quite different for the department's other new hire, medical director Dr. Bill Letson: In May alone, 41 cases of communicable diseases were reported and investigated. They ranged from salmonella and viral meningitis to varicella. In addition, two cases of active tuberculosis were suspected.
Letson has held several director positions with the Centers for Disease Control and Prevention (CDC) in the areas of epidemiology and immunization.
More details on Letson and Law's experience are available in the release below:
Contact: Danielle Oller, Communication Specialist
(719) 575-8985 office; (719) 465-6186 cell
El Paso County Board of Health Appoints Director, Welcomes New Medical Director
El Paso County, CO - The El Paso County Board of Health on Monday unanimously appointed Jill Law to the position of El Paso County Public Health Director, and welcomed Dr. Bill Letson as the agency's Medical Director.
Both Law and Letson come to their new roles with decades of public health experience.
Law has served as the interim public health director since March 2012 following the retirement of executive director Kandi Buckland. Law received her registered nurse degree from Beth-El College of Nursing at the University of Colorado, Colorado Springs. Law also has a master's degree in health administration from the University of Phoenix. She previously served as the agency's division director of health services, overseeing six departments and was the division director of clinical and health services. Prior to joining El Paso County Public Health in 1995, she worked at Cedar Springs Psychiatric Unit and the Penrose Hospital Surgical Unit.
In addition to her years of clinical expertise, throughout her 17 years with El Paso County Public Health Law has held leadership roles in the areas of strategic planning, policy development, program evaluation, coalition and community organization, budget and finance, emergency preparedness, marketing, agency management and resource development.
Dr. G. William (Bill) Letson has dedicated his career to research and implementation activities in public health, pediatrics and infectious diseases. He holds a bachelor's degree in environmental biology from the University of Colorado, Boulder; a master's degree in zoology from Colorado State University; and a Doctor of Medicine from the University of Colorado School of Medicine, Denver. He has been published widely and served several faculty appointments.
From 2011 to 2012, he worked as a global public health consultant, serving as International Vaccine Institute Vaccinology Faculty, and is a member of the Independent Review Committee for the Global Alliance for Vaccines and Immunizations. Previously, Dr. Letson was Deputy Director of Pediatric Dengue Vaccine Initiative (PDVI) and Director of Vaccine Evaluation Program, PDVI/International Vaccine Institute in Seoul Korea, a post he held for four and a half years.
Dr. Letson has held several director positions with the Centers for Disease Control and Prevention (CDC) in the areas of epidemiology and immunization. In addition, he served as the Wyoming's State Health Officer and Director of their Maternal Child Health program and Maternal Child Health Consultant/Epidemiologist at Colorado Department of Public Health and Environment.
Tags: El Paso County board of health, Dr. Bill Letson, pediatrics and infectious disease, Centers for Disease Control and Prevention (CDC), Jill Law, Kandi Buckland, Image
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redpajama_set_name: RedPajamaCommonCrawl


Paul Potts' debut CD "One Chance" was released shortly after this audition was first broadcast. The album has a been a huge success globally, reaching number one in some 14 countries. It is still available in stores in the US, Europe and around the world or on line.
Having both won the competition and released his debut album, Paul is exhaustively touring, engaging in live performances in the UK and TV appearances worldwide. For details of live performances, check out his website.
Please do not promote any other websites in the comments. Any such comments will be deleted.
Paul Potts has won the final of this competition. The prize is £100,000 and the chance to perform at the Royal Variety Performance on 3rd December 2007.
Paul Potts sings Nessun dorma on UK talent show "Britain's got Talent" featuring Simon Cowell as a judge.
Comments about the performance only please. All others will be removed and blocked.
Aerosmith's "I don't want to miss a thing" is also featured in the clip, for those that keep asking!


The modern kitchen is characterized by its elegance, well-arrangement uniqueness, and simplicity. To have a modern kitchen, try to find a definite theme to apply on your kitchen's design, color, and tiles. The tiles of the floor, countertop, back splash, ceiling and wall have to be harmonized with the main theme.
Tiles have different kinds, designs and colors to match your kitchen's parts. The limestone tile can fit nearly every part in your kitchen. It is a natural stone that suit both of modern and traditional designs because it comes in the colors range between beige and brown. Another natural and durable kind is the faux proclaim tiles that can look like the real wood but more durable. Likely, the vinyl tiles have many usages, textures, colors, and designs. They are very hardy, waterproofed, and easy to clean. The vinyl's solution could be found easily in the market.
You could use other kinds of tiles that have less durability but still popular and long lasting. The reinforced glass tiles are suitable for nearly all of your kitchen parts. They are made of recycled glass but able to give your modern kitchen a sleek, sparkling, and shiny look. They could be used in the backsplash or even the floor for those who have no children at home. The clear glass will match your large window, glass doors of the cabinet, and the light wood or white painted cabinet.
The inexpensive tiles are widely used also because they are long lasting. The metal tiles either they are stainless steel or copper could be used for your countertop and backsplash because they are safe, but when applied to the floor, the wet metal may cause slipping. The ceramic also is inexpensive and durable if it is installed in a right way. You can use the colored grout to keep the ceramic clean all the time. The mosaic tiles could be adhered to the wall and ceiling to protect your kitchen. The broken terracotta and the slate tiles can give your kitchen a marvelous colorful modern look. The marble, brick, and granite will make you feel the freedom of nature in your own kitchen. Now it's your turn to tell us what kind of above tiles more suitable to your kitchen's theme and nature.


text:
Michael Kimmel
State University of New York, Stony Brook
STI Publications
Gender Identities in a Globalized World
Michael Kimmel: "Gender Equality is Probably the Best Thing that Has Ever Happened for Men" 5 years ago
Michael S. Kimmel is Professor of Sociology at SUNY at Stony Brook. His books include Changing Men (1987), Men Confront Pornography (1990), Men's Lives (6th edition, 2003) Against the Tide: Profeminist Men in the United States, 1776-1990 (1992), The Politics of Manhood (1996), Manhood in America: A Cultural History(1996, 10th anniversary second edition, 2006), and The Gendered Society (2nd edition, 2003). He co-edited The Encyclopedia on Men and Masculinities (2004) and the Handbook of Studies on Men and Masculinities (2004). His most recent publications are the bestselling study of young (16-26-year-old) men, Guyland: The Perilous World Where Boys Become Men (2008), and Misframing Men (2010).
Dr. Kimmel is the founder and editor of Men and Masculinities, the field's premier scholarly journal, a book series on Gender and Sexuality at New York University Press, and the editor of the Sage Series on Men and Masculinities. He is the Spokesperson for the National Organization for Men Against Sexism (NOMAS) and lectures extensively in corporations and on campuses in the U.S. and abroad.
Read an excerpt from Guyland in USA TODAY
Read Dr. Kimmel's article "A Black Woman Took My Job," in which he argues that men should work for gender equality, in the New Internationalist.
Read a PBS interview with Dr. Kimmel in which he discusses violence against women.


text:
South Porcupine Hockey Leagues Stay on the Ice
It was game on in South Porcupine, Ontario, as the 2020/21 season played through with enhanced health and safety protocols. The Carlo Cattarello arena, known locally as 'The Barn,' is home to Northern Ontario's largest recreational hockey league. The Barn Recreational League consists of 16 teams, and the Oldtimers League has eight teams.
Hockey is an important part of the community. The hard work of volunteers like Burt St. Amour, administrator for the two leagues, Owen Rigg, the Referee in Chief, the officials, and the many team reps, keeps these leagues successful year after year. Through Burt's leadership, having a season during the pandemic became a possibility.
Burt St. Amour (left) and Owen Rigg (right)
"Burt did a ton of work during the summer that I wasn't even aware of," said Owen. "I wasn't really aware that this was going on in the background at the time, so for him to get the approvals to be able to make it a go was pretty special because there wasn't anything around in Northern Ontario, and I think from a mental health standpoint this was very important for the guys in town."
Fortunate to be in an area where the case numbers were low, Burt worked closely with the local health unit and consulted with CARHA Hockey on their return to play guidelines. He adjusted the league operations, enforced contact tracing, and implemented a plan to provide a safe sport for all the participants and league officials, which helped ensure the community's safety. With this being a very social league, the support from team reps was crucial to ensuring teams followed the rules.
"It sure opened my eyes, but we had a lot of support from the local health unit too, and that was good for us. Just happened to have a girl there who played hockey, and that helped us quite a bit."
A typical season would have had 24-26 games each week, spread out over multiple nights. To accommodate for the modified season, adjustments included condensing the schedule, adding games during the week, and getting creative. Owen recognizes the team reps' efforts and gives credit to the officials and timekeepers who were there every night. There were very few cancellations, and it was a team effort.
Referred to as a broken-up season, it was like two seasons in one. Play ran until Christmas, then paused for a month and a half due to provincial lockdown. They resumed in February, with players coming back slightly out of shape but in good spirits to continue. The Oldtimers League successfully completed in early April, while the Rec League ended a week and a half before their intended end date of April 10 due to the province's new round of lockdown measures. While they came short of a full season, no league play was impacted by a local positive COVID case, and Burt and Owen are grateful to have been able to provide their community with an outlet during these uncertain times.
"Every time a week went by, we put a checkmark on the calendar and said we survived another week," Burt shared.
"It's pretty cool to see, and kudos has to be the team reps because there were reminder emails, and we would send the reminders about things," said Owen. "The referees were on them about spitting and things that are pretty typical to the game of hockey, and changing that dynamic was tough at first, but the guys followed the rules and were understanding of them and doing what they had to continue to play, and I was super impressed by how it went down this year, it really went well."
A positive they saw out of the challenging season was getting all the names on game sheets. Contact tracing mandated this, which also made Owen's job easier when following up on game infractions. In addition, it has helped make the league more professional and something they plan to continue asking teams to provide. "Just to get to the point where we almost finished the season was still pretty incredible, and a lot of recognition goes to Burt for pulling it off," said Owen. "To see our league be as successful as it was and not have any cases of COVID from it is really indicative of the fact that all the protocols were followed. Everyone worked hard, everyone did the cleaning. I've never seen the arena so clean, it was amazing to see. It really worked well, and we should be proud of that aspect of it."