text:
\section{Introduction}
Bill Thurston has had a tremendous impact on shaping the geometric imagination of many mathematicians. His new ways of thinking about geometric shapes changed the way we now think about hyperbolic surfaces, three-manifolds, and higher dimensional hyperbolic manifolds.

There are many interesting geometries besides hyperbolic geometry, as for example complex hyperbolic geometry, projective geometry or other higher rank geometries, i.e. geometries whose transformation groups
are semisimple Lie groups of higher rank, as for example
$\mathrm{SL}(n,\mathbf{R})$, $n\geq 3$, or $\mathrm{Sp}(2n,\mathbf{R})$, $n\geq2$.
In recent years several developments led to the discovery of
interesting classes of discrete subgroups of Lie groups, which parametrize non-trivial deformation spaces of geometric structures, which are governed by higher rank Lie groups.
One such development is the theory of Anosov representations of word hyperbolic groups which has been introduced by Fran\c{c}ois Labourie \cite{Labourie_anosov}.
The notion of Anosov representations is defined for representation of a finitely generated word hyperbolic group into any semisimple (even reductive) non-compact Lie group.
However, in this article we restrict our attention to representations into the symplectic groups $\mathrm{Sp}(2n,\mathbf{R})$ and $\mathrm{Sp}(2n,\mathbf{C})$.

The aim of this article is to describe, in a rather informal way and in a special case, some of the geometric aspects of Anosov representations, which lead us to think about manifolds modelled on higher rank
geometries and try to build a geometric imagination for such geometric structures.

We are dearly missing Bill Thurston's imagination. We will have to rely on and build our own.
\section{Preliminaries}
\subsection{The symplectic group}
A symplectic vector space is a pair $(V_\mathbf{K}, \omega_\mathbf{K})$, where $V_\mathbf{K}$ is a $2n$-dimensional $\mathbf{K}$-vector space, for  $\mathbf{K} = \mathbf{R}$ or $\mathbf{C}$, and $\omega_\mathbf{K}$ is a non-degenerate skew-symmetric bilinear form on $V$.
The linear (real or complex) symplectic group of $(V_\mathbf{K},\omega_\mathbf{K})$ is
$$
\mathrm{Sp}(V_\mathbf{K}, \omega) = \{ g \in \mathrm{GL}(V_\mathbf{K})\, |\, \omega_\mathbf{K}( gv, gw) = \omega_\mathbf{K}(v,w) \text{ for all } v,w \in V_\mathbf{K}\}
$$
If we choose a linear isomorphism $V_\mathbf{K} \cong \mathbf{K}^{2n}$ and represent $\omega_\mathbf{K}$ by the matrix $ \begin{pmatrix}
  0 & \mathrm{Id} \\
  -\mathrm{Id} & 0
 \end{pmatrix}$, i.e.
$
\omega_\mathbf{K} ( v, w) = v^T \begin{pmatrix}
  0 & \mathrm{Id} \\
  -\mathrm{Id} & 0
 \end{pmatrix} w,$
 then \[\mathrm{Sp}(V_\mathbf{K}, \omega) \cong \mathrm{Sp}(2n,\mathbf{K}) = \left\{g \in \mathrm{GL}(2n,\mathbf{K}) \,|\,  g^T\begin{pmatrix}
  0 & \mathrm{Id} \\
  -\mathrm{Id} & 0
 \end{pmatrix} g = \begin{pmatrix}
  0 & \mathrm{Id} \\
  -\mathrm{Id} & 0
 \end{pmatrix}\right\}.\]

We will sometimes denote $V_\mathbf{R}$ simply by $V$ and $\omega_\mathbf{R}$ by $\omega$. If $V_\mathbf{C}$ is the complexification of $V$ and $\omega_\mathbf{C}$ is the complex linear extension of $\omega$, we have a natural embedding of Lie groups
\[\mathrm{Sp}(2n,\mathbf{R})\cong\mathrm{Sp}(V, \omega) \rightarrow \mathrm{Sp}(V_\mathbf{C}, \omega) \cong \mathrm{Sp}(2n,\mathbf{C}).\]

The symplectic group $\mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})$ acts transitively on various homogeneous spaces:
\begin{enumerate}
\item The Riemannian symmetric space $X_\mathbf{R}(n)$ associated to $\mathrm{Sp}(2n,\mathbf{R})$, which can be identified with \[X_\mathbf{R}(n) \cong\mathrm{Sp}(2n,\mathbf{R})/ \mathrm{U}(n).\]
\item The Riemannian symmetric space $X_\mathbf{C}(n)$ associated to $\mathrm{Sp}(2n,\mathbf{C})$, which can be identified with \[X_\mathbf{C}(n) \cong\mathrm{Sp}(2n,\mathbf{C})/ \mathrm{Sp}(n).\]
\item The affine symmetric spaces $X_{p,q}$,  $p+q = n$, which can be identified with
\[X_{p,q}\cong \mathrm{Sp}(2n,\mathbf{R})/ \mathrm{U}(p,q).\]
The extreme cases being $X_{0,n} = X_\mathbf{R} (n) = X_{n,0}$.

\item The space $\mathrm{Is}_i(V_\mathbf{K})$ of $\omega_\mathbf{K}$-isotropic subspace of $V_\mathbf{K}$ of dimension $i$. We can identify
 \[ \mathrm{Is}_i(V_\mathbf{K}) \cong \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})/ Q_1(V_\mathbf{K}),\] where $Q_i(V_\mathbf{K})$ denotes the stabilizer of an isotropic i-dimensional subspace of $V_\mathbf{K}$
Of particular interest to us will be the projective space $ \mathbf{P}(V_\mathbf{K})$, which can be identified with  $\mathrm{Is}_1(V_\mathbf{K})$, and the
the space of Lagrangians $ \mathrm{Lag}(V_\mathbf{K})$, which can be identified with $\mathrm{Is}_n(V_\mathbf{K})$.

\end{enumerate}

\begin{remark}
\begin{enumerate}
\item
Recall that a subspace
$L\subset V_\mathbf{K}$ is $\omega_\mathbf{K}$-isotropic if $L \subset L^{\perp_{\omega_\mathbf{K}}}$, where $L^{\perp_{\omega_\mathbf{K}}}$ is the orthogonal complement with respect to $\omega_\mathbf{K}$. Since $\omega_K$ is skew-symmetric, any one-dimensional subspace is isotropic. An isotropic subspace $L$ is maximal if it is of dimension $n$, or equivalently if $L = L^{\perp_{\omega_\mathbf{K}}}$. In this case $L$ is called a Lagrangian subspace.
\item Two elements $F, F' \in \mathrm{Is}_i(V_\mathbf{K})$ are said to be {\em transverse} if $F \oplus F'^{\perp_{\omega_\mathbf{K}}} = V_\mathbf{K} = F' \oplus F^{\perp_{\omega_\mathbf{K}}} $.
\item For every isotropic subspace  $Z \in \mathrm{Is}_i(V_\mathbf{K})$, the symplectic form $\omega_\mathbf{K}$ induces a non-degenerate skew-symmetric bilinear form on the vector space $Z^{\perp_{\omega_\mathbf{K}}}/ Z$, turning  $Z^{\perp_{\omega_\mathbf{K}}}/ Z$ into a $2(n-i)$-dimensional symplectic vector space.

\end{enumerate}

\end{remark}

\subsection{Decomposition of the space of complex Lagrangian subspaces}
Let $(V, \omega)$ be a symplectic vector space and  $(V_\mathbf{C}, \omega_\mathbf{C})$ its complexification Then the space
of complex Lagrangians $\mathrm{Lag}(V_\mathbf{C})$ decomposes into several $\mathrm{Sp}(V,\omega)$-orbits.

First, there is a decomposition with respect to the real structure: for every $i = 0, \cdots, n$ we set
$$\mathcal{R}_i = \{ W \, |\, dim(W \cap \overline{W} ) = i\},$$ where $\overline{W}$ is the complex conjugation with respect to $V \subset V_\mathbf{C}$.

\begin{proposition}\label{prop:orbits}
The set $\mathcal{R}_i$ fibers over the space $\mathrm{Is}_i(V)$ of $i$-dimensional  isotropic subspaces in $V$ with fiber isomorphic to $\bigcup_{p'+q' = n-i} X_{p',q'}$.
The set $\mathcal{R}_n$ is the unique closed $\mathrm{Sp}(V,\omega)$ orbit and identifies with $\mathrm{Lag}(V)$.  The set $\mathcal{R}_0$ is the union of the $n+1$ open $\mathrm{Sp}(V,\omega)$-orbits, $\mathcal{R}_0 = \bigcup_{p=0, \cdots, n} H_{p,q}$, where $H_{p,q} \cong X_{p,q}$.
\end{proposition}
\begin{proof}
We sketch the proof, leaving some verifications to the reader. A more detailed description of the embedding $X_\mathbf{R}(n)$  into $\mathrm{Lag}(V_\mathbf{C})$ can be found for example in \cite{Satake_book}.

Note that the imaginary part of $\omega_\mathbf{C}$ gives rise to a non-degenerate Hermitian form $h$ of signature $(n,n)$ on $V_\mathbf{C}$,
\( h(v,w) = i \omega_\mathbf{C}(\overline{v}, w) \)
This Hermitian form is preserved by $\mathrm{Sp}(V,\omega)$.

Let us first describe $\mathcal{R}_0$.
It is an easy calculation that if $W \in \mathcal{R}_0$, then the restriction of $h$ to $W\times W$ is non-degenerate and of signature $(p,q)$ with $p+q = n$.
The set  $\mathrm{R}_0$ then decomposes into the disjoint union
 \[\mathcal{R}_0 = \bigcup_{p=0, \cdots, n} \mathcal{H}_{p,q},\]
 where
 \[
 \mathcal{H}_{p,q} = \{ W \in\mathcal{R}_0\, |\, h|_{W\times W} \text{ is of signature } (p,q)\}.
 \]
On each $ \mathcal{H}_{p,q} $ the group $\mathrm{Sp}(V,\omega)$ acts transitively, and the stabilizer of $W\in \mathcal{H}_{p,q} $ is $U(W, h|_{W\times}) \cong U(p,q)$. Thus $ \mathcal{H}_{p,q} \cong X_{p,q}$.

If $W \in \mathcal{R}_i$, then $Z= W\cap \overline{W}$ is an $i$-dimensional (complex) isotropic subspace of $V_\mathbf{C}$. Since $\overline{Z} = Z$, it is the complexification of a real $i$-dimensional isotropic subspace $Z'$ of $V$. This defines the projection $\mathcal{R}_i \rightarrow \mathrm{Is}_i(V)$.

 To describe the fiber consider $M=Z^{\perp_{\omega_\mathbf{C}}}/Z$. The symplectic form $\omega_\mathbf{C}$ induces a symplectic form on $M$, the real structure induces a real structure, and $h$ induces a non-degenerate Hermitian form of signature $(n-i, n-i)$. Note that any $W \in \mathcal{R}_i$ for which  $W\cap \overline{W} = Z$ is uniquely determined by a maximal isotropic (Lagrangian) subspace  $Y$ of $M$, which satisfies $Y\cap \overline{Y} = \{0\}$. Thus the description of $\mathcal{R}_0$ above  gives the description of the fiber.
 In particular, $\mathrm{R}_i$ also decomposes into several $\mathrm{Sp}(V,\omega)$-orbits $H^i_{p', q'}$, where $p' + q' = n-i$ and $H^i_{p', q'}$ is the component of $\mathcal{R}_i$ with fiber isomorphic to $X_{p',q'}$.

If $W \in \mathcal{R}_n$, then $W = \overline{W}$ and hence $W$ is the complexification of a real maximal $\omega$-isotropic subspace of $V$. The fiber of the projection $\mathcal{R}_n \to \mathrm{Lag}(V)$ is trivial.
This gives the identification $\mathcal{R}_n \cong \mathrm{Lag}(V)$. In particular, $\mathcal{R}_n$ is a compact, and hence closed $\mathrm{Sp}(V,\omega)$-orbit.
\end{proof}

Proposition~\ref{prop:orbits} gives explicit embeddings
\[X_{p,q} \hookrightarrow H_{p,q} \subset \mathrm{Lag}(V_\mathbf{C}).\] Since $\mathrm{Lag}(V_\mathbf{C})$ is compact, the closure of $H_{p,q} \subset \mathrm{Lag}(V_\mathbf{C})$ gives rise to a compactification of the affine symmetric spaces $X_{p,q}$. The closure of $H_{p,q}$ has the following structure
\[cl(H_{p,q}) = \bigcup_{p' \leq p, \, q' \leq q } H^i_{p',q'},\] with $H_{0,0} := \mathcal{R}_n$.

For the Riemannian symmetric space $X_\mathbf{R}(n)$  the embedding $X_\mathbf{R}(n) \hookrightarrow \mathrm{Lag}(V_\mathbf{C})$ is called the Harish-Chandra-embedding. The described compactification is isomorphic to a (minimal) Satake-Furstenberg compactification, and to the bounded symmetric domain compactification of $X_\mathbf{R}(n)$.

\begin{remark}
\begin{enumerate}
\item
In fact the bounded symmetric domain realization of $X$ is usually derived from the above embedding: Choosing two transverse Lagrangian subspaces $V_+$ and $V_-$ of $V_\mathbf{C}$, such that $h$ is positive definite on $V_+$ and negative definite on $V_-$, and such that $V_+$ and $V_-$ are orthogonal with respect to $h$,  any other space $W \in H_{n,0}$ is transverse to $V_-$, and thus can be written as a graph of a linear map from $V_+ \rightarrow V_-$. The condition that $W$ is Lagrangian implies that this map is symmetric and the condition that $h$ is positive definite on $W$ translates into the condition that $Id - \overline{Z} Z$ is positive definite. For details we refer to \cite{Satake_book}.
\item
When $n=1$ the space $ \mathrm{Lag}(V_\mathbf{C}) $ is just the projective space $ \mathbf{C}\mathbf{P}^1$, which  decomposes into three $\mathrm{SL}(2,\mathbf{R})$-orbits: upper hemisphere, lower hemisphere, equator.
\end{enumerate}
\end{remark}

\subsection{Symplectic Anosov representations}
We shortly recall here the notion of Anosov representations. We focus on representations into the symplectic group which are Anosov with respect to the stabilizer of an isotropic subspace of $V_\mathbf{K}$.
 The definition we give relies on a characterization of Anosov representations given in \cite{GGKW_anosov}, which can also be obtained from \cite{KapovichLeebPorti, KapovichLeebPorti14}. For general discussions of Anosov representations we refer to \cite{Labourie_anosov, Guichard_Wienhard_DoD, GGKW_anosov, KapovichLeebPorti, KapovichLeebPorti14, KapovichLeebPorti14_2}.

Let $\Gamma$ be a finitely generated word hyperbolic group. Its boundary $\partial_\infty \Gamma$ is a compact space, equipped with a natural action of $\Gamma$ as a uniform convergence group. In particular, every element $\gamma \in \Gamma$ of infinite order has a unique attracting fix point $\gamma^+ \in \partial_\infty \Gamma$ and a unique repelling fix point $\gamma^- \in  \partial_\infty \Gamma$ such that for every $x \in \partial_\infty \Gamma \backslash \{\gamma^-\}$ we have $\lim_{n\to \infty} \gamma^n \cdot x =  \gamma^+$. In the following we will assume for simplicity that $\Gamma$ is torsion free.

The symmetric spaces $X_\mathbf{K}$ of $\mathrm{Sp}(2n,\mathbf{K})$  are symmetric spaces of non-compact type of rank $n$. In such spaces the Riemannian distance function only partially determines the relative position of two points. More precise information is given by their distance vector. For group elements $g \in \mathrm{Sp}(2n,\mathbf{K})$ the distance vector between a fixed base point $x_0 \in X_\mathbf{K}$ and $g \cdot x_0$ is measured by a Cartan decomposition with respect to $x_0$.
To describe this in more detail, let $K$ be a maximal compact subgroup of $\mathrm{Sp}(2n,\mathbf{K})$, i.e. $K = \mathrm{U}(n)$ if $\mathbf{K} = \mathbf{R}$ and $K = \mathrm{Sp}(n)$ if $\mathbf{K} = \mathbf{C}$.  Let
$\mathfrak{a} \subset \mathfrak{s}\mathfrak{p}(2,\mathbf{K})$ be the subset of diagonal matrices with real entries, i.e.
\[\mathfrak{a} = \{ \mathrm{diag}(\lambda_1, \cdots \lambda_n, - \lambda_n, \cdots - \lambda_1) \, |\, \lambda_i \in \mathbf{R} \}.\]
Then $\mathrm{Sp}(2n,\mathbf{K})$ admits a decomposition  $\mathrm{Sp}(2n,\mathbf{K}) = K \exp({\mathfrak{a}})K$, i.e.  any element $g \in \mathrm{Sp}(2n,\mathbf{K}) $ can be written as $g = k_1 \exp{(a_g)} k_2$, with $a_g \in \mathfrak{a}$ and  $k_1, k_2 \in K$. This decomposition is in general not unique.
But if we consider the cone
\[\mathfrak{a}^+ = \{ diag(\lambda_1, \cdots \lambda_n, - \lambda_n, \cdots - \lambda_1) \in \mathfrak{a}\, |\, \lambda_1\geq \lambda_2\geq \cdots \geq \lambda_n\geq 0 \} \subset \mathfrak{a},\]
then  any element $g \in \mathrm{Sp}(2n,\mathbf{K}) $ can be written as $g = k_1 \exp{(a_g)} k_2$, with an unique $a_g \in \mathfrak{a}^+$ and  $k_1, k_2 \in K$. This decomposition
\[\mathrm{Sp}(2n,\mathbf{K}) = K \exp({\mathfrak{a}^+})K\] is called a Cartan decomposition.

It induces the {\em Cartan projection}
\[
\mu: \mathrm{Sp}(2n,\mathbf{K}) \rightarrow \mathfrak{a}^+, \, g \mapsto \mu(g) := a_g.
\]

The Cartan projection is a continuous, proper surjective map.

We denote by $\alpha_i \mathfrak{a}^*$, $ i = 1, \cdots, n$ the linear map given by  $\alpha_i = \epsilon_i - \epsilon_{i+1}$, where $\epsilon_i$ is the linear form which sends a diagonal matrix to its $i$-th eigenvalue.
Note that $\alpha_n = \epsilon_n - \epsilon_{n+1} = 2 \epsilon_n$.

\begin{definition}\label{def:anosov}
Let $\Gamma$ be a finitely generated word hyperbolic group and $\partial_\infty \Gamma$ its boundary.  A representation $\rho: \Gamma \to \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})$ is $Q_i$-Anosov, $i= 1, \cdots, n$ if the following holds
\begin{enumerate}
\item There exist a {\em continuous} $\rho$-equivariant map $\xi_i: \partial_\infty \Gamma \to \mathrm{Is}_i(V_\mathbf{K})$.
\item The map $\xi_i$ is {\em dynamics preserving}, i.e. for every $\gamma \in \Gamma$ of infinite order, the images of the attracting resp. repelling fix point $\gamma^\pm$ under $\xi_i$ is the attracting resp. repelling fix point of $\rho(\gamma)$ in $\mathrm{Is}_i(V_\mathbf{K})$.
\item The map $\xi_i$ is {\em transverse}, i.e. for every $t \neq t'$ in $\partial_\infty \Gamma$, the images under $\xi_i$ satisfy
\[\xi_i(t) \oplus \xi_i(t')^{\perp_{\omega_\mathbf{K}}} = V_\mathbf{K} = \xi_i(t') \oplus \xi_i(t)^{\perp_{\omega_\mathbf{K}}}.\]
\item For every diverging sequence $\gamma_n$ in $\Gamma$ we have
\[
\mathrm{lim}_{n\to \infty} \alpha_i(\mu(\gamma_n)) = \infty.
\]
\end{enumerate}
\end{definition}
\begin{remark}
\begin{enumerate}
\item
The original definition \cite{Labourie_anosov, Guichard_Wienhard_DoD} uses the geodesic flow space of the hyperbolic group $\Gamma$. This definition is very useful in several applications, see for example \cite{Sambarino, Bridgeman_Canary_Labourie_Sambarino}.  Characterizations not involving the geodesic flow space are given in \cite{KapovichLeebPorti, KapovichLeebPorti14, KapovichLeebPorti14_2, GGKW_anosov}. Some of the characterizations in \cite{KapovichLeebPorti, KapovichLeebPorti14, KapovichLeebPorti14_2} do not require the group $\Gamma$ to be assumed to be word hyperbolic a priori.
\item One can define $P$-Anosov representations with respect to any parabolic subgroup $P< \mathrm{Sp}(2n,\mathbf{R})$, but this notion essentially reduces to the notion of $Q_i$-Anosov representations above, because a representation is $P$-Anosov if it $Q_i$-Anosov for all maximal parabolic subgroups $Q_i$ which contain $P$.  For example, a representation $\rho: \Gamma \to \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})$ is $B$-Anosov, for $B<\mathrm{Sp}(2n,\mathbf{K})$ the minimal parabolic subgroups, if and only if it is $Q_i$-Anosov for every $i= 1,\cdots, n$.
\item
Under the embedding $\mathrm{Sp}(2n,\mathbf{R}) \to \mathrm{Sp}(2n,\mathbf{C})$, a $Q_i$-Anosov representation $\rho: \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{R})$ gives rise to a $Q_i$-Anosov representation $\rho_\mathbf{C}:  \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{C})$  into $\mathrm{Sp}(2n,\mathbf{C})$.
\end{enumerate}
\end{remark}

We recall, without proof several of the remarkable properties Anosov representations have.
\begin{proposition}\label{prop:properties_individual}\cite{Labourie_anosov, Guichard_Wienhard_DoD}
Let $\rho: \Gamma \to \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})$ be a $Q_i$-Anosov representation.
Then
\begin{enumerate}
\item $\rho$ has finite kernel and discrete image.
\item the map $\Gamma \rightarrow X_\mathbf{K}(n), \, \gamma \mapsto \gamma x_0$, for $x_0\in X_\mathbf{K}(n)$ fixed,  is a quasi-isometric embedding with respect to the word metric $|\cdot |_\Gamma$ on $\Gamma$ (with respect to a finite generating set) and the Riemannian distance $d_X$ on $X_\mathbf{K}$, i.e. there exists constants $L,l>0$ such that for all $\gamma \in \Gamma$
\[
\frac{1}{L} |\gamma|_\Gamma - l \leq  d_X(\gamma x_0, x_0)\leq L |\gamma|_\Gamma + l,
\]
where $x_0 \in X_\mathbf{K}$ is a fixed base point.
\item Every element of infinite order $\gamma \in \Gamma$ acts proximal on $\mathrm{Is}_i(V_\mathbf{K})$, i.e. there exists $F_\gamma^+$, $F_\gamma^-$ in $\mathrm{Is}_i(V_\mathbf{K})$ such that for every $F$ transverse to $F_\gamma^-$ the sequence
$\rho(\gamma)^n F$ converges to $F_\gamma^+$, and for every $F'$ transverse to $F_\gamma^+$ the sequence  $\rho(\gamma^{-n}) F'$ converges to $F_\gamma^-$. In fact, for a $Q_i$-Anosov representation we have $F_\gamma^\pm = \xi_i(\gamma^\pm)$.
\end{enumerate}
\end{proposition}

Also the set of Anosov representations inside the representation variety has remarkable properties.
\begin{proposition}\label{prop:properties_set}\cite{Labourie_anosov, Guichard_Wienhard_DoD, Canary_survey, GGKW_anosov}
\begin{enumerate}
\item
The set $\mathrm{Hom}_{Q_i-Anosov}(\Gamma, \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}))$  of $Q_i$-Anosov representation is open in $\mathrm{Hom}(\Gamma, \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}))$.
\item The  group $\mathrm{Out}(\Gamma)$ of outer automorphisms acts properly discontinuously on
$\mathrm{Hom}_{Q_i-Anosov}(\Gamma, \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}))/\mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})$, the set of conjugacy classes of Anosov representations.
\item The set of Anosov representations $\mathrm{Hom}_{Q_i-Anosov}(\Gamma, \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}))$ descends to a well defined subset of the character variety.
\end{enumerate}
\end{proposition}

\subsection{Examples}
In this subsection we describe several examples of Anosov representations

\subsubsection{Hitchin representations}
Let $\pi_1(\Sigma)$ be the fundamental group of a closed surface of genus $g\geq 2$, and let $\iota: \pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{R})$ be a discrete embedding. We denote by $\pi_{2n}: \mathrm{SL}(2,\mathbf{R})  \to \mathrm{Sp}(2n,\mathbf{R})$ the $2n$-dimensional irreducible representation of $\mathrm{SL}(2,\mathbf{R})$. It is unique up to conjugation in $\mathrm{Sp}(2n,\mathbf{R})$.  We call the composition \[\rho_{irr}: \pi_{2n} \circ \iota: \pi_1(\Sigma) \to Sp(2n,\mathbf{R})\] an irreducible Fuchsian representation. A {\em Hitchin representation} is any representation $\rho: \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{R})$ which can be continuously deformed to an irreducible Fuchsian representation. Hitchin representation were introduced by Hitchin in \cite{Hitchin}, who showed that there are $2\times2^{2g}$ connected components of Hitchin representations into $\mathrm{Sp}(2n,\mathbf{R})$ which are all homeomorphic to a ball of dimension $(2g-2)\dim{\mathrm{Sp}(2n,\mathbf{R})}$. The $2\times2^{2g}$ connected components come from the fact that there are $2^{2g}$ different ways of lifting a discrete embedding $\iota: \pi_1(\Sigma) \to \mathrm{PSL}(2,\mathbf{R})$ to $\mathrm{SL}(2,\mathbf{R})$ and $2$ distinct connected components of the space of discrete embeddings $ \pi_1(\Sigma) \to \mathrm{PSL}(2,\mathbf{R})$.

Hitchin representations were extensively studied by Labourie in \cite{Labourie_anosov}, who in particular showed that Hitchin representations are $Q_i$-Anosov for all $i= 1, \cdots, n$.

\subsubsection{Maximal representations}
The Toledo number is a characteristic number of a representation $\rho: \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{R})$. It satisfies $|T(\rho)|\leq n |\chi(\Sigma)|$, where $\chi(\Sigma)$ is the Euler characteristic of $\Sigma$. Representations for which $T(\rho) = n |\chi(\Sigma)|$ are called {\em maximal representations}. Maximal representations are $Q_n$-Anosov, in particular they are discrete embeddings \cite{Goldman_88, Burger_Iozzi_Wienhard_toledo, Burger_Iozzi_Labourie_Wienhard}.

The set of maximal representations is a union of connected components, which contains the set of Hitchin representations as a proper subset. By definition any Hitchin representation can be deformed to a representation that factors through $\mathrm{SL}(2,\mathbf{R})$. This does not hold for maximal representations. In fact the set of maximal representations into $\mathrm{Sp}(4,\mathbf{R})$ contains $2g-4$ connected components in which every representation is Zariski dense. We refer the reader to \cite{Gothen, GarciaPrada_Mundet, GarciaPrada_Gothen_MundetPubli, Guichard_Wienhard_InvaMaxi, Bradlow_GarciaPrada_Gothen_SP4} for more details on the number of connected components and topological invariants of maximal representations into the symplectic group.

Maximal representations are in general only $Q_n$, not $Q_1$-Anosov. Since some of the constructions we describe below are specifically for $Q_1$-Anosov representations, it is interesting to ask.
\begin{question}
Are there maximal representations $\rho: \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{R})$ outside of the Hitchin component that are $Q_1$-Anosov or even $B$-Anosov? What is the set of maximal representation that are also $Q_1$-Anosov?
\end{question}

\subsubsection{Deformations of $\mathrm{SL}(2,\mathbf{R})$ embeddings}
Examples of $Q_i$-Anosov representations that are not maximal (and in particular not Hitchin) can easily be constructed by considering other embeddings of $\mathrm{SL}(2,\mathbf{R})$.

For example, consider a symplectic vector space $(\mathbf{R}^2, \omega_0)$ and let $(V, \omega) = (\mathbf{R}^2 \oplus \mathbf{R}^2, \omega_0\oplus \epsilon \omega_0)$, where $\epsilon\in \{-1, +1\}$, and $\iota: \pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{R})$ a discrete embedding.
\begin{enumerate}
\item The representation $(\iota, 1): \pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{R}) \times \mathrm{SL}(2,\mathbf{R}) \subset \mathrm{Sp}(V, \omega)$ is $Q_1$-Anosov, but not maximal. Since the set of $Q_1$-Anosov representations is open, the same holds for small deformations of this representation.
\item For $\epsilon = -1$ the representation $(\iota, \iota): \pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{R}) \times \mathrm{SL}(2,\mathbf{R}) \subset \mathrm{Sp}(V, \omega)$ is $Q_n$-Anosov, but not maximal. They same holds true for small deformations. Note that for $\epsilon = 1$ the representation $(\iota, \iota): \pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{R}) \times \mathrm{SL}(2,\mathbf{R}) \subset \mathrm{Sp}(V, \omega)$ is maximal.
\end{enumerate}

\subsubsection{$\mathrm{SL}(2,\mathbf{C})$ embeddings and their deformations}
Consider a two-dimensional complex symplectic vector space $(V_0^\mathbf{C}, \omega_0^\mathbf{C})$. Then $\mathrm{Sp}(V_0^\mathbf{C}, \omega_0^\mathbf{C})  \cong \mathrm{SL}(2,\mathbf{C})$.
If we forget the complex structure, we can identify $V_0^\mathbf{C} = V$ with the real four-dimensional vector space $V$, and the real part of $\omega_0^\mathbf{C}$ defines a symplectic form $\omega$ on $V$.
This gives a natural embedding $\pi: \mathrm{SL}(2,\mathbf{C})  \to \mathrm{Sp}(4,\mathbf{R})$.

Composing a quasi-Fuchsian representation $\iota: \pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{C})$ with $\pi$ we get a $Q_n$-Anosov representation $\rho: \pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{C}) \to \mathrm{Sp}(4,\mathbf{R})$.

This embedding also gives interesting examples when $\Gamma$ is the fundamental group of a closed hyperbolic $3$-manifold, and $ \iota: \Gamma \to \mathrm{SL}(2,\mathbf{C})$ is the embedding of a cocompact lattice.
Then $\rho:=\pi \circ \iota: \Gamma \to \mathrm{Sp}(4,\mathbf{R})$ is a $Q_n$-Anosov representation. It was explained to me by Gye-Seon Lee, that under the local isomorphism between $\mathrm{SL}(2,\mathbf{C})$ and $\mathrm{SO}(1,3)$ , and between $\mathrm{Sp}(4,\mathbf{R})$ and $\mathrm{SO}(2,3)$, the above embedding $\mathrm{SL}(2,\mathbf{C}) \hookrightarrow \mathrm{Sp}(4,\mathbf{R})$ corresponds to the natural embedding $\mathrm{SO}(1,3)  \hookrightarrow \mathrm{SO}(2,3)$. It is then  a consequence of the work of Barbot \cite{BarbotDefAdSQF} that any deformation of $\rho$ is $Q_n$-Anosov.

\section{Domains of Discontinuity}
In this section we recall a construction of domains of discontinuity for $Q_i$-Anosov representations into $\mathrm{Sp}(2n,\mathbf{K})$, $i = 1$ or $n$,  which was introduced in \cite{Guichard_Wienhard_DoD}.
We denote by $\hat{}$ the nontrivial involution of the set $\{1,n\}$, i.e. $\hat{1} = n$ and $\hat{n}= 1$.

\subsection{Construction and statement}
Let $\rho: \Gamma \to \mathrm{Sp}(2n,\mathbf{K})$ be a $Q_i$-Anosov representation, $ i = 1$ or $n$ and $\xi_i: \partial_\infty \Gamma \to \mathrm{Is}_i(V_\mathbf{K})$ its continuous $\rho$-equivariant boundary map.
We are going to use $\xi_i$ to exhibit a set $\mathcal{K}_{\xi_i} \subset \mathrm{Is}_{\hat{i}} (V_\mathbf{K})$, such that the action of $\Gamma$  (via $\rho$)
 on the complement of $\mathcal{K}_{\xi_i} $ in $\mathrm{Is}_{\hat{i}} (V_\mathbf{K})$ is proper.

 Given an (isotropic) line $l \in \mathbf{P}(V_\mathbf{K})$, we set
  \[
 K_l:= \{ L\in \mathrm{Lag}(V_\mathbf{K}) \, |\, l \subset L\}.
 \]
 In words, for a line $l$, $K_l$ is the set of Lagrangian subspaces containing $l$. Note that we can identify $K_l$ with the set $ \mathrm{Lag} (l^{\perp_{\omega_\mathbf{K}}}/l)$.

 Given a Lagrangian subspace $L \in \mathrm{Lag}(V_\mathbf{K})$ we set
  \[
 K_L:= \{ l\in \mathbf{P}(V_\mathbf{K})\, |\, l \subset L\}.
 \]
 In words, for a Lagrangian subspace $L$, $K_L$ is the set of lines contained in $L$. Note that we can identify $K_L$ with $\mathbf{P} (L)$.

\begin{proposition}\cite[Proposition~8.1.]{Guichard_Wienhard_DoD}\label{thm:dod}
Let $\rho: \Gamma \to \mathrm{Sp}(2n,\mathbf{K})$ be a $Q_i$-Anosov representation, $ i =1$ or $n$ and $\xi_i: \partial_\infty \Gamma \to \mathrm{Is}_i(V_\mathbf{K})$ its continuous $\rho$-equivariant boundary map.
Set
\[\mathcal{K}_{\xi_i} := \bigcup_{t\in \partial_\infty \Gamma} K_{\xi_i(t)} \subset \mathrm{Is}_{\hat{i}} (V_\mathbf{K}).\]

The complement
\[\Omega_{\xi_i} = \mathrm{Is}_{\hat{i}} (V_\mathbf{K})\backslash\mathcal{K}_{\xi_i}\] is a $\Gamma$-invariant open subset. The action of $\Gamma$ (via $\rho$) on  $\Omega_{\xi_i}$ is properly discontinuous and cocompact.

\end{proposition}

\begin{remark}
\begin{enumerate}
\item Note that $\mathcal{K}_{\xi_i}$ is in fact the {\em disjoint} union of $ K_{\xi_i(t)}$, $t\in \partial_\infty \Gamma$.
\item
In general the domain of discontinuity $\Omega_{\xi_i}$ could be empty, and in fact this happens for example if $\Gamma = \pi_1(\Sigma)$ is a surface group, $n = 1$, and $\mathbf{K} = \mathbf{R}$, or if $\Gamma$ is the fundamental group of a closed hyperbolic $3$-manifold, $n= 1$, and $\mathbf{K} = \mathbf{C}$.
When $\Gamma$ is a free group, then $\Omega_{\xi_i}$ is always nonempty. In general,  $\Omega_{\xi_i}$ is nonempty, as soon as $n$ is bigger than the dimension of $\partial_\infty \Gamma$.
\end{enumerate}
\end{remark}

We refer the reader to \cite{Guichard_Wienhard_DoD} for a proof of Theorem~\ref{thm:dod}. Here we would like to illustrate how the construction of the domains of discontinuity can be used to deduce further information on proper actions on homogeneous space and compactifications of the corresponding quotient manifolds. More results in this direction are proved in \cite{Guichard_Wienhard_DoD,GGKW_anosov, GGKW_compactif}.

\subsection{Complexifying $Q_1$-Anosov representations}\label{sec:Q0}
Let $\rho: \Gamma \to \mathrm{Sp}(V, \omega)$ be a $Q_1$-Anosov representation with boundary map $\xi: \partial_\infty \Gamma \to \mathbf{P}(V)$. Let $(V_\mathbf{C}, \omega_\mathbf{C})$ be the complexification of $(V,\omega)$ and $\rho_\mathbf{C}: \Gamma \to \mathrm{Sp}(V_\mathbf{C}, \omega_\mathbf{C})$ the induced representation, which is also $Q_1$-Anosov. We denote by $\xi_\mathbf{C}: \partial_\infty \Gamma \to \mathbf{P}(V_\mathbf{C})$ its boundary map.
Since $\xi_\mathbf{C}$ is the complexification of $\xi$ we have
\begin{equation}\label{eq:cc}
\overline{\xi_\mathbf{C}(t)} = \xi_\mathbf{C}(t)
\end{equation} for all $t \in \partial_\infty \Gamma$.

By Theorem~\ref{thm:dod} we obtain a domain of discontinuity $\Omega_{\xi_\mathbf{C}} \subset \mathrm{Lag}(V_\mathbf{C})$ on which $\Gamma$ acts (via $\rho_\mathbf{C}$) properly discontinuous and cocompact.
Let us denote the compact complex quotient manifold $\Gamma \backslash \Omega_{\xi_\mathbf{C}}$ by $M_{\rho_\mathbf{C}}$.

\begin{question}\label{question:M}
What is the homeomorphism type of $M_{\rho_\mathbf{C}}$? What are the properties of $M_{\rho_\mathbf{C}}$ as complex manifold?
\end{question}

In order to describe the relation of $\Omega_{\xi_\mathbf{C}} $ with the $\mathrm{Sp}(V,\omega)$-orbits described above, let us analyze in more detail the structure of the set $\mathcal{K}_{\xi_\mathbf{C}}$. By definition
\[\mathcal{K}_{\xi_\mathbf{C}} = \bigcup_{t \in \partial_\infty\Gamma} K_{\xi_\mathbf{C}(t)} =  \bigcup_{t \in \partial_\infty\Gamma} \{ L \in  \mathrm{Lag}(V_\mathbf{C}) \, |\, \xi_\mathbf{C}(t) \subset L\}.
\]

\begin{lemma}\label{lem:R0}
The domain of discontinuity $\Omega_{\xi_\mathbf{C}} $ contains $\mathcal{R}_0$.
\end{lemma}

\begin{proof}
Let $L\in \mathcal{K}_{\xi_\mathbf{C}}$, then $L$ contains $\xi_\mathbf{C}(t)$ for some $t\in \partial_\infty \Gamma$. By \eqref{eq:cc}, this implies that $\xi_\mathbf{C}(t) \subset \overline{L} \cap L$, and in particular $L \notin \mathcal{R}_0$, hence $\mathcal{R}_0 \subset\Omega_{\xi_\mathbf{C}} $
\end{proof}

\begin{corollary}\label{cor:prop}
Let  $ \rho: \Gamma \to \mathrm{Sp}(V, \omega)$ be a $Q_1$-Anosov representation. Then $\Gamma$ acts (via $\rho$) properly discontinuous on $X_{p,q}$.
\end{corollary}

Analogously to Question~\ref{question:M} one can ask
\begin{question}\label{question:X}
What is the homeomorphism type of $N^{p,q}_\rho = \Gamma \backslash X_{p,q}$? What are the properties of $N^{p,q}_\rho$ as complex manifold?
\end{question}

Note that the non-compact manifolds $N^{p,q}_\rho = \Gamma \backslash X_{p,q}$ embed into $M_{\rho_\mathbf{C}}$, and since $M_{\rho_\mathbf{C}}$ is compact, the closure of $N^{p,q}_\rho$ in $M_{\rho_\mathbf{C}}$ provides a compactification of the locally (affine) symmetric manifold $\Gamma \backslash X_{p,q}$.

Let us describe in a bit more detail this compactification for the locally Riemannian symmetric space $\Gamma \backslash X_\mathbf{R}(n)$.  Since the compactification of $N_\rho = \Gamma \backslash X(n)$ obtained by taking the closure in $M_{\rho_\mathbf{C}}$ is the quotient of  $cl(H_{n,0}) \cap \Omega_{\xi_\mathbf{C}} $ by the action of $\Gamma$ (via $\rho$), we describe the set $cl(H_{n,0}) \cap \Omega_{\xi_\mathbf{C}} $.

The compactification of $X_\mathbf{R}(n)$ is given by
\[cl(H_{n,0}) = \bigcup_{n' \leq n} H^{i}_{n',0},\] where $i = n-n'$.
The set
$H^{i}_{n',0}$ is a fiber bundle over the space  $\mathrm{Is}_i(V)$ of $i$-dimensional isotropic subspaces of $V$ with fiber isomorphic to $X_\mathbf{R}(n')$.

Given $l \in \mathbf{P}(V)$ we set
\[S^i(l) = \{ Z \in \mathrm{Is}_i(V) \, |\, l \subset Z\}.\]

\begin{lemma}
The intersection $\mathcal{K}_{\xi_\mathbf{C}}\cap H^{i}_{n',0}$ is the restriction of the bundle $H^{i}_{n',0}$ to the subset $S^i_\xi := \bigcup_{t\in \partial_\infty \Gamma} S^i(\xi(t))$.
\end{lemma}
\begin{proof}
Consider a complex Lagrangian $W \in H^{i}_{n',0}$ and let $Z' \in\mathrm{Is}_i(V)$ be the real $i$-dimensional isotropic subspace such that $Z'_\mathbf{C} = W \cap \overline{W}$. Then $W$  is in $\mathcal{K}_{\xi_\mathbf{C}}$ if and only if there exists $t\in \partial_\infty \Gamma$  with $\xi_\mathbf{C}(t) \subset W\cap \overline{W} = Z'_\mathbf{C}$. This happens by \eqref{eq:cc} if and only if $\xi(t) \in Z'$, i.e. $Z' \in S^i(\xi(t))$.
\end{proof}

\subsubsection{Deforming real $Q_1$-Anosov representation}
In the previous section we analysed the structure of the domain of discontinuity of a $Q_1$-Anosov representation $\rho': \Gamma \to \mathrm{Sp}(V_\mathbf{C}, \omega_\mathbf{C})$ with boundary map $\xi': \partial_\infty \Gamma \to \mathbf{P}(V_\mathbf{C})$
in the special case, when $\rho' = \rho_\mathbf{C}$  is the complexification of a real $Q_1$-Anosov representation $\rho: \Gamma \to \mathrm{Sp}(2n,\mathbf{R})$. When we deform $\rho_\mathbf{C}$ to a $Q_1$-Anosov representation $\rho': \Gamma \to \mathrm{Sp}(V_\mathbf{C}, \omega_\mathbf{C})$ which does not preserve the real structure, the domains of discontinuity and the quotient manifolds
$M_{\rho_\mathbf{C}}$ and $M_{\rho'}$ are homeomorphic \cite[Theorem 9.12]{Guichard_Wienhard_DoD}, but $M_{\rho'}$ is not anymore invariant under complex conjugation, and there is no equivariant decomposition of $M_{\rho'}$ with respect to the real structure.

The problem of gaining a better understanding of $M_{\rho'}$ is of particular interest when $\Gamma = \pi_1(\Sigma)$ is a surface group.

In the case when $\mathrm{dim}(V) = 2$, $Q_1$-Anosov representations of $\pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{R}) \hookrightarrow \mathrm{SL}(2,\mathbf{C})$ are discrete embeddings,
i.e. a Fuchsian representations. The set $\mathcal{K}_{\xi_\mathbf{C}}$ we remove is $\mathbf{R}\mathbf{P}^1 \subset \mathbf{C}\mathbf{P}^1$, and the domain of discontinuity $\Omega_{\xi_\mathbf{C}} = \Omega^+ \cup \Omega^-$ is the union of the upper and the lower hemisphere. The  quotient manifold $M_{\rho_\mathbf{C}}$ is the union of two Riemann surfaces which are homeomorphic to $\Sigma$ and carry - up to an orientation reversing isometry the same conformal structure.
A  $Q_1$-Anosov representation $\rho':\Gamma \to \mathrm{SL}(2,\mathbf{C})$  is precisely a quasi-Fuchsian representation of $\pi_1(\Sigma)$. In this case $\mathcal{K}_{\xi_\mathbf{C}} \subset \mathbf{C}\mathbf{P}^1$ is a quasi-circle.
The quotient manifold $M_{\rho' }$ is still the union of two Riemann surfaces, now with different conformal structures.

In this case, $n=1$, the Ahlfors-Bers simultaneous uniformization theorem, gives a parametrisation of the space of $Q_1$-Anosov representations of $\pi_1(\Sigma) \to \mathrm{SL}(2,\mathbf{C})$ by two copies of the Teichm\"uller space of $\Sigma$.
It is an intriguing question to ask whether an analogue of the simultaneous uniformization theorem holds for deformations of Hitchin representations when $\mathrm{dim}(V) > 2$.
A question along these lines was first raised by Fock and Goncharov \cite[Section~11]{Fock_Goncharov}.

In order to make this question more precise we define

\begin{definition}\label{defi:QH}
The set of {\em Quasi-Hitchin representations} is the set of representation $\rho': \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{C})$ which are $B$-Anosov with respect to a full isotropic flag, and which can be deformed to a
representation \[\rho_\mathbf{C}: \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{R}) \hookrightarrow \mathrm{Sp}(2n,\mathbf{C}),\] which is the complexification of a Hitchin representation into $\mathrm{Sp}(2n,\mathbf{R})$.

The set of {\em $Q_1$-Quasi-Hitchin representations} is the set of representation $\rho': \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{C})$ which are $Q_1$-Anosov, and which can be deformed to a
representation $\rho_\mathbf{C}: \pi_1(\Sigma) \to \mathrm{Sp}(2n,\mathbf{R}) \hookrightarrow \mathrm{Sp}(2n,\mathbf{C})$, which is the complexification of a Hitchin representation into $\mathrm{Sp}(2n,\mathbf{R})$.
\end{definition}

\begin{question}
Is there a meaningful generalisation of the simultaneous uniformization theorem to Hitchin representations into $\mathrm{Sp}(2n,\mathbf{R})$?
Is there a parametrisation of the set of Quasi-Hitchin representations (or of the set of $Q_1$-Quasi-Hitchin representations) in terms of Hitchin components for $\mathrm{Sp}(2n,\mathbf{R})$?
\end{question}

Note that the domain of discontinuity $\Omega_{\xi_\mathbf{C}}$ always contains both the (upper) Siegel space $H_{n,0}$, as well as the (lower) Siegel space $H_{0,n}$,  which are mapped to each other under complex conjugation. When $n=1$ these are just the upper and the lower hemisphere. However, in the general case   $\Omega_{\xi_\mathbf{C}}$ is connected, and the various open quotient manifolds $\Gamma \backslash H_{p,q}$ are glued along their boundaries in $M_{\rho_\mathbf{C}}$.

\begin{question}
How can one exhibit deformations of these open quotient manifolds $\Gamma \backslash H_{p,q}$ in $M_{\rho'}$ for a complex deformation $\rho'$ of $\rho_\mathbf{C}$.
\end{question}

\subsection{Products}
Let us describe another example, where the construction of the domain of discontinuity for $Q_1$-Anosov representation leads to some interesting consequences. We refer to \cite{GGKW_anosov, GGKW_compactif} for more details and proofs.

Consider the symplectic vector space $(V_\mathbf{K}, \omega_\mathbf{K})$. Let \[(V', \omega') = (V_\mathbf{K} \oplus V_\mathbf{K}, \omega_\mathbf{K} \oplus -\omega_\mathbf{K}),\] and let
\[ \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}) \times \mathrm{Sp}(V_\mathbf{K}, -\omega_\mathbf{K})  \hookrightarrow  \mathrm{Sp}(V', \omega')\] be the corresponding embedding.

Assume that  $\rho_0 : \Gamma \to \mathrm{Sp}(V_\mathbf{K},\omega_\mathbf{K})$ is a $Q_1$-Anosov with boundary map $\xi_0: \partial_\infty \Gamma \to \mathbf{P}(V_\mathbf{K})$ and consider the representation

\[\rho = (\rho_0, 1) : \Gamma \to \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}) \times \mathrm{Sp}(V_\mathbf{K}, - \omega_\mathbf{K})  \hookrightarrow \mathrm{Sp}(V', \omega'),\]
where $1$ is the trivial representation. Then $\rho$ is $Q_1$-Anosov as a representation into $\mathrm{Sp}(V', \omega')$ with boundary map $\xi:\partial_\infty \Gamma \to \mathbf{P}(V')$ given by
\[t \mapsto \xi_0(t) \oplus 0 \in \mathbf{P}(V_\mathbf{K} \oplus 0) \subset \mathbf{P} (V').\]

So the above construction provides a domain of discontinuity $\Omega_\xi \subset \mathrm{Lag}(V')$, obtained as the complement of
\[
\mathcal{K}_\xi = \bigcup_{t\in \partial_\infty \Gamma} K_\xi(t)  = \bigcup_{t\in \partial_\infty \Gamma}\{ W \in \mathrm{Lag}(V')\, |\, \xi(t) \subset W\}.
\]

With respect to the symplectic splitting $V' = V_\mathbf{K} \oplus V_\mathbf{K}$, the space  $\mathrm{Lag}(V')$ decomposes into several $\mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}) \times \mathrm{Sp}(V_\mathbf{K}, - \omega_\mathbf{K})$.

The orbits are
\begin{equation}\label{eq:orbits}
\mathcal{T}_i := \{ W \in \mathrm{Lag}(V')\, |\, \mathrm{dim}(W \cap V_\mathbf{K}\oplus 0) = i\}.
\end{equation}
The open orbit is $\mathcal{T}_0$. It contains in particular the diagonal
\[\Delta(V_\mathbf{K}) = \{ (v,v) \in V'\, |\, v \in V_\mathbf{K}\},\]
whose $\mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}) \times \mathrm{Sp}(V_\mathbf{K}, - \omega_\mathbf{K})$-orbit identifies with
\[\mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}) \times \mathrm{Sp}(V_\mathbf{K}, - \omega_\mathbf{K})/ \Delta(\mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})) \cong \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}),\]
where $\Delta: \mathrm{Sp}(V_\mathbf{K}, \omega_KK) \to \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K}) \times \mathrm{Sp}(V_\mathbf{K}, - \omega_\mathbf{K})$ denotes the diagonal embedding.

\begin{proposition}
The domain of discontinuity $\Omega_\xi \subset \mathrm{Lag}(V')$ contains $\mathcal{T}_0$ as an open and dense submanifold. The quotient manifold $N_\rho= \Gamma \backslash \mathcal{T}_0$, which identifies with $\Gamma\backslash \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})$, embeds into $M_\rho = \Gamma\backslash \Omega_\xi$ as an open and dense submanifold. In particular, the manifold $M_\rho$ provides a compactification of $ N_\rho=\Gamma\backslash \mathrm{Sp}(V_\mathbf{K}, \omega_\mathbf{K})$.
\end{proposition}

\subsection{$Q_n$-Anosov representations}
Let $\rho: \Gamma \to \mathrm{Sp}(2n,\mathbf{R})$ be a $Q_n$-Anosov representation with boundary map $\xi: \partial_\infty \Gamma \to \mathrm{Lag}(\mathbf{R}^{2n})$.
Then the construction of the domain of discontinuity above provides a domain of discontinuity $\Omega_\xi \subset \mathbf{P}(\mathbf{R}^{2n})$. In particular, the quotient manifold
 $M_\rho: \Gamma \backslash \Omega_\xi$ is endowed with a projective structure. For Hitchin representations $\rho: \pi_1(\Sigma) \to \mathrm{Sp}(4,\mathbf{R})$ this domain of discontinuity was considered in detail in \cite{Guichard_Wienhard_Duke}.

It is however also of interest to consider subsets in $\mathrm{Lag}(\mathbf{R}^{2n})$ on which the image of a $Q_n$-Anosov representation acts properly.

Let us consider the 2n-dimensional symplectic vector space $(V_0, \omega_0)$ and set $(V, \omega) = (V_0 \oplus V_0, \omega_0 \oplus -\omega_0)$. Let
$\mathrm{Sp}(V_0, \omega_0) \times \mathrm{Sp}(V_0, -\omega_0) \hookrightarrow  \mathrm{Sp}(V, \omega)$ be the corresponding embedding.

Assume that  $\rho_1, \rho_2 : \Gamma \to \mathrm{Sp}(V_0,\omega_0)\cong \mathrm{Sp}(2n,\mathbf{R})$ are $Q_n$-Anosov representations,
 then the representation
\[\rho = (\rho_1, \rho_2): \Gamma \to \mathrm{Sp}(V_0,\omega_0)\times \mathrm{Sp}(V_0, -\omega_0) \hookrightarrow  \mathrm{Sp}(V, \omega)\cong \mathrm{Sp}(4n,\mathbf{R})\]
 is $Q_{2n}$-Anosov.
Let $\xi: \partial_\infty \Gamma \to \mathrm{Lag}(V)$ be the corresponding boundary map.
Instead of the domain of discontinuity $\Omega_\xi \subset \mathbf{P}(V)$ constructed above, let us consider the set $\mathcal{D}_\xi \subset \mathrm{Lag}(V)$ defined as follows.

For a Lagrangian $L \in \mathrm{Lag}(V)$  we consider \[N_L = \{ L' \in \mathrm{Lag}(V)\, |\, L' \cap L \neq \{ 0\}\},\] the set of all Lagrangians which are not transverse to $L$, and set
\[
\mathcal{N}_\xi = \bigcup_{t\in \partial_\infty \Gamma} N_\xi(t).
\]
Note that here $N_\xi(t)$ and $N_\xi(t')$ for $t \neq t'$ will in general intersect, and the union is {\em not} disjoint.

We set $\mathcal{D}_\xi $ to be the complement of $\mathcal{N}_\xi$ in $\mathrm{Lag}(V)$.
The action of $\rho$ on $\mathcal{D}_\xi $ is properly discontinuous, but will not be cocompact.

When $n=1$, then the Toledo number of the representation $\rho = (\rho_1, \rho_2): \Gamma \to \mathrm{Sp}(4,\mathbf{R})$ is zero.
In that case$\mathcal{T}_0 \cong \mathrm{SL}(2,\mathbf{R})$ can be identified with Anti-de Sitter space, and the set $\mathcal{D}_\xi \cap \mathcal{T}_0\subset \mathrm{Lag}(V)$ is described in detail in \cite{BarbotDefAdSQF}, where also  its relation to the domain of discontinuity in Anti-de Sitter space defined by Mess in \cite{Mess} is described.

\begin{question}
Develop a description of the set $\mathcal{D}_\xi \cap \mathcal{T}_0\subset \mathrm{Lag}(V)$ when $n\geq 2$.
\end{question}

\section{Some questions}
Let us mention a few other questions concerning Anosov representations.

\subsection{Connected components}
There are several examples known, where entire connected components of the space of homomorphisms $\mathrm{Hom}(\Gamma, G)$ consist of Anosov representations. In the case when $\Gamma$ is a surface group
this happens for example for the set of Hitchin representations or the set of maximal representations. But this phenomena also occurs when $\Gamma$ is the fundamental group of a closed hyperbolic manifold of dimension $n$. In this case connected component which exist entirely of Anosov representations have been described by Benoist \cite{Benoist_CD1} and by \cite{BarbotDefAdSQF, Barbot_Merigot_fusionPubli}.

\begin{question}
For which finitely generated word hyperbolic groups $\Gamma$ and for which non compact simple Lie groups $G$ are there entire connected components in $\mathrm{Hom}(\Gamma, G)$ which consist entirely of Anosov representations?
Is there a common underlying structure that makes this possible?
\end{question}

In some cases the set of Anosov representation consists of essentially one connected component, for example for $Q_1$-Anosov representations into $\mathrm{SL}(2,\mathbf{K})$, or for the Anosov representations considered in \cite{BarbotDefAdSQF}.
But in general the set of Anosov representations has several connected components, which contain representations of rather different geometric flavour.
\begin{question}
Investigate the set of connected components of the space of Anosov representations.
\end{question}

\subsection{Quasi-isometry invariance}
Recently, Haissinsky \cite{Haissinsky} proved that the class of convex cocompact Kleinian groups is quasi-isometrically rigid, i.e. if $\Gamma$ is a finitely generated group, which is quasi-isometric to a discrete convex cocompact subgroup $\Lambda < \mathrm{PSL}(2,\mathbf{C})$, then there exists a finite index subgroup $\Gamma'$ of $\Gamma$ and an isomorphism of $\Gamma'$ onto a discrete convex cocompact subgroup in $\mathrm{PSL}(2,\mathbf{C})$.

For $\mathrm{PSL}(2,\mathbf{C})$ and $\Gamma$ a finitely generated hyperbolic group, a representations $\rho: \Gamma \to \mathrm{PSL}(2,\mathbf{C})$ is Anosov if and only if $\rho$ has finite kernel and the image of $\rho$ is a discrete convex cocompact subgroup in $\mathrm{PSL}(2,\mathbf{C})$.
The following questions are thus a natural generalisation of Haissinsky's theorem
\begin{question}
Is the class of finitely generated groups that admit a $Q_i$-Anosov representation into $\mathrm{Sp}(2n,\mathbf{K})$ quasi-isometrically rigid?
Is the class of finitely generated groups that admit any Anosov representation into a non compact simple Lie group quasi-isometrically rigid?

\end{question}

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redpajama_set_name: RedPajamaArXiv


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Wash away all of your stresses and worries in a place that offers an experience that is both delightful and relaxing. Located in Koramangala, Bengaluru, Magical Touch Destress Spa offers customers a variety of spa services like head therapies, foot therapies, a range of full body massages and much more for both men & women. The therapists are highly skilled and they make sure the customers are de-stressed at the end of the session while the ambiance is pleasing and serene.


It was released on September 19, 2011 and produced by Inna's longtime producing team, Play & Win. It is organised by Pod Concerts and Festival Republic, who purchased the majority shareholding in 2009. In early 2010, the song charted in Romania at No.18 of the official Romanian Top 100. So Fresh: The Hits of Summer 2012 + The Best of 2011 is a two disc compilation that was released on 18 November 2011. The Radio România Actualităţi Awards 2011 was held on March 13, 2011 in the Sala Radio of the Romanian Radio Broadcasting Company. Ultra Music is an electronic dance music record label in North America, comprising Ultra Records, Ultra Music Publishing, and Empire Artist Management. Starting 21 August 2012, the name was changed to Décompte MusiquePlus. It is the third installment of the Dance Central series and was announced at E3 2012 during Microsoft's press conference. I Am the Club Rocker is the second studio album by Romanian recording artist Inna.
Although they lean towards R&B/Hip-Hop, Wired also adds Rhythmic Pop tracks to their playlist. The company has around 100 employees, having offices in Bucharest, Iaşi, Cluj and Timişoara. He has performed with such American acts as Xzibit, Strong Arm Steady and Trey Songz. Antonia Clara Iacobescu, commonly known as Antonia, is a Romanian singer, performer, and model. The following is a list of the MTV Europe Music Award winners and nominees for Best European Act. "Danza Kuduro" is a Spanish/Portuguese-language hit song from Don Omar's collaborative album Meet the Orphans released as the lead single from the album on August 15, 2010 through Machete Music and VI Music. Now That's What I Call Music! 17 was released on September 22, 2011. It features 41 songs from various artists that were either recent hits or hits throughout the year on the ARIA Charts.
It is a ninety-minute music video program block, that airs on French Canadian music television station MusiquePlus. Addiction is a song by Danish singer Medina from her international debut studio album Welcome to Medina. Season fifteen of Dancing with the Stars, called Dancing with the Stars: All-Stars, premiered on September 24, 2012. Saxobeat" is an up-tempo Eurodance song with house elements. All Around the World Productions Limited is a record label from Blackburn, United Kingdom. The 2011 MTV EMAs were held in Northern Ireland's capital Belfast, on Sunday, November 6, 2011, at the Odyssey Arena in the Titanic Quarter area of the city and were hosted by Selena Gomez. The Beirut Central District or Centre Ville is the name given to Beirut's historical and geographical core, the "vibrant financial, commercial, and administrative hub of the country." At the heart of Lebanon's capital, Beirut Central District is an. Tantsud tähtedega 2011 is the fifth season of the Estonian version of Dancing with the Stars, and was broadcast on the Estonian television channel Kanal 2 starting on 9 October 2011. The song was released on 12 June 2012 as the lead single from Clichè, the re-issue of her debut album Saxobeats.


text:
Tag: Cartilage sugars
Cartilage saccharides and bioactive compounds from sea cucumbers
Ólafur Friðjónsson, Varsha Kale, Jón Óskar Jónsson, Sesselja Ómarsdóttir, Hörður Kristinsson, Margrét Geirsdóttir, Patricia Y. Hamaguchi, Guðlaugur Sighvatsson, Sigfús Snorrason, Kári P. Ólafsson, Guðmundur Ó. Hreggviðsson
Tækniþróunarsjóður, AVS
Ólafur H. Friðjónsson
Research Group Leader
olafur@matis.is
In recent years, Matís, in collaboration with the Faculty of Pharmacy at the University of Iceland, IceProtein ehf and Reykofninn ehf, has conducted research on cartilage sugars (chondroitin sulphate) from shark cartilage and sea urchins from Icelandic smelters (Cucumaria frondosa). Studies around the world have shown a wide range of bioactivity of cartilage sugars, in vitro and in vivo, and such sugars are used as a dietary supplement, usually with glucosamine to treat osteoarthritis. Studies have also shown that cutting cartilage sugars into smaller units (oligosaccharides) can potentially increase their bioactivity in vitro. The research of Matís and partners, which was supported by the AVS Fisheries Research Fund and the Technology Development Fund, showed that cartilage sugars can be produced from shark cartilage and coarsely refined cartilage sugars from simple edema. It is also possible to produce oligosaccharides from shark cartilage with specific biocatalysts, which were prepared in the research project. The cartilage sugars show considerable bioactivity in vitro and cartilage sugars from Icelandic sea otters are particularly interesting as they show antioxidant activity, immune-regulating activity and hypoglycemic activity. The molecular structure of cartilage sugar from sea urchins is complex compared to shark cartilage sugars as they contain side chains composed of different types of sugars. The production of refined cartilage sugars from sea edema is therefore a complex process and it is anticipated that such sugars will be expensive on the market. Matís and IceProtein in collaboration with Reykofninn are now preparing further production of coarsely refined cartilage sugars from seaweed in sales and promotional activities.
In recent years, Matís ohf, The Faculty of Pharmaceutical Sciences, University of Iceland, IceProtein and Reykofninn ehf have collaborated in a research project on cartilage saccharides (chondroitin sulfate) isolated from shark and sea cucumbers from waters around Iceland (Cucumaria frondosa). The project results indicate that processing of the chondroitin sulfate from shark cartilage is a simple procedure and production of disaccharides with recombinant biocatalyst, evolved in the project, may be profitable. The chondroitin sulfate shows considerable bioactivity. Fractions of chondroitin sulfate purified from sea cucumbers, are especially interesting as they display immunomodulating activity and anti ‐ diabetic properties. However, the structure of the sea cucumber chondroitin sulfate is complex as they contain side chains composed of fucoside residues. Hence, the production and purification of chondroitin sulfate from Icelandic sea cucumbers will be a complicated procedure. Nevertheless, the results indicate that production of crude chondroitin sulfate from sea cucumber can be viable procedure.
Report closed until 01.07.2015
Tags Bioactivity, Cartilage sugars, Chondroitin sulfate, Enzymes, Low sugars, Shark, Catalysts, Bioactivity, Oligosaccharides, Sea edema, Sea cucumbers, Shark, Disaccharides

redpajama_set_name: RedPajamaCommonCrawl


text:
15th to 21st October 2018
NEXT WEEKS EVENTS
THE FOLLOWING IS AN EXTRACT FROM BRISTOL AND WEST COUNTRY GEOLOGY CALENDARS
MORE DETAILS CAN BE FOUND IN THE BRISTOL AND THE WEST COUNTRY CALENDARS AND ON THE WEB SITES OF THE RELEVANT SOCIETY OR ORGANISATION.
MENDIP HILLS AONB HAVE A SERIES OF EVENTS WHICH HAVE ONLY RECENTLY COME TO MY NOTICE. THEY CAN BE DISCOVERED HERE. I HAVE INSERTED SOME OF THEM BELOW. VISIT THE WEBSITE FOR BOOKING DETAILS.
MONDAY 15TH
Course - Geology of the Bristol and Gloucester Region.
Mon, 15 October, 19:30 – 21:30
Wynstones School, Stroud Road, Whaddon, Gloucester (map)
Geology of the Bristol and Gloucester Region.
One of the most varied regions of Britain, spanning the major divisions of the Variscan Front/Bristol Channel-Bray Fault, and the Malvern fault/ Worcester Graben, incorporating rocks varying in age from late PreCambrian to Cretaceous, and three of the terranes making up the geological "jigsaw" of Britain. The course will cover roughly the area covered by the eponymous BGS publication in the British Regional Geology series (3rd edition 1992). Starts Mon 17th September for 10 weeks (not 22nd or29th Oct or 5th Nov), until 10th December Held at Wynstones School, Stroud Road, Whaddon, Gloucester from 7.30-9.30pm on Mondays. Cost £75 (including tea, coffee etc at breaktime!).
Contact Dave Green by email at davegeostudies@gmail.com, by phone at 01594 960858 and by post at Dave Green, Joys Green Farm, Lydbrook, Gloucestershire, GL17 9QU
Tuesday 16th
Mendip Rocks Festival - Batts Combe Quarry Tour - A Limestone Legacy
Tue, 16 October, 10:00 – 12:00
Meet at Batts Combe Quarry office car park, Warrens Hill, Cheddar, BS27 3LR. (map)
Batts Combe Quarry Tour - A Limestone Legacy
A guided tour to with Hanson's UK to view inside this Burrington Oolite quarry, a mineral very valuable as a pure limestone for animal feed, toothpaste and many industrial fillers. Wear boots, long sleeves, no shorts, safety wear provided. Meet at Batts Combe Quarry office car park, Warrens Hill, Cheddar, BS27 3LR. Limited places, 10 per tour. Charge £5pp. Book am or pm tour.
This tour is repeated - same day 2 to 4PM
SMFS Evening Meeting
The Friends' Meeting House, 1A, Ordnance Rd, Southampton. SO15 2AZ (map)
Quartz by Gary Morse.
Members' Display Table: Quartz varieties.
Wednesday 17th
Bath Royal Literary & Scientific Institution - Lecture
Wed, 17 October, 19:30 – 21:00
16 Queen Square, Bath, BAI 2HN (map)
BRLSI Collection Talk
Charles Moore and his Fossils
Matt Williams, BRLSI Collections Manager
ln 1854 charles Moore joined the BRLS| and deposited his
already substantial collections of fossils to be exhibited ,for the
Public Benefit'. on his death in 1BgL, the BRLSr purchased
Moore's collection from his widow for more than f 1000. Now
recognised as internationally important, this unique collection
is still yielding new scientific discoveries and, now well cared
for; is sure to inspire a deeper appreciation of georogy and the
history of life in future generations.
Visitors welcome £4, Members/Students £2
Bristol Nats AGM and Lecture
Westbury on Trym Methodist Church Hall (map)
Dr Aaron Hunter will talk on "Lost Jurassic Sea Lillies". Please note that this will take place at Westbury on Trym Methodist Church Hall; not the University. It is hoped that Geology Section Members will make every effort to support this talk
Thursday 18th
Mendip Rocks Festival - Halecombe Quarry Geology Tour
Thu, 18 October, 10:00 – 12:00
Somerset Earth Science Centre, Moons Hill Quarry, Stoke St Michael, BA3 5JU (map)
Halecombe Quarry Geology Tour - Somerset Earth Science Centre, Moons Hill Quarry, Stoke St Michael, BA3 5JU.
A rare chance to see inside this working limestone quarry and explore the rock exposures with Dr Ian Brewer, Senior Geologist with Tarmac.
For 16yrs and over. Charge £5 pp. Wear boots, long sleeves, no shorts, safety wear provided. Limited places 10 per tour. Parking at SESC, for minibus tour. Please arrive 15 mins prior to start promptly.
Info: Gill Odolphie 01749 840156
Please Note that this tour is repeated - same day 2 to 4PM
Course - Understanding Geology and Scenery around Ross-on-Wye
Understanding Geology and Scenery around Ross-on-Wye
Starts September 21st (not 25th Oct or 8th Nov), finishes 6th December.
This 10 week course aims to introduce you to the ways in which the varied and beautiful landscape round our area has come into being. The effects of folds, faults and tectonic dip in exposing different rock types, and the past and present processes of uplift, weathering, erosion and deposition that have acted upon them, will be examined by reference to local scenery. Use will be made of local maps and cross-sections across them, both geological and topographic; together with specimens of local rocks. Contact Paul Mason on 01989 760399. Cost £60 Enrol before 15th September to ensure that the course runs.
Thornbury Geology Group, 7.30pm, The Chantry, Thornbury
Thornbury Geology Group, 7.30pm at The Chantry, Thornbury, and every 3rd Thursday in the month.
Field Course - The Geology of Northern Cyprus and NW Turkey
19 – 31 Oct 2018
Field Course: The Geology of Northern Cyprus and NW Turkey.
Dates will be for a week to 10 days trip around a period from 19th to 31st October (half term in Glos). Half the time will be spent in N Cyprus and half in Turkey, in the area around and to the south west of Istanbul. A wide variety of geology is exposed; from the Kyrenia Terrane of N Cyprus, spectacularly exposed in the narrow, and still rising, mountain range that forms the entire northernmost coast, but deeply buried beneath the younger sediments of the Mesoria Basin to the south. Jurassic and Cretaceous sediments were folded and metamorphosed during subduction in the late Cretaceous, then exhumed, eroded and covered by latest Cretaceous-Palaeogene sediments and volcanics, then by an Eocene melange; the whole being thrust and folded by Eocene collision with Troodos. Turkey was not a single entity until the Early Tertiary, when several continental fragments with independent Palaeozoic and Mesozoic geological histories were assembled during a complex sequence of events leading to the collision of Gondwana and Laurasia. We shall examine the Istanbul and Sakarya terranes, which were detached parts of Laurasia, undergoing Triassic subduction, collision and obduction/exhumation of ophiolites, including some of the best high pressure metamorphic rocks in the world.
Effects of the Variscan Orogeny in South West England and South Wales
Sat, 20 October, 14:30 – 14:30
Bournemouth Natural Science Society, 39 Christchurch Road, Bournemouth, Dorset, BH1 3NS, United Kingdom (map)
Lecture: Effects of the Variscan Orogeny in South West England and South Wales
Speaker: Alan Holiday
Alan is an ex geology lecturer, and Chairman of the Dorset's Important Geological Sites (DIGS) group and the Dorset Geologists' Association Group (DGAG).
Event date: 20th October 2018 at: 2.30pm
Sunday 21st
Explore Ebbor Gorge Family Fun Day
Sun, 21 October, 10:00 – 16:00
Ebbor Gorge National Nature Reserve, Ebbor Gorge National Trust Car Park, Somerset, United Kingdom (map)
Fun packed family day of sand pit 'fossil digs', story-telling, guided walks to gorge and caves and much more to reveal the reserve's unique geology and rare archaeological finds. With Mendip Hills AONB, SESC, The Mendip Society and Natural England. Family activities/rocks and fossils. Refreshments.
Ebbor Gorge National Nature Reserve, ST 520 484.
Car park on Priddy to Wookey road.
No need to book.
An Earth Science Week Event
Jurassic Coast Art Exhibition
Sun, 21 October, 10:00 – Wed, 24 October, 17:00
The Tithe Barn, The Tithe Barn, Symondsbury, Dorset, United Kingdom (map)
Colin Bentley returns with the second year of his three-year artistic collaboration with the Jurassic Coast Trust.
This exhibition will be held across four days in October (Sunday 21st – Wednesday 24th) at the beautiful Symondsbury Tithe Barn, just outside of Bridport, and is proudly supported by Symondsbury Estate.
Colin will be showcasing a stunning selection of landscapes drawn from the middle section of the Jurassic Coast World Heritage Site between Axmouth and the Isle of Portland, encompassing Lyme Regis, West Bay and Chesil Beach among others.
Colin's landscapes will be accompanied by another selection of Jurassic Coast Ambassadors' portraits.
Entry is free to the public and the exhibition will be accompanied by talks and other activities. For more information.
For private view and exhibition enquiries please contact: contact@colinbentley.com
Crack Lane Quarry conservation
Crack Lane Quarry, Crack Lane, Langton Matravers, Dorset (map)
An opportunity to study a geologically and historically important site.
In the morning we shall carry out conservation work on this exposure of Purbeck Limestone of the Lower Cretaceous. This is a shelly limestone (Broken Shell Limestone or Burr) used for building and the famous Purbeck Marble was extracted.
Lunch will be taken in Worth Matravers where Mr Charlie Newman will demonstrate how he recreates replica oilshale decorative armlets of the Bronze Age and Iron Age using a pole lathe and flint tools.
In the afternoon we shall walk from Renscombe Farm via the coastal footpath to view Chapman's Pool and the coast to the west. Then we shall continue to Emmets Hill and St Aldhelm's Head and return via St Aldhelm's Quarry. The walk is 3.5 km along footpaths with some steep sections. There is one section of about 50m (150ft) with steps.
This is an opportunity to help conserve an important site and have some exciting geology later.
The field trip leader, Alan Holiday, taught geography and geology in Weymouth. He is now retired but spends much of his time leading field trips! Over the years he has been involved in a range of OUGS activities as well as now being Chairman of the Dorset GA Group and Dorset RIGS Group.
Further details and Booking HERE
29th October to 11th November 2018
Down to Earth Extra - November
Festival of Geology
22nd to 28th October
Dead Sea Sinkholes
8th to 14th October 2018
Down to Earth extra October 2018
Mendips consultation


When Rick Massa agreed to join the Bob Hope Airport Police Department, the 60-year-old was obligated to work just two days a week.
But the 35-year LAPD veteran, 25 of them in Special Weapons And Tactics, ended up giving a great deal more.
After joining Chief Ed Skvarna's department two years ago, it didn't take long for the recipient of the Los Angeles Police Department Medal of Valor to start in with large-scale improvements to policy and training. He appeared before the Burbank-Glendale-Pasadena Airport Authority six months into his tenure with a stern warning: They were ill-equipped and unprepared to handle an active shooter.
Every officer was soon trained on the M-16 rifle, complete with a new policy, earning Massa the department's Officer of the Year award.
"Now we have a police department here that has the weapons and the training that if something were to happen, we would be able to respond," Skvarna said.
Massa's next request for Joint-SWAT Aircraft Training, which the authority also granted, would put Bob Hope Airport and its police department on the map with agencies across the West Coast, he said.
The session last year included about 90 officers from SWAT teams with the Los Angeles County Sheriff's Department, LAPD and FBI training on three aircraft donated by Southwest Airlines and JetBlue Airways.
This year, the exercise grew to 300 officers from more agencies.
"Bob Hope Airport has now become a training ground for tactical teams that are able to train on airplanes and get extensive training on airplanes," Massa said.
For his Officer of the Year award, Massa was flown to the national memorial for peace officers in Washington, D.C.
There, he was able to reconnect with fallen SWAT officers Randy Simmons, Robert J. Cottle, a Marine reserve and the first officer killed in combat in Afghanistan, and longtime LAPD Chief Daryl Gates, for whom he cooked his famous macaroni and cheese.
"With those three officers gone, it was very important that I was able to go back there," Massa said. "I was able to see Randy's name on the wall. They had put Gates' name on the wall. And I was able to go to R.J.'s grave site over at Arlington National Cemetery. This is a family. And it's because of that family that I got to say goodbye."
Massa was involved in dozens of high-profile incidents in Los Angeles. In 1997, he was one of the officers who responded to the North Hollywood shootout at Bank of America.
But what went unnoticed was his work as the unit's expert on high-risk hostage negotiations, said Burbank Police Capt. Mike Albanese, a retired SWAT commander.
"He was a very effective negotiator in hostage-taking incidents and with folks who were on the threshold of killing themselves," Albanese said.
Massa also got to know Wolfgang Puck's team of chefs by setting up security of the Governor's Ball for the Academy Awards.
"They asked me to come by and cook," Massa said. "And at first I said, 'No way.' It's very intimidating. You're talking five-star dining, and at that time I was a backyard burger flipper."
He took classes for a year, and when the chefs extended the next offer he agreed, whipping up 10 gallons of sauces in the back.
The next year, after receiving more formal training, he agreed to cook alongside seven executive chefs on the Oscar hotline, turning out 350 of Puck's classic baked potatoes in 45 minutes, each topped with crème fraiche and caviar.
He surprised everyone again after winning $25,000 on the Food Network's "Ultimate Recipe Showdown" with his gourmet macaroni and cheese.
"People often say, 'How do you do it,'" Massa said. "I say, from cop to cook, it's really no big deal."


Michelle Anne Holman was born and raised in Texarkana, Texas. Holman is an artist who creates large scale oil painted portraits that stylistically use irregular paint application mixed with controlled brushstrokes and refined areas.
Recently, Holman has become interested in researching and creating work about her vision. Visual Snow/Static is a rare eye condition where she sees colorful, moving, television-like static throughout her entire field of vision. She is currently creating work using tools and materials exploring how humans see, with the addition of this moving static.
She attended Oklahoma State University where she obtained a Bachelor of Fine Arts with an emphasis in painting in December of 2014. Holman graduated from School of Visual Arts in New York City and achieved her Masters of Fine Arts, May 2017. Michelle interned at ULAE on Long Island, NY and has exhibited works in New York City, New Jersey, Oklahoma, and Texas.


I am a professional nature photographer, specializing in photographing the amazing diversity of wildlife and landscapes of the Greater Yellowstone Ecosystem. I am a Jackson Hole native, having lived in the valley for over 40 years. Every day I go out, I strive to make compelling images, enjoy myself, appreciate the beauty that surrounds me, and behave respectfully in the field.
I haven't always been a photographer. In anticipation of the birth of my first child, I became interested in learning to use a camera to be able to make nice images of the baby. As I learned to use the camera and started trying to see things as a photographer, I began re-connecting with the beauty that I was surrounded by living in Jackson Hole. With a new vision, I enjoy now more than ever, the magical place that is my home (and that I had taken for granted for so many years of my life).
I was awarded the Grand Prize in the 2018 Audubon Photography Awards, and also I won the Audubon 2017 Professional Division. I currently have a photograph of a great gray owl in the Nature's Best Photography display at the Smithsonian Museum in Washington D.C. I have also had winning images recognized in several other contests including Birds As Art and Wyoming Wildlife Magazine. My photography has been published in Wyoming Wildlife Magazine, Audubon Magazine, Natures' Best Photography Magazine, 5280 magazine, Delta Sky Magazine, Explore Yellowstone, and numerous newspapers around the United States.


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text:
from django.conf.urls.defaults import *
import os
CURRENT_PATH = os.path.dirname(__file__)

urlpatterns = patterns('django.views',
    (r'^$', 'generic.simple.direct_to_template', {'template': 'yz_djs_demo.html', 'extra_context' : {
        'comments' : [
            {
                'comment_title': 'I really digg this tool',
                'author'       : 'Wisste Nicht',
                'comment_body' : 'Now I just which somebody else could feel the same way'
            },
            {
                'comment_title': 'Save bandwidth and server load by off loading template construction to client side',
                'author'       : 'Bu Zhidao',
                'comment_body' : 'One template, client and server sides, one parser to unit them all'
            }
        ],
            'server_or_client_side' : 'server'
    }}),
    (r'^js/(?P<path>.*)$', 'static.serve',{'document_root': CURRENT_PATH + '/generated_js'}),
)

redpajama_set_name: RedPajamaGithub


CARFAX One-Owner. Clean CARFAX. 2016 Nissan NV200 S FWD 2.0L 4-Cylinder DOHC 16V AM FM, CD Player, Bluetooth, Traction Control, ABS Brakes, Rear Defrost, Climate Control, Power Windows, Back Door Glass Defroster, Back Door Glass Package, Interior Rear View Mirror, Wheels: 15" Steel w/Center Caps.


text:
The Mystery of Hoboken's Domino's Pizza Will Never be Solved
Anyone who grew up in the New York City metro area knows one thing: Pizza. We may not all be doctors or lawyers, we may not be in sync with the rest of the country on a long list of issues and political questions, but we know good pizza because our ancestors invented good pizza, then perfected it. And we continue to taunt every other area of the world with our pizza superiority.
Which is why it was always kind of a mystery as to why there was a Domino's Pizza in town. We have Benny Tudino's, we have Grimaldi's, we have so many other fantastic pizza places in town, and Domino's is legendary for producing a pizza-shaped object that looks and smells like pizza but is certainly not in any way real pizza. When you base a national advertising campaign around how terrible your pizza was, you know you've got a problem. Although some folks seemed to think the new Domino's recipes were much improved, the local Domino's was still rocking a 1.5 Star average on Yelp! Which does nothing to explain its presence.[caption id="attachment_9917" align="aligncenter" width="298"]
1.5 Stars![/caption]While we can accept that some folks who aren't local maybe grew up on Domino's and thus think it's real pizza, or that some people simply have a guilty pleasure thing going on with it, the fact is we may never really solve the riddle, because Domino's is following Taco Truck out of town.
Mo' Condos Less Pizza
Domino's closed its doors without explanation in early 2017, and now we know why: 462 Newark Street is being redeveloped. The current one-story building will be torn down, and a five-story building with 4 apartments and a street-level retail space will be built in its place.No, we don't know if it's supposed to be a tiny Shake Shack.No one's going to chain themselves to that ugly white building for historical preservation or because of their devotion to Domino's pizza, so the redevelopment is a good thing overall. And if you're the big Domino's fan that's been keeping them in business here all these years, your closest Domino's is now about one mile away at 3682 JFK Boulevard in Jersey City. They don't deliver to Hoboken, but it's only a 15-minute drive.For the rest of us, the mystery of a Domino's in Hoboken may never be solved, but at least our access to real pizza is unchanged.So what's your guilty pleasure when it comes to food in Hoboken?


This recipe for healthier Buffalo Chicken Tenders with Blue Cheese Dip is finger-lickin' good and better for you than your average basket of wings.
These tangy, spicy buffalo chicken tenders are the perfect dish for setting out before a big game or at your next potluck.
You can drench these babies in your favorite hot wing sauce dip 'em in our creamy blue cheese sauce and easily forget they're only 158 calories per serving!
These tangy, spicy chicken tenders dipped in creamy blue cheese sauce are delicious enough to make you forget they're only 158 calories per serving.
Combine 1 Tbsp. hot sauce, vinegar, garlic powder, paprika, and cayenne (if desired) in a small bowl; mix well.
While chicken is baking, combine yogurt and cheese in a small bowl; mix well. Set aside.
Add veggies to make this a great snackional.
*Note on equivalents: Per serving, the amount of yogurt and cheese found in this recipe is not enough to count toward a day's red or blue allowance. As such, these ingredients (and calories) have been calculated into the tsp. equivalent.


text:
Newsroom Home News Stream Council Enhances Oversight of Administrative Office of the Courts
Council Enhances Oversight of Administrative Office of the Courts
Changes geared to improve transparency, accountability, and efficiency
SAN FRANCISCO—At its public meeting today, the Judicial Council adopted recommendations from its Executive and Planning Committee to terminate, maintain, or modify specific delegations of authority that the council has issued to the Administrative Director of the Courts or Administrative Office of the Courts (AOC) since 1998. The delegations represent the Judicial Council's authorization for the Administrative Director or AOC to act on the council's behalf.
The council's Executive and Planning Committee reviewed the delegations in conjunction with the council's restructuring directives to the AOC. Those directives stemmed from the final report of the Strategic Evaluation Committee (SEC), appointed by California Chief Justice Tani G. Cantil-Sakauye in March 2011 to conduct an in-depth review of the AOC with a view toward promoting transparency, accountability, and efficiency.
Placer County Assistant Presiding Judge Charles Wachob, who chaired the SEC and is now an advisory member of the Judicial Council, was one of the first council members to respond after hearing a detailed report from Justice Douglas Miller, the chair of the Executive and Planning Committee. "From an SEC perspective, this is fabulous," Judge Wachob said. "If you look at the whole picture, the mantra of the SEC was transparency, accountability, and efficiency. We're seeing those components of evaluation of the Administrative Office of the Courts, the Judicial Council, and the branch coming together as a whole. I think it's exciting and I'm encouraged."
Merced County Presiding Judge Brian McCabe, who also was a member of the SEC and is now an advisory member of the Judicial Council agreed. "I've talked to my counterparts from a number of states, and they are awed and frightened by the prospect of what our Chief Justice has done," he said. "This self-assessment effort is extraordinary, unique, and unparalleled anywhere in the United States. We are very proud of what has been done so far and are very pleased with the process."
The council agreed to terminate 54 delegations of authority and approved recommendations that 20 delegations be modified to update, strengthen, or reinforce the council's oversight and authority in areas such as finance and budgeting, courthouse construction and maintenance, and judicial branch education.
Other actions taken by the council:
AOC Restructuring: Center for Families, Children & the Courts: The council received an update on changes to the AOC Center for Families, Children & the Courts (CFCC). The changes are responsive to the council's restructuring directives to the AOC, which include increased oversight to ensure transparency, accountability, and efficiency in the AOC's operations and practices. This review and resulting changes will help ensure that CFCC continues to have an organizational structure and staff that enable it to provide its services to courts and court users.
Judicial Council Directives on AOC Restructuring: The council received an update and informational report on overall implementation of the restructuring directives for the AOC, as approved by the Judicial Council on August 31, 2012. As of a year later, 85 directives have been implemented.
Budget Allocations for Trial Court Facility Modifications and Planning: The council approved the report and recommendation of the Trial Court Facility Modification Advisory Committee to allocate the $50 million appropriated by the Legislature for trial court facility modifications in the fiscal year 2013–2014 budget. The funding will allow the branch to address emergency and critical projects that arise every day in court facilities. However, current funding will not be enough to prevent further deterioration of aging courthouses in need of upgrades to such critical components as HVAC systems, elevators, and roofs.
California's Language Access Plan: The council received an informational report from the Joint Working Group for California's Language Access Plan, which comprises members of both the council's Court Interpreters Advisory Panel and Access and Fairness Advisory Committee. The working group was established in June 2013 to create a comprehensive statewide language access plan for California's limited-English-proficient court users. The informational report provided an update on the working group's goals, timeline, and next steps in the development of the plan, anticipating a phased-in approach to increasing language access in the courts based on public and stakeholder input and available resources. The working group expects to propose a draft plan to the council at the council's June 2014 meeting. That proposed plan would be subject to further public and stakeholder comment and input, with a final plan expected for consideration by the council in December 2014.


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Blantyre is a city in the Shire Highlands, in southern Malawi. It adjoins the town of Limbe and is known for its British colonial architecture and nearby parks and reserves. This is Malawi's largest conurbation with a population of over half a million people. It is the centre of Malawi's industry and commerce with many businesses being based here. It is named after David Livingstone's birthplace in Scotland! Chileka International Airport is an airport located 16 km from Blantyre's city centre.


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Magister Wigbold († 1401 in Hamburg; auch Wygbold, Wycholt), auch "Magister der Sieben Künste" genannt, war ein Seeräuber und einer der Anführer der Vitalienbrüder. Zusammen mit Klaus Störtebeker, Hennig Wichmann, Klaus Scheld und Gödeke Michels, ebenfalls Anführer der Vitalienbrüder, machte er Ende des 14. Jahrhunderts die Nord- und Ostsee unsicher.

Der Name Wigbold kommt von wig (= Streit) und bold (= tapfer). Es ist nicht bekannt, ob Wigbold sein echter Name oder ein Spitzname war.

Leben
Über das Leben Wigbolds ist wenig bekannt und zeitgenössische Quellen fehlen aus seiner Lebenszeit. Um seine Person ranken sich zahlreiche Legenden, die größtenteils nicht historisch belegt sind.

Er soll schon früh im Kloster aufgenommen und dort in den unterschiedlichsten Wissensbereichen unterrichtet worden sein. Dann soll er die Hochschule zu Rostock besucht haben, wo er später als Magister der Weltweisheit tätig gewesen sein soll. Ludwig Bühnau schreibt hingegen, Wigbold habe (bei John Wyclif) in Oxford studiert. Bühnau nennt allerdings keine Quellen für seine Behauptung. In A.B. Emdens Biographical Register of the University of Oxford to A.D. 1500 ist Wigbolds Name nicht angegeben. Es ist möglich, dass er unter einem anderen Namen studiert hat oder dass alle Aufzeichnungen über seine Zeit in Oxford verloren gegangen sind. Heinz Gundlach behauptet, er habe die Lehren John Wyclifs und John Balls, die eine maßgebliche Rolle spielten in dem großen Bauernaufstand, der 1381 England erschütterte, verbreitet. Gundlach führt hierfür jedoch ebenfalls keine Quellen auf.

Er soll keine besonders imposante Erscheinung gewesen sein und wurde als "der listige Zwerg" oder "das teuflische Gehirn" beschrieben. Er soll einem bürgerlichen Gelehrten ähnlich gesehen haben, er sei lang und hager gewesen und habe ein dunkles Samtwams getragen.

Wigbold war nach der Darstellung in der Lübischen Chronik von Johannes Rufus (1406/30) ein gelehrter, geistlicher und weltlicher Meister ("mester an den seven kunsten", Magister der sieben [freien] Künste, das heißt, der sieben Buchgelehrsamkeiten). Er soll in Verbindung zu dem englischen Radikalreformer John Wyclif gestanden haben.

Er war angeblich nicht, wie Gödeke Michels oder Klaus Störtebeker, aktiv in die Kämpfe verwickelt und zog es vor, Verhandlungen zu führen, um somit wenige Verluste zu erleiden. Leonhard Wächter nennt Wigbold "Hauptmann Wigbold, einen rostockschen Magister Philosophiä ..., der seinen Stand auf dem Catheder gegen den auf dem Schiffskastell vertauscht hatte".

In der Literatur wird Wigbold den Hauptleuten der Likedeeler (Vitalienbrüder) zugerechnet (Unterfeldherr unter Michels) und gelegentlich als "Freund" Störtebekers bezeichnet. Gödeke Michels und Wigbold und neben ihnen Wichmann und Störtebeker werden in der Lübischen Chronik zum Jahre 1395 als die Häuptlinge jener Seeräuber genannt, die sich nach der Befreiung König Albrechts von Schweden in den Städten Rostock und Wismar nicht mehr sicher fühlten und deshalb anderswo Zuflucht suchten.

Die Norder Autorin Gudrun Anne Dekker stellt fest: Möglicherweise waren es die Schieringer und Vitalienbrüder Junker Johann Sissingh(a) van Groningae und Wigbold (vermutlich Geschlecht von Ewsum aus Oert?). Letzterer war höchstwahrscheinlich als "Magister artium" für die Erstellung der Gesetze der Groninger "Liekedeeler" nach denen der Fratres Devoti verantwortlich.

Die Vitalienbrüder oder (seit 1398) auch Likedeeler ("Gleichteiler"; weil sie ihre Beute gerecht verteilten) machten lange die Nordsee unsicher, wo sie von friesischen Häuptlingen unterstützt wurden, bis die Hanse zum Gegenschlag ausholte und die Gruppe um Störtebeker zerschlug. Michels und Wigbold entkamen zunächst nach Norwegen. Doch kurz nach dem Tod Störtebekers am 21. Oktober 1401 wurden auch Michels und Wigbold nebst etwa 200 Gefolgsleuten auf der Weser (und Jade) gefangen genommen und noch im gleichen Jahr in einer zweiten Hinrichtungswelle (insgesamt 80 Seeräuber) auf dem Grasbrook vor Hamburg hingerichtet.

Störtebekers Hinrichtung und die seiner Gesellen Hennig Wichmann, Magister Wigbold und Gödeke Michels 1401 sind nur durch zwei Zeilen in einer alten Chronik belegt. Die kurze Hamburgische Chronik von 1457 berichtet nämlich: "Anno 1402 ward Wichmann unde Störtebeker afgehouwen altohand na Feliciani. Anno 1403 ward Wikbolt unde Goedeke Michael afgehouwen."

Auf einem Flugblatt zum 300-jährigen Jubiläum der Gefangennahme, gedruckt bei Nicolaus Sauer, Hamburg 1701, heißt es (zu 1401): "Im selbigen Jahr sind abermahl 80. See-Räuber aufgebracht / deren Hauptleute waren Gödecke Micheel und Gottfried Wichold / promovirter Magister Artium, sie wurden gleichfals auf dem Brocke enthauptet / und ihre Köpffe auf Pfähle / zu den vorigen gestecket."

Im Historischen Taschenbuch von Friedrich von Raumer (1840) heißt es ergänzend: "Magister Wigbold hatte das Schicksal, alle seine Mitgesellen vor sich enthaupten zu sehen; er war der Letzte, den das Richtschwert traf."

Literatur

Wissenschaftliche Literatur und Sachbücher
 Matthias Blazek: Seeräuberei, Mord und Sühne – Eine 700-jährige Geschichte der Todesstrafe in Hamburg 1292–1949. ibidem, Stuttgart 2012, ISBN 978-3-8382-0457-4.
 Ortwin Pelc: Seeräuber auf Nord- und Ostsee: Wirklichkeit und Mythos. Boyens, Heide, 2005, ISBN 3804211704.
 Matthias Puhle: Die Vitalienbrüder: Klaus Störtebeker und die Seeräuber der Hansezeit. Campus Verlag, Frankfurt, 2012, ISBN 9783593398013.

Belletristik
 Thomas Einfeldt: Störtebekers Kinder. Ravensburger Buchverlag, Ravensburg 2002, ISBN 3-473-58200-X.
 Gustav Schalk: Klaus Störtebeker. Ueberreuter, Wien 2002, ISBN 3-8000-2876-X.
 Indrek Hargla: Apotheker Melchior und das Rätsel der Olaikirche. Ein Hansekrimi. Aus dem Estnischen übersetzt von Uta Kührt. Europäische Verlagsanstalt, Hamburg 2014, ISBN 978-3-863-93056-1

Einzelnachweise

Pirat
Hingerichtete Person (Heiliges Römisches Reich)
Hingerichtete Person (15. Jahrhundert)
Vitalienbrüder
Geboren im 14. Jahrhundert
Gestorben 1401
Mann

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