text:
\section{Introduction, and summary of results}

The main purpose of this article
is to study aspects of `locality' for the connective constants of
Cayley graphs of finitely presented groups. The locality
question may be posed as follows: if
two Cayley graphs are locally isomorphic in the
sense that they agree on a large ball centred at the identity,
then are their connective constants close in value?
The current work may be viewed as a continuation of
the study of locality for connective constants of quasi-transitive graphs
reported in \cite{GL-loc}.
The locality of critical points is a well developed topic
in the theory of disordered systems, and the reader is referred,
for example,  to
\cite{bnp,mt,psn} for related work about percolation on Cayley graphs.

The self-avoiding walk (SAW) problem was introduced to
mathematicians in 1954 by Hammersley and Morton \cite{hm}.
Let $G$ be an infinite, connected, transitive graph.
The number of $n$-step SAWs on $G$
from a given origin grows in the manner of
$\mu^{n(1+\o(1))}$ for some growth rate $\mu=\mu(G)$ called the
\emph{connective constant} of the graph $G$.
The value of $\mu(G)$ is not generally known,
and a substantial part of
the literature on SAWs is targeted at
properties of connective constants. The
current paper may be viewed in this light, as a continuation of the
series of papers   \cite{GrLrev, GrL2, GrL1,GL-loc, GrL3}.

The principal result of \cite{GL-loc} is as follows.
Let $G$, $G'$ be infinite, transitive graphs,
and write $S_K(v,G)$ for the $K$-ball around the vertex $v$ in $G$.
If $S_K(v,G)$ and $S_K(v',G')$
are isomorphic as rooted graphs, then
\begin{equation}\label{eq:loc1}
|\mu(G)-\mu(G')| \le \eps_K(G),
\end{equation}
where $\eps_K(G)\to 0$ as $K \to\oo$.
This is proved subject to a
condition on  $G$ and  $G'$, namely
that they support so-called \lq \ughf s'.

Cayley graphs of finitely generated groups  provide a category
of transitive graphs of special interest.
They possess an algebraic structure in addition to
their graphical structure, and this algebraic structure provides
a mechanism for the study of their graph height functions.
A necessary and sufficient condition is
given in Theorem \ref{thm3} for the existence of a so-called
\lq \GHF', and it is pointed out that a \GHF\ is a \ughf, but not \emph{vice versa}.
The class of Cayley groups that possess  \GHF s
includes all infinite, finitely generated, free solvable groups and free nilpotent groups,
and groups with fewer relators than generators.

We turn briefly to the topic of harmonic functions.
The study of the existence and structure of non-constant harmonic functions on Cayley graphs has
acquired prominence in geometric group theory through the work of
Kleiner  and others, see \cite{kle10,st}.  The group height functions
of Section \ref{sec:Cay} are harmonic with linear growth.
Thus, one aspect of the work reported in this paper is the construction,
on certain classes of Cayley graphs, of
linear-growth harmonic functions with the additional property of
having differences that are invariant under the action of a subgroup
of automorphisms. Such harmonic functions do not appear to contribute
to the discussion of the Liouville property (see, for example, \cite[Defn 2.1.10] {Kum}),
since both their positive and negative parts are unbounded.
For recent articles on the related aspect of geometric group theory,
the reader is referred to \cite{MY14,T14}.

This paper is organized as follows.
Graphs, self-avoiding walks, and Cayley graphs are introduced in Section \ref{sec:not}.
Graph height functions and the locality theorem of \cite{GL-loc} are reviewed
in Section \ref{sec:sec2}, and a principal tool is presented at Theorem \ref{prop:indep}.
Graphs with non-unimodular automorphism subgroups may be handled by similar means
(see Theorem \ref{cor:nonunim}), but the resulting \ghf s need not
be harmonic.

Group height functions are the subject of Section \ref{sec:Cay}, and a necessary and
sufficient condition is presented in Theorem \ref{thm3} for the existence of a \GHF.
Section \ref{sce} is devoted to existence conditions for height functions,
leading to existence theorems for virtually solvable groups.
In Section \ref{sec:conv} is presented a theorem
for the convergence of connective constants subject to the addition of further relators.
This parallels the Grimmett--Marstrand theorem \cite{gm} for the critical percolation probabilities of
slabs of $\ZZ^d$ (see also \cite[Thm 5.2]{GrL3}).
Sections \ref{sec:pf-}--\ref{sec:nonunim} contain the proofs of Theorems
\ref{prop:indep} and \ref{cor:nonunim}, \resp.

\section{Graphs, self-avoiding walks, and groups}\label{sec:not}

The graphs $G=(V,E)$ considered here are
infinite, connected, and usually simple.
An undirected edge $e$ with endpoints $u$, $v$ is
written as $e=\langle u,v \rangle$, and if directed from $u$
to $v$ as $[u,v\rangle$.
If $\langle u,v \rangle \in E$, we call $u$ and $v$ \emph{adjacent}
and write $u \sim v$. The set of neighbours of $v \in V$
is denoted $\pd v:=\{u: u \sim v\}$.

The \emph{degree} $\deg(v)$ of vertex $v$ is the number of edges
incident to $v$, and $G$ is called \emph{locally finite} if
every vertex-degree is finite.
The \emph{graph-distance} between two vertices $u$, $v$ is the number of edges
in the shortest path from $u$ to $v$, denoted $d_G(u,v)$.

The automorphism group of the graph $G=(V,E)$ is
denoted $\Aut(G)$. A subgroup $\Ga \le \Aut(G)$ is said to
\emph{act transitively} on $G$
if, for $v,w\in V$, there exists $\g \in \Ga$ with $\g v=w$.
It is said to  \emph{act quasi-transitively} if there is a finite
set $W$ of vertices
such that, for $v \in V$, there exist
$w \in W$ and $\g \in \Ga$ with $\g v =w$.
The graph is called (\emph{vertex}-)\emph{transitive}
(\resp, \emph{quasi-transitive}) if $\Aut(G)$ acts transitively
(\resp, quasi-transitively).
For $\Ga \le \Aut(G)$ and a vertex $v \in V$, the orbit of $v$ under $\Ga$ is written
$\Ga v$.

An (\emph{$n$-step}) \emph{walk} $w$ on $G$ is
an alternating sequence $(w_0,e_0,w_1,e_1,\dots, e_{n-1}, w_n)$,
where $n \ge 0$,  of vertices $w_i$
and edges $e_i=\langle w_i, w_{i+1}\rangle$, and its \emph{length} $|w|$  is the number of its edges.
The walk $w$ is called \emph{closed} if $w_0=w_n$.
A \emph{cycle} is a closed walk $w$ satisfying $n \ge 3$ and
$w_i\ne w_j$ for $1 \le i < j \le n$.

An (\emph{$n$-step}) \emph{self-avoiding walk} (SAW)
on $G$ is  a walk containing $n$ edges
no vertex of which appears more than once.
Let $\Si_n(v)$ be the set of $n$-step SAWs starting at $v$, with
cardinality $\si_n(v):=|\Si_n(v)|$.
\emph{Assume that $G$ is transitive},
and select a vertex of $G$ which we call the \emph{identity}
or \emph{origin}, denoted $\id=\id_G$, and let
$\si_n=\si_n(\id)$.
It is standard (see \cite{hm,ms}) that
\begin{equation}\label{eq:sisub}
\si_{m+n} \le \si_m\si_n,
\end{equation}
whence, by the subadditive limit theorem,  the \emph{connective constant}
$$
\mu=\mu(G) :=\lim_{n\to\oo} \si_n^{1/n}
$$
exists. See \cite{bdgs,ms} for recent accounts of the theory of SAWs.

We turn now to finitely generated groups and their Cayley graphs.
Let $\Ga$ be a group with
generator set $S$ satisfying $|S|<\oo$
and $\id\notin S$, where
$\id=\id_\Ga$ is the identity element.
We write $\Ga =\langle S \mid R\rangle$ with $R$ a set of relators,
and we adopt the following convention for the inverses of generators.
For the sake of concreteness, we consider $S$ as a set of symbols,
and any information concerning inverses is encoded in the relator set;
it will always be the case that, using this information, we may identify the inverse
of $s \in S$ as another generator $s'\in S$. For example,
the free abelian group of rank $2$ has presentation $\langle x,y,X,Y\mid xX, yY, xyXY\rangle$,
and the infinite dihedral group $\langle s_1,s_2\mid s_1^2,s_2^2\rangle$.
Such a group is called \emph{finitely generated} (in that $|S|<\oo$), and \emph{finitely presented}
if, in addition,  $|R|<\oo$.

The \emph{Cayley graph} of $\Ga=\langle S \mid R\rangle$ is the simple graph $G=G(S,R)$
with vertex-set $\Ga$, and an (undirected) edge $\langle \g_1,\g_2\rangle$ if and only
if $\g_2=\g_1s$ for some $s\in S$.
Further properties of Cayley graphs are presented as needed in Section \ref{sec:Cay}.
See \cite{bab95} for an account of Cayley graphs, and \cite{LyP} for a short account.
The books \cite{dlH} and \cite{LedW,Rob} are devoted
to geometric group theory, and general group theory,
\resp.

The set of integers is written $\ZZ$, the natural numbers as
$\NN$, and the rationals as $\QQ$.

\section{Graph height functions}\label{sec:sec2}

We recall from \cite{GL-loc} the definition
of a \ghf\ for a transitive graph, and then we review the locality theorem (the
proof of which may be found in \cite{GL-loc}).
This is followed by Theorem \ref{prop:indep}, which is
one of the main tools of this work.

Let $\sG$ be the set of all infinite, connected, transitive, locally finite, simple graphs,
and let $G=(V,E)\in\sG$.
Let $\sH$ be a subgroup of $\Aut(G)$.
A function $F:V \to \RR$ is said to be \emph{\hdi} if
\begin{equation}\label{eq:hdi2}
F(v)-F(w)=F(\g v)-F(\g w), \qq v,w\in V,\ \g\in \sH.
\end{equation}

\begin{definition} \label{def:height}

A \emph{graph height function} on $G$ is a pair $(h,\sH)$, where
$\sH\le \Aut(G)$ acts quasi-transitively on $G$ and  $h:V \to \ZZ$, such that
\begin{letlist}
\item $h(\id)=0$,
\item $h$ is \hdi,
\item for  $v\in V$,
there exist $u,w \in \pd v$ such that
$h(u) < h(v) < h(w)$.
\end{letlist}
The \ghf\ $(h,\sH)$ is called \emph{unimodular} if
$\sH$ is unimodular.
\end{definition}

We remind the reader of the definition of the unimodularity
of a subgroup $\sH \le \Aut(G)$.
The ($\sH$-)\emph{stabilizer}
 $\Stab_v$ ($=\Stab_v^\sH$) of a vertex $v$ is the set of all $\g\in \sH$ for which $\g v = v$.
As shown in \cite{Trof} (see also \cite{BLPS,LyP,SoW}), when viewed as a topological group
with the  topology of pointwise convergence, $\sH$
is unimodular  if and only if
\begin{equation}\label{g804}
|\Stab_u v| = |\Stab_v u|, \qquad v\in V,\ u \in \sH v.
\end{equation}
We follow
\cite[Chap.\ 8]{LyP} by \emph{defining} $\sH$ to be \emph{unimodular} (on $G$)  if
\eqref{g804} holds.

We sometimes omit the reference to $\sH$ and refer to such $h$ as a
\ghf. In Section \ref{sec:Cay} is defined the related concept of a \emph{\GHF}
for the Cayley graph of a finitely presented group. We shall see that every \GHF\ is
a graph height function, but not \emph{vice versa}.

\begin{remark}[Linear growth]
A \ghf\ $(h,\sH)$ on $G$ has linear growth in that
$h(\gamma^n \id) = nh(\g \id)$ for $\g\in\sH$.
\end{remark}

Associated with the \ghf\ $(h,\sH)$ are two integers $d$, $r$  given as follows.
We set
\begin{equation}\label{eq:defd}
d=d(h)=\max\bigl\{|h(u)-h(v)|: u,v\in V,\ u \sim v\bigr\}.
\end{equation}
If $\sH$ acts transitively, we set
$r=0$. Assume $\sH$ does not act transitively, and
let $r=r(h,\sH)$ be the least integer $r$ such that the following holds.
For $u,v \in V$ in different orbits of $\sH$, there
exists $v' \in \sH v$
such that $h(u)<h(v')$, and a SAW $\nu(u,v')$ from $u$
to $v'$, with length $r$ or less, all of whose vertices $x$, other than
its endvertices,  satisfy $h(u)<h(x)< h(v')$. We recall \cite[Prop.\ 3.2]{GL-loc}
where it is proved, \emph{inter alia}, that $r<\oo$.

We state next the locality theorem for transitive graphs.
The \emph{ball} $S_k=S_k(G)$, with centre $\id=\id_G$ and radius $k$, is
the subgraph of $G$ induced by the set of its vertices within
graph-distance $k$ of $\id$. For $G,G'\in \sG$, we write
$S_k(G) \simeq S_k(G')$ if there exists a  graph-isomorphism from $S_k(G)$ to
$S_k(G')$ that maps $\id_G$ to $\id_{G'}$, and we let
$$
K(G,G') = \max\bigl\{k: S_k(G) \simeq S_k(G')\bigr\}, \qq G,G' \in \sG.
$$
For $D\ge 1$ and $R \ge 0$, let $\sG_{D,R}$ be the set of
all $G\in\sG$ which possess a \ughf\
$h$ satisfying $d(h)\le D$ and $r(h,\sH)\le R$.

For $G \in \sG$ with a given \ughf\ $(h,\sH)$, there is a subset of SAWs called \emph{bridges}
which are useful in the study of the geometry of SAWs on $G$.
The SAW $\pi=(\pi_0,\pi_1,\dots,\pi_n)\in \Si_n(v)$ is called a \emph{bridge} if
\begin{equation}\label{eq:bridge}
h(\pi_0)<h(\pi_i) \leq h(\pi_n), \qq 1\leq i\leq n,
\end{equation}
and the total number of such bridges is denoted $b_n(v)$.
It is easily seen (as in \cite{HW62})  that $b_n:=b_n(\id)$ satisfies
\begin{equation}\label{eq:b-subadditive}
b_{m+n} \ge b_m b_n,
\end{equation}
from which we deduce the existence of the \emph{bridge constant}
\begin{equation}\label{eq:bexists}
\be  = \be(G) = \lim_{n\to\oo} b_n^{1/n}.
\end{equation}
The definition of $\be$ depends on the choice of height function,
but it turns out that, under reasonable conditions, its value does not.

\begin{theorem}[Bridges and locality for transitive graphs, \cite{GL-loc}]\label{thm2}
\mbox{}
\begin{letlist}
\item
If $G \in \sG$ supports a \ughf\ $(h,\sH)$, then
$\be(G)= \mu(G)$.
\item
Let $D \ge 1$, $R \ge 0$,
and let $G \in \sG$ and $G_m \in \sG_{D,R}$ for $m \ge 1$ be such that
$K(G,G_m)  \to \oo$ as $m \to\oo$.
Then $\mu(G_m) \to \mu(G)$.
\end{letlist}
\end{theorem}

The main thrust of the current paper is to identify classes of finitely generated groups
whose Cayley graphs support \ghf s, and one of our main tools is
the following theorem, of which the proof is given in Section \ref{sec:pf-}.

\begin{theorem}\label{prop:indep}
Let $G=(V,E)\in \sG$.
Suppose there exist
\begin{letlist}
\item  a subgroup $\Ga\le \Aut(G)$ acting
transitively on $V$,
\item  a normal subgroup $\sH\normal\Ga$ satisfying
$[\Ga:\sH] < \oo$,  which is unimodular,
\item  a function $F:\sH\id \to \ZZ$ that is \hdi\ and non-constant.
\end{letlist}
Then,
\begin{romlist}
\item there exists a unique harmonic, \hdi\ function $\psi$ on $G$ that agrees with $F$ on $\sH\id$.
\item there exists  a harmonic, \hdi\ function $\psi'$ that increases
everywhere, in that every $v \in V$ has
neighbours $u$, $w$ such that $\psi'(u)<\psi'(v)<\psi'(w)$,
\item the function $\psi$ of part (i) takes rational values, and
the $\psi'$ of part (ii) may be taken to be rational also; therefore, there exists a harmonic,
\ughf\ of the form $(h, \sH)$.
\end{romlist}
\end{theorem}

The first part of condition (c) is to be interpreted as saying that \eqref{eq:hdi2} holds
for $v,w\in \sH\id$ and $\g \in \sH$.
Since $G$ is transitive, the choice of origin $\id$ is arbitrary, and hence
the orbit $\sH\id$ may be replaced by any orbit of $\sH$.

One application of Theorem \ref{prop:indep}, or more precisely
of its method of proof, is the proof of the existence of \ghf s for graphs
with quasi-transitive, non-unimodular automorphism subgroups.
See Section \ref{sec:nonunim} for the proof of the following.

\begin{theorem}\label{cor:nonunim}
Let $G=(V,E)\in\sG$.
Suppose there exist a subgroup $\Ga\le \Aut(G)$ acting
transitively on $V$, and a normal subgroup $\sH\normal\Ga$ satisfying
$[\Ga:\sH] < \oo$,   such that $\sH$ is  non-unimodular.
Then $G$ has a \ghf\ $(h, \sH)$, which is not generally harmonic.
\end{theorem}

The proofs of Theorems \ref{prop:indep} and
\ref{cor:nonunim} are inspired in part by the proofs of
\cite[Sect.\ 3]{LeeP} where, \emph{inter alia}, it is explained
that some graphs support harmonic maps, taking values in a function space, with a
property of equivariance in norm. In this paper,  we study \hdi, integer-valued harmonic functions.

\section{Group height functions}\label{sec:Cay}

We consider  Cayley graphs of
finitely generated groups next, and a type of graph height function
called a \lq \GHF'.
Let $\Ga$ be a finitely generated group with presentation $\langle S\mid R\rangle$,
as in Section \ref{sec:not}. A \GHF\ on a Cayley graph $G$ of $\Ga$ may in fact be defined
as a function on the group $\Ga$ itself, but it will be convenient that it acts
on the same domain as a \ghf\ (that is, on $G$ rather than on $\Ga$).
When viewed as a function on the group $\Ga$,
a \GHF\ is essentially a surjective homomorphism to $\ZZ$, and such functions are
of importance in group theory (see Remark \ref{rem:betti}).

Each relator $\rho\in R$ is a
word of the form $\rho = t_1t_2\cdots t_r$ with $t_i \in S$ and $r\ge 1$, and we define the vector
$u(\rho) = (u_s(\rho): s\in S)$ by
$$
u_s(\rho) = |\{i: t_i = s\}| , \qq s\in S.
$$
Let $C$ be the $|R| \times |S|$ matrix with row vectors $u(\rho)$, $\rho \in R$,
called the \emph{coefficient matrix} of the presentation $\langle S\mid R \rangle$.
Its null space $\sN(C)$ is the set of column vectors $\g=(\g_s: s\in S)$ such that $C \g=\bzero$.
Since $C$ has integer entries, $\sN(C)$ is non-trivial if and only if it contains
a non-zero vector of integers (that is, an integer vector
other than the zero vector $\bzero$).
If $\g\in\ZZ^S$ is a non-zero element of $\sN(C)$, then $\g$ gives rise to
a function $h:V\to\ZZ$ defined as follows. Any $v \in V$ may be expressed as a word in the
alphabet $S$, which is to say that $v = s_1 s_2 \cdots s_m$ for some $s_i \in S$ and $m\ge 0$. We set
\begin{equation}\label{eq:gheight}
h(v) = \sum_{i=1}^m \g_{s_i}.
\end{equation}
Any function $h$ arising in this way is called a \emph{\GHF} of the presentation (or of the Cayley graph).
We see next that a \GHF\ is well defined by \eqref{eq:gheight},
and is indeed a graph height function in the sense of Definition
\ref{def:height}. A graph height function, even if unimodular,
need not be a \GHF\ (see, for
instance, Example (d) following Remark \ref{rem:betti}).

\begin{theorem}\label{thm3}
Let $G$ be the Cayley graph of the finitely generated group
$\Ga=\langle S\mid R\rangle$, with
coefficient matrix $C$.
\begin{letlist}
\item
Let $\g=(\g_s: s\in S)\in\sN(C)$ satisfy $\g\in\ZZ^S$, $\g \ne \bzero$.
The \GHF\ $h$ given by \eqref{eq:gheight} is well defined,
and gives rise to a \ughf\ $(h,\Ga)$ on $G$.
\item
The Cayley graph $G(S,R)$ of the presentation $\langle S\mid R\rangle$ has a \GHF\
if and only if $\rank(C) < |S|$.
\item
A \GHF\ is a group invariant in the  sense that,
if $h$ is a \GHF\ of $G$, then it is also a \GHF\ for
the Cayley graph of any other presentation of $\Ga$.
\item
A \GHF\ $h$ of $G$ is harmonic, in that
$$
h(v) = \frac1{\deg(v)}\sum_{u \sim v} h(u), \qq v \in V.
$$
\end{letlist}
\end{theorem}

Since the \GHF\ $h$ of \eqref{eq:gheight} is a graph height function, and $\Ga$
acts transitively,
\begin{equation}\label{def:d}
d(h) = \max\{\g_s: s \in S\},
\end{equation}
in agreement with \eqref{eq:defd}.
In the light of Theorem \ref{thm3}(c), we may speak of a group possessing a \GHF.

\begin{remark}\label{rem:betti}
The quantity $b(\Ga):=|S|-\rank(C)$ is in fact
an invariant of $\Ga$, and may be
called the \emph{first Betti number} since it
equals the power of $\ZZ$ in the abelianization $\Ga/[\Ga,\Ga]$
(see, for example, \cite[Chap.\ 8]{LedW}).
Group height functions are a standard tool of group theorists, since they are
(when the non-zero $\g_s$ are coprime) surjective homomorphisms
from  $\Ga$ to $\ZZ$. This fact is used, for example,  in the proof of \cite[Thm 4.1]{GL-amen},
which asserts that the Cayley graph of an infinite, finitely generated, elementary amenable group
possesses a harmonic (unimodular) \ghf.
Further details may be found in \cite{Hill94, Hill09}.

Although some of the arguments of the current paper are standard within group theory,
we prefer to include sufficient details to aid readers from other backgrounds.
\end{remark}

It follows in particular from Theorem \ref{thm3} that $G$ has a \GHF\ if $|R| < |S|$,
which is to say that the presentation $\Ga=\langle S \mid R\rangle$ has strictly
positive deficiency (see \cite[p.\ 419]{Rob}). Free groups
provide examples of such groups.

Consider for illustration the examples of \cite[Sect.\ 3]{GL-loc}.
\begin{letlist}
\item
The \emph{hypercubic lattice} $\ZZ^n$ is the Cayley group of an abelian
group with $|S|=2n$, $|R|=n+\binom n2$, and $\rank(C)=n$.
It has  a  \GHF\ (in fact, it has many, indexed by the non-zero, integer-valued
elements of $\sN(C)$).

\item The \emph{$3$-regular tree} is the Cayley graph of the
group with $S=\{s_1,s_2,t\}$, $R=\{s_1t,s_2^2\}$, and $\rank(C)=2$.
It has a \GHF.

\item The \emph{discrete Heisenberg group} has $|S|=|R|=6$
and $\rank(C)=4$. It has a \GHF.

\item
The \emph{square/octagon lattice} is
the Cayley graph of a finitely presented group with $|S|=3$ and
$|R|=5$, and this
does not satisfy the hypothesis of Theorem \ref{thm3}(b) (since $\rank(C)=3$).
This presentation has no \GHF. Neither
does the lattice have a graph height function with automorphism
subgroup acting transitively, but nevertheless
it possesses a \ughf\ in the sense of
Definition \ref{def:height}, as explained in
\cite[Sect.\ 3]{GL-loc}.

\item The \emph{hexagonal lattice} is the Cayley graph of
the finitely presented group with $S=\{s_1,s_2,s_3\}$ and
$R=\{s_1^2,s_2s_3, s_1s_2^2s_1s_3^2\}$. Thus, $|R|=|S|=3$,
$\rank(C)=2$, and the graph has a \GHF.

\end{letlist}

A discussion is presented in Section \ref{sce} of certain types of infinite groups
whose Cayley graphs have group or graph height functions. We describe next
some illustrative examples and a question.
The next proposition is extended in Theorem \ref{iq}.

\begin{proposition}\label{prop:abel}
Any finitely generated group which is infinite and abelian has a
\GHF\ $h$ with $d(h)=1$.
\end{proposition}

\begin{example}\label{ex:ladder}
The \emph{infinite dihedral group}
$\dih_\oo=\langle s_1,s_2\mid  s_1^2,s_2^2 \rangle$ is an example of an infinite,
finitely generated group $\Ga$ which has no \GHF.
The Cayley graph of $\Ga$  is the
line $\ZZ$ with (harmonic) \ughf\ $(h,\sH)$, where $h$ is
the identity and $\sH$ is the group of shifts.
This example of a solvable group is extended in
Theorem \ref{thm:vs}.
\end{example}

\begin{example}\label{ex:higman}
The \emph{Higman group}  $\Ga$  of \cite{gh} is an infinite, finitely presented group with presentation $\Ga=\langle S \mid R\rangle$ where
\begin{align*}
S&=\{ a,b,c,d,a',b',c',d'\},\\
R&=\{aa',bb',cc',dd'\}\cup \{a'ba(b')^2,b'cb(c')^2,c'dc(d')^2,
d'ad(a')^2\}.
\end{align*}
The quotient of $\Ga$ by its maximal proper normal
subgroup is an infinite, finitely generated, simple group.
By Theorem \ref{thm3}(b), $\Ga$ has no \GHF.
\end{example}

\begin{remark}\label{rem:grig0}
Since writing this paper, the authors have shown
in \cite{GL-amen} that the Cayley graph of  the Higman group
does not possess a \ghf.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm3}]
(a) Let $\g$ be as given. To check that $h$ is well defined by \eqref{eq:gheight},
we must show that $h(v)$ is independent of the chosen representation
of $v$ as a word. Suppose that
$v=s_1\cdots s_m=u_1\cdots u_n$ with $s_i, u_j \in S$,
and extend the definition of $\g$ to the directed edge-set of $G$ by
\begin{equation}\label{eq:ga}
\g([g,gs\rangle) = \g_s, \qq g \in \Ga,\ s \in S.
\end{equation}
The
walk $(\id,s_1,s_1s_2,\dots,v)$ is denoted as $\pi_1$, and
$(\id,u_1,u_1u_2,\dots,v)$ as $\pi_2$, and the
latter's reversed walk as $\pi_2^{-1}$.
Consider the walk $\nu$ obtained by
following $\pi_1$, followed by $\pi_2^{-1}$.
Thus $\nu$ is a closed walk of $G$ from $\id$.

Any $\rho\in R$ gives rise to a directed cycle in $G$ through
$\id$, and we write $\Ga R$ for the set of images of such
cycles under the action of $\Ga$.
Any closed walk lies in the vector space over $\ZZ$ generated by
the directed cycles of  $\Ga R$ (see, for example, \cite[Sect.\ 4.1]{Ham}).
The sum of the $\g_s$ around any $g \rho\in \Ga R$ is zero, by
\eqref{eq:ga} and the fact that $C\g=\bzero$. Hence
\begin{equation*}
\sum_{i=1}^m \g_{s_i}-  \sum_{j=1}^n \g_{u_j}=0,
\end{equation*}
as required.

We check next that $(h,\Ga)$ is a \ghf. Certainly, $h(\id)=0$.
For $u,v\in V$, write $v=ux$ where $x=u^{-1}v$,
so that $h(v)-h(u) = h(x)$ by \eqref{eq:gheight}.
For $g\in \Ga$, we have that $gv = (gu)x$, whence
\begin{equation}\label{eq:extra1}
h(gv)-h(gu) = h(x) = h(v)-h(u).
\end{equation}
Since $\g \ne \bzero$, there exists $s\in S$ with $\g_s>0$.
For $v \in V$, we have $h(vs^{-1}) < h(v) < h(vs)$.

(b) The null space $\sN(C)$ is non-trivial if and only if $\rank(C)<|S|$.
Since $C$ has integer entries and $|S|<\oo$, $\sN(C)$ is non-trivial if and only if it contains a non-zero
vector of integers.

(c) See Remark \ref{rem:betti}. This may also be proved directly, but we omit
the details.

(d)
We do not give the details of this, since a more general fact is proved in
Proposition \ref{prop11}(b). The current proof follows that of the latter proposition
with $\sH=\Ga$, $F=h$, and $\Ga$ acting on $V$ by left-multiplication. Since
this action of $\Ga$
has no non-trivial fixed points, $\Ga$ is unimodular.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:abel}]
See Remark \ref{rem:betti} for an elementary group-theoretic explanation.
Since $\Ga$ is infinite and abelian, there exists a generator, $\si$ say,
of infinite order. For $s\in S$, let
\begin{equation}\label{eq:unit}
\g_s = \begin{cases} 1 &\text{if } s=\si,\\
-1 &\text{if } s=\si^{-1},\\
0 &\text{otherwise}.
\end{cases}
\end{equation}
Since any relator must contain equal numbers of appearances of $\si$ and $\si^{-1}$,
we have that $\g \in \sN(C)$. Therefore, the function $h$ of \eqref{eq:gheight}
is a \GHF.
\end{proof}

\begin{comment}
\begin{proof}[Proof of Theorem \ref{q3}]
This is almost immediate from the fact that a \GHF\
acts on the \emph{vertices} of the Cayley graph, that is,
on the group $\Ga$ itself. Let $h:\Ga\to\ZZ$ be a \GHF\
of the Cayley graph $G$ of $\langle S \mid R\rangle$, and let $G'$
be the Cayley graph of the (possibly different) finite presentation  $\langle S' \mid R'\rangle$
of $\Ga$. We write $\g_s=h(s)$ for $s \in S$.

We define $\g'= (\g'_{s'}: s'\in S')$ by $\g_{s'}=h(s')$.
Since $h$ is non-constant and satisfies \eqref{eq:extra1}, $\g' \ne \bzero$.
A relator $\rho'\in R'$ is a product of elements $s'_i\in S'$, and each such $s'_i$
is a product of elements $s^i_j \in S$. Thus the sum of $\g_{s'}'$
around the directed cycle of $G'$ corresponding to $\rho'$ equals the sum
of the corresponding $\g_{s^i_j}$. Since this is a sum around a closed walk of $G$,
it equals $0$ as in the proof of Theorem \ref{thm3}. Therefore,
$h$ is a \GHF\ of $G'$.
\end{proof}
\end{comment}

\section{Cayley graphs with height functions}\label{sce}

The main result of this section is as follows. The associated definitions are presented later,
and the proofs of the next two theorems are at the end of this section.

\begin{theorem}\label{thm:vs}
Let $\Ga$ be an infinite, finitely generated group with a normal
subgroup $\Ga^*$ satisfying $[\Ga:\Ga^*]<\oo$.
Let $q=\sup\{i: [\Ga^*: \Ga^*_{(i)}]<\oo\}$ where $(\Ga^*_{(i)} : i \ge 1)$
is the derived series of $\Ga^*$.
If $q<\oo$ and $[\Ga^*_{(q)},\Ga^*_{(q+1)}]=\oo$,
then every Cayley graph of $\Ga$ has a \ughf\
of the form $(h,\Ga^*_{(q)})$ which is harmonic.
\end{theorem}

The theorem may be applied to any finitely generated,
virtually solvable group $\Ga$, and more generally whenever the derived series
of $\Ga^*$ terminates after finitely many steps at a \emph{finite} perfect group.

In preparation for the proof, we present  a general
construction of a height function
for a group having  a normal subgroup.
Part (a) extends Proposition \ref{prop:abel} (see also Remark \ref{rem:betti}).

\begin{theorem}\label{iq}
Let $\Ga$ be an infinite, finitely generated group, and let $\Ga'\normal \Ga$.
\begin{letlist}
\item If the quotient group $\Ga/\Ga'$ is infinite and abelian,
then  $\Ga$ has a \GHF\ $h$ with $d(h)=1$.
\item If the quotient group $\Ga/\Ga'$ is finite, and $\Ga'$ has
a \GHF, then every Cayley graph of $\Ga$ has a harmonic, \ughf\ of the form $(h, \Ga')$.
\end{letlist}
\end{theorem}

Recall that $\Ga/\Ga'$ is abelian if and only if $\Ga'$ contains the commutator
group $[\Ga,\Ga]$, of which the definition follows.
An example of  Theorem \ref{iq}(b) in action is the special linear group
$\SL_2(\ZZ)$ of the forthcoming Example \ref{ex:sl2z} (see \cite[p.\ 66]{dlH}).

We turn now towards solvable groups.
Let $\Ga$ be a group with identity $\id_\Ga$.
The \emph{commutator} of the pair  $x,y\in \Ga$ is the group element
$[x,y]:=x^{-1}y^{-1}xy$.
Let $A$, $B$ be subgroups of $\Ga$. The \emph{commutator subgroup} $[A,B]$ is defined to be
\begin{equation*}
[A,B]=\bigl\langle [a,b] : a\in A,\ b\in B\bigr\rangle,
\end{equation*}
that is, the subgroup generated by all commutators $[a,b]$ with $a\in A$, $b\in B$.
The \emph{commutator subgroup} of $\Ga$ is the subgroup $[\Ga,\Ga]$.
It is standard that $[\Ga,\Ga]\normal\Ga$, and the quotient group
$\Ga/[\Ga,\Ga]$ is abelian. The group $\Ga$ is called \emph{perfect}
if $\Ga=[\Ga,\Ga]$.

Here is an explanation of the terms in Theorem \ref{thm:vs}.
Let $\Ga_{(1)}=\Ga$. The \emph{derived series} of $\Ga$ is given recursively by the formula
\begin{equation}\label{eq:deriv}
\Ga_{(i+1)}=[\Ga_{(i)},\Ga_{(i)}], \qq i \ge 1.
\end{equation}
The group $\Ga$ is called \emph{solvable} if there exists an integer $c\in\NN$
such that $\Ga_{(c+1)}=\{\id_\Ga\}$. Thus, $\Ga$ is solvable if
there exists $c \in \NN$ such that
\begin{equation*}
\Ga=\Ga_{(1)}\normalgr \Ga_{(2)}\normalgr \cdots \normalgr \Ga_{(c+1)}=\{\id_{\Ga}\}.
\end{equation*}
A \emph{virtually solvable group} is a group $\Ga$ for which there exists a normal subgroup
$\Ga^*$  which is solvable and satisfies $[\Ga: \Ga^*] < \oo$.
The reader is referred to \cite{LedW,Rob} for general accounts of group theory.

\begin{example}\label{ex:lampl}
Here is an example of a finitely generated but not finitely presented group with a \GHF.
The \emph{lamplighter group} $L$ has presentation
$\langle S\mid R\rangle$ where $S=\{a,t,u\}$ and $R=
\{a^2, tu\}\cup\{[a,t^nau^{n}]: n\in \ZZ\}$.
It has a \GHF\ since the rank of its coefficient matrix is $2$.
A recent reference to linear-growth harmonic functions on $L$ is \cite{BDCKY}.
\end{example}

\begin{example}\label{ex:sl2z}
The special linear group $\Ga:=\SL_2(\ZZ)$ has a presentation
\begin{equation} \label{sl2z}
\Ga=\langle x,y,u,v\mid xu, yv, x^4, x^2v^{3} \rangle,
\end{equation}
where
\begin{equation*}
x=\begin{pmatrix}0&-1\\1 &0\end{pmatrix},\qq
 y=\begin{pmatrix}0&-1\\1&1\end{pmatrix}.
\end{equation*}
The presentation has no \GHF.

The commutator subgroup $\Ga^{(2)}:=[\Ga,\Ga]$ is a normal
subgroup of $\Ga$ with index $12$, and $\Ga^{(2)}$ is free of rank $2$.
(See  \cite{Kon}
and \cite[p.\ 66]{dlH}.) By Theorem \ref{iq}(b),
every Cayley group of $\Ga$ has a harmonic, \ughf.
\end{example}

\begin{proof}[Proof of Theorem \ref{iq}]
(a) This is an immediate consequence of Remark \ref{rem:betti}. A detailed argument
may be outlined as follows. Let $\Ga=\langle S\mid R\rangle$.
If $Q:=\Ga/\Ga'$ is infinite and abelian, it is generated
 by the cosets $\{\ol s:=s\Ga': s \in S\}$,
and its relators are the
words $\ol s_1\ol s_2\cdots \ol s_r$ as  $\rho=s_1s_2\cdots s_r$ ranges
over $R$.  Choose $\si\in S$ with infinite order, and let
\begin{equation}\label{eq:unit2}
\g_s=\begin{cases} 1 &\text{if } s\in \ol\si,\\
-1 &\text{if } s^{-1} \in  \ol\si,\\
0 &\text{otherwise}.
\end{cases}
\end{equation}
It may now be checked that $C\g=\bzero$ where $C$ is the coefficient
matrix.

\noindent
(b)
Let $G$ be a Cayley graph of $\Ga$, and let $\Ga'\normal \Ga$
satisfy $[\Ga:\Ga']<\oo$.
By assumption, $\Ga'$ has a \GHF\ $h'$.
The subgroup $\Ga'$ of $\Ga$ acts on $G$ by left-multiplication, and it
is unimodular since its elements act with no non-trivial fixed points.
We apply Theorem \ref{prop:indep} with $\sH=\Ga'$ and $F=h'$
to obtain a harmonic, \ughf\ $(h,\Ga')$ on $G$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:vs}]
Since $q<\oo$, we have that $[\Ga:\Ga^*_{(q)}]<\oo$, and in particular
$\Ga^*_{(q)}$ is finitely generated.
Now, $\Ga^*_{(q)}$ is characteristic in $\Ga^*$, and $\Ga^*\normal \Ga$, so that
$\Ga^*_{(q)} \normal \Ga$.

By applying Theorem \ref{iq}(a) to the pair $\Ga_{(q+1)}^* \normal \Ga^*_{(q)}$,
there exists a \GHF\ $h^*_q$
on $\Ga^*_{(q)}$.
We apply Theorem \ref{iq}(b) to the pair $\Ga^*_{(q)}\normal\Ga$ to obtain
a harmonic, \ughf\ $(h, \Ga^*_{(q)})$ on $\Ga$.
\end{proof}

\section{Convergence of connective constants of Cayley graphs}\label{sec:conv}

Let $\Ga=\langle S \mid R\rangle$ be finitely presented with coefficient matrix $C$
and Cayley graph $G=G(S,R)$.
Let $t \in \Ga$ have infinite order. We consider in this section the effect of adding a new relator
$t^m$, in the limit as $m \to\oo$. Let $G_m$ be the Cayley graph of
the group $\Ga_m=\langle S\mid R\cup\{t^m\}\rangle$.

\begin{theorem}\label{torus}
If $\rank(C) < |S|-1$, then
 $\mu(G_m) \to \mu(G)$ as $m\to\oo$.
\end{theorem}

\begin{proof}
The coefficient matrix  $C_m$ of $G_m$ differs from $C_1$ only in  the multiplicity
of the row corresponding to the new relator, and therefore $\sN(C_1)=\sN(C_m)$.
Since $\Ga_1$ has only one relator more than $G$, $\rank(C_1) \le \rank(C)+1$.
If $\rank(C)<|S|-1$, then
$\rank(C_1)<|S|$. By Theorem \ref{thm3}, we may find $\g=(\g_s:s\in S)\in\sN(C_1)$ such that
$\g\in\ZZ^S$, $\g\ne\bzero$. By the above, for $m \ge 1$, $\g\in\sN(C_m)$,
so that  $G_m$ has a corresponding
\GHF\ $h_m$. By \eqref{def:d},
$d(h)=d(h_m) = :D$ for all $n$,
so that $G_m \in \sG_{D,0}$ for all $m$.

The group $\Ga_m$ is obtained as the quotient group
of $\Ga$ by the (normal) subgroup generated by $t^m$.
We apply \cite[Thm 5.2]{GL-loc} with $\sA_m$ the cyclic group generated by $t^m$.
The condition of the theorem holds since $t$ has infinite order.
\end{proof}

\begin{remark}[Approximating $\mu(G)$]\label{rem:app}
The question is posed in \cite{GrL3} of whether one can obtain rigorous sequences
of bounds for $\mu(G)$ which are sharp. Such upper bounds are provided
by the subadditive argument of \eqref{eq:sisub}, namely $\mu\le \si_m^{1/m}$ for $m \ge 1$.
Theorem \ref{torus}, taken together with \cite[Thm 3.8]{GrL3}, provides lower bounds.
It is however preferable to use the improved lower bound $\mu \ge b_m^{1/m}$,
where $b_m$ is the number of $m$-step bridges.
The latter inequality is asymptotically sharp whenever $G$ has a \ughf\ (see \cite[Remark 4.5]{GL-loc}),
and this is a less restrictive condition than that of Theorem \ref{torus}.
\end{remark}

As examples of finitely generated
groups satisfying the conditions of Theorem \ref{torus}, we mention
free groups, abelian groups, free nilpotent groups, free solvable groups, and, more widely,
nilpotent and solvable groups $\Ga$
with presentations $\langle S\mid R\rangle$
whose coefficient matrix $C$ satisfies
$b(\Ga)=|S|-\rank(C) >1$. Here is an
example where Theorem \ref{torus} cannot be applied,
though the conclusion is valid.

\begin{example}\label{ex:dih2}
Let $G$ be the Cayley graph of the \emph{infinite dihedral group}
$\dih_\oo = \langle s_1,s_2 \mid s_1^2,s_2^2\rangle$ of Example \ref{ex:ladder}.
As noted there, $G$ has no \GHF, though it has a \ughf\ $(h,\sH)$ with $d(h)=1$.
Let $\Ga_m=\dih_\oo \times J_m$ where
$m \ge 3$ and $J_m=\langle a,b\mid ab, a^m\rangle$
is the cyclic group $\{\id,a,a^2,\dots, a^{m-1}\}$. Thus, $\Ga_m$ is finitely presented
but, by Theorem \ref{thm3}(b), it
has no \GHF. In particular, Theorem \ref{torus} may not be applied.

The Cayley graph $G$  is isomorphic to $\ZZ$.
Therefore, we may define a \ughf\ $(h',\sH')$ on the Cayley graph $G_m$
of $\Ga_m$ by
$h'(\g, a^k)=h(\g)$, with $\sH'$ generated by the shifts $(\g,a^k)\mapsto (\g+1,a^k)$ and
$(\g,a^k)\mapsto (\g,a^{k+1})$. Furthermore,
$d(h') = d(h)=1$ and $r(h',\sH')=r(h,\sH)=0$.
By \cite[Thms 5.1, 5.2]{GL-loc}, $\mu(G_m)\to\mu(\ZZ^2)$ as $m\to\oo$.
\end{example}

\section{Proof of Theorem \ref{prop:indep}}\label{sec:pf-}

Assume that assumptions (a)--(c) of Theorem \ref{prop:indep} hold.
There are two steps in the proof,
namely of the following.
\begin{Alist}
\item (Prop.\ \ref{102})
There exists $\psi:V \to \QQ$ which is \hdi, harmonic,
non-constant, and takes values in the rationals.
\item (Prop.\ \ref{prop12})
There exists a \ghf\ which is harmonic on $G$.
\end{Alist}

The vertex $\id$ may appear to play a distinguished role in this section.
This is in fact not so: since $G$ is assumed transitive, the following is valid with any choice
of vertex for the label $\id$.
The approach of the proof is inspired in part by the proof of
\cite[Cor.\ 3.4]{LeeP}.
Let $X=(X_n: n=0,1,2,\dots)$ be a simple random walk on $G$, with transition matrix
\begin{equation*}
P(u,v)=\PP_u(X_1=v)=\frac{1}{\deg(u)}, \qq u,v\in V,\  v\in\pd u,
\end{equation*}
where $\PP_u$ denotes the law of the random walk starting at $u$.

Let $V_1=\sH\id$ be the orbit of the identity under $\sH$, and
let $P_1$ be the transition matrix of the induced random walk on $V_1$, that is
\begin{equation*}
P_1(u,v)=\PP_u(X_{\tau}=v), \qq u,v \in V_1,
\end{equation*}
where $\tau=\min\{n\geq 1:X_n\in V_1\}$. It is easily seen that
$\PP_u(\tau<\oo)=1$ since, by the quasi-transitive action of $\sH$,
there exist $\a>0$ and $K<\oo$ such that
\begin{equation}\label{eq:bnd}
\PP_u(X_k\in V_1\text{ for some $1\le k \le K$}) \ge \a, \qq u\in V.
\end{equation}
We note for later use that, by \eqref{eq:bnd}, there exist $\a'=\a'(\a,K)\in(0,1)$ and $A=A(\a,K)$ such that
\begin{equation}\label{eq:bnd2}
\PP_u(\tau\ge m) \le A(1-\a')^m, \qq m \ge 1, \ u\in V.
\end{equation}

Since $\sH\le \Aut(G)$, $P_1$ is invariant under $\sH$ in the sense that
\begin{equation}\label{eq:inv}
P_1(u,v) = P_1(\g u, \g v), \qq \g \in \sH,\ u,v\in V_1.
\end{equation}

\begin{proposition}\label{prop11}
\mbox{}
\begin{letlist}
\item
The transition matrix $P_1$ is symmetric, in that
$$
P_1(u,v)=P_1(v,u), \qq u,v\in V_1.
$$
\item Let $F:V_1\to \ZZ$ be \hdi.
Then $F$ is $P_1$-harmonic in that
$$
F(u) = \sum_{v\in V_1} P_1(u,v)F(v), \qq u \in V_1.
$$
\end{letlist}
\end{proposition}

\begin{proof}
(a)
Since $P$ is reversible with respect to the measure $(\deg(v): v \in V)$,
and $\deg(v)$ is constant on $V_1$, we have that
$$
P(u_0,u_1)P(u_1,u_2)\cdots P(u_{n-1},u_n) =
P(u_n,u_{n-1})P(u_{n-1},u_{n-2})\cdots P(u_1,u_0)
$$
for $u_0,u_n \in V_1$, $u_1,\dots,u_{n-1} \in V$. The symmetry of $P_1$ follows
by summing over appropriate sequences $(u_i)$.

\noindent
(b) It is required to prove that
\begin{equation}\label{poh}
\sum_{v\in V_1}P_1(u,v)[F(u)-F(v)]=0, \qq u\in V_1,
\end{equation}
and it is here that we shall use assumption (b) of Theorem \ref{prop:indep},
namely, that $\sH$ is unimodular.
Since $F$ is \hdi, there exists $D<\oo$ such that
$$
|F(u)-F(v)| \le D d_G(u,v), \qq u,v\in V_1.
$$
By \eqref{eq:bnd},
the random walk on $V_1$ has finite mean step-size.
It follows that the summation in \eqref{poh} converges absolutely.

Equation \eqref{poh} may be proved by a cancellation of summands, but it is shorter to
use the mass-transport principle.  Let
\begin{equation*}
m(u,v) = P_1(u,v)[F(u)-F(v)],\qq u,v\in V_1.
\end{equation*}
The sum $\sum_{v\in V_1}m(u,v)$ is absolutely convergent as above, and
$m(\g u,\g v)= m(u,v)$ for $\g \in \sH$. Since $\sH$ is unimodular,
by the mass-transport principle (see, for example, \cite[Thm 8.7, Cor.\ 8.11]{LyP}),
\begin{equation}\label{eq:mtp1}
\sum_{v\in V_1} m(u,v) = \sum_{w\in V_1} m(w,u), \qq u \in V_1.
\end{equation}
Now,
\begin{align*}
\sum_{w\in V_1} m(w,u) &= \sum_{w\in V_1} P_1(w,u)[F(w)-F(u)]\\
&= -\sum_{w\in V_1} P_1(u,w)[F(u)-F(w)] \qq\text{by part (a)},
\end{align*}
and \eqref{poh} follows by \eqref{eq:mtp1}.

It is usual to assume in the mass-transport principle that $m(u,v) \ge 0$, but
it suffices that $\sum_v m(u,v)$ is absolutely convergent.
\end{proof}

Let $\be>1$, and let $f:V\to\RR$. We write $f=\O(\beta^n)$ if there exists $B$ such that
\begin{equation}\label{eq:expgrowth}
|f(v)|\le B\beta^n\qq \text{if } d_G(\id,v)\le n, \text{ and } n \ge 1.
\end{equation}

\begin{proposition}\label{102}
Let $F:V_1 \to \ZZ$ be \hdi, and
let
\begin{equation}\label{eq:psidef}
\psi(v)=\EE_v[F(X_T)],\qq v \in V,
\end{equation}
where $T=\inf\{n\ge 0: X_n\in V_1\}$. Then,
\begin{letlist}
\item the function $\psi$ is \hdi,  and agrees with $F$ on $V_1$,
\item  $\psi$ is harmonic on $G$, in that
\begin{equation}\label{eq:harm2}
\psi(u) = \sum_{v\in V} P(u,v)\psi(v), \qq u \in V,
\end{equation}
and, furthermore, $\psi$ is the unique harmonic function
that agrees with $F$ on $V_1$ and satisfies $\psi=\O(\beta^n)$
with any $1\le \beta< 1/(1-\a')$, where $\a'$ satisfies \eqref{eq:bnd2},
\item $\psi$ takes rational values.
\end{letlist}
\end{proposition}

\begin{remark}\label{rem:unique}
By Proposition \ref{102}(a,\,b), any $\O(\be^n)$ harmonic extension of $F$ (with suitable $\be$)
is \hdi. Conversely, any \hdi\ function $f$ satisfies $f=\O(\be^n)$
for all $\beta>1$, whence the function $\psi$
of \eqref{eq:psidef} is the unique harmonic extension of $F$ that
is \hdi.
\end{remark}

\begin{proof}
(a) The function $\psi$ is \hdi\ since the law of
the random walk is $\sH$-invariant, and
$$
\psi(v)-\psi(w) = \EE_v[F(X_T)] -\EE_w[F(X_T)].
$$
It is trivial that $\psi\equiv F$ on $V_1$.

\noindent
(b) By conditioning on the first step,
$\psi$ is harmonic at any $v \notin V_1$.
For $v\in V_1$, it suffices to show that
$$
\psi(v) = \sum_{w\in V}P(v,w)\psi(w).
$$
Since $\psi\equiv F$ on $V_1$, and $F$ is $P_1$-harmonic
(by Proposition \ref{prop11}), this may be written as
\begin{equation*}
\sum_{w\in V_1}P_1(v,w)\psi(w) = \sum_{w\in V} P(v,w)\psi(w), \qq v \in V_1.
\end{equation*}
Each term equals $\EE_v[\psi(W(X_1))]$, where $X_1$ is the
position of the random walk after one step,  and $W(X_1)$ is
the first element of $V_1$ encountered having started at $X_1$.

To establish uniqueness, let $\phi$ be a harmonic function with
$\phi=\O(\beta^n)$ where
$1\le\beta < 1/(1-\a')$, such that $\phi\equiv F$ on $V_1$.
Then $Y_n:= \phi(X_n)$ is a martingale, and furthermore
$T$ is a stopping time with tail satisfying \eqref{eq:bnd2}.
By the optional stopping theorem (see, for example, \cite[Thm 12.5.1]{GS})
and \eqref{eq:psidef},
$$
\phi(u)=\EE_u(Y_T) = \EE_u(F(X_T)) = \psi(u),
$$
so long as $\EE_u(|Y_n|I_{\{T\ge n\}})\to 0$ as $n \to\oo$,
where $I_E$ denotes the indicator function of an event $E$.
To check the last condition, note by \eqref{eq:expgrowth} and \eqref{eq:bnd2}  that
\begin{align*}
\EE_u(|Y_n|I_{\{T\ge n\}}) &\le B\beta^{n+|u|}\PP_u(T\ge n)\\
&\le (AB\beta^{|u|})\beta^n(1-\a')^n\to 0 \qq\text{as } n\to\oo,
\end{align*}
where $|u|=d_G(\id,u)$.

\noindent
(c) The quantity $\psi(v)$
has a representation as a sum of values of the unique
solution of a finite set of linear equations with integral coefficients and boundary
conditions, and thus $\psi(v)\in\QQ$. Some further details follow.

Let  $\vG=(V, \vE)$ be the directed graph obtained from $G=(V,E)$ by
replacing each $e\in E$ by two edges $\vec e$, $-\vec e$
with the same endpoints and opposite orientations.
Suppose $\de: \vE\to \RR$ satisfies the linear equations
\begin{alignat}{2}
\de(-\vec e) +\de(\vec e)&=0, \qq &&e \in E,\label{eq:negsym}\\
\sum_{\vec e \in W} \de(\vec e)&=0, \qq && W \in \sW(G),\label{eq:cycles2}\\
\sum_{v \sim  u}\de([ u, v\rangle) &= 0,\qq  &&u \in V,\label{eq:harm}\\
\de(\a \vec e) &= \de(\vec e),\q &&e \in E,\ \a \in \sH,\label{eq:alv}
\end{alignat}
where $\sW(G)$ is the set of directed closed walks of $G$.
(Equation \eqref{eq:cycles2} may be viewed as including \eqref{eq:negsym}.)
By \eqref{eq:cycles2}, the sum
$$
\De(v) := \sum_{\vec e \in \ell_v} \de(\vec e),\qq v \in V,
$$
is well defined, where $\ell_v$ is an arbitrary (directed) walk from $\id$ to $v\in V$.
Equation \eqref{eq:harm} requires that $\De$ be harmonic, and \eqref{eq:alv}
that $\De$ be \hdi.

Since $\sH$ acts quasi-transitively, by \eqref{eq:alv},
the linear equations \eqref{eq:negsym}--\eqref{eq:harm} involve only
 finitely  many variables.  Therefore, there
exists a finite subset of equations, denoted as $\Eq$, of \eqref{eq:negsym}--\eqref{eq:harm}
such that $\de$ satisfies \eqref{eq:negsym}--\eqref{eq:alv} if
only if $\de$ satisfies $\Eq$ together with \eqref{eq:alv}.
In summary, any harmonic, \hdi\ function $\De$, satisfying $\De(\id)=0$,  corresponds to
a solution to the finite collection $\Eq$ of linear equations.

With $F$ as given, let $\psi$ be given by \eqref{eq:psidef}.
By Remark \ref{rem:unique},
equations \eqref{eq:negsym}--\eqref{eq:alv} have a
unique solution  satisfying
\begin{equation}\label{eq:ne}
\sum_{\vec e \in \ell_v} \de(\vec e) =F(v)-F(\id), \qq v \in V_1.
\end{equation}
By \eqref{eq:cycles2}, it suffices in \eqref{eq:ne} to
consider only the finite set $V_1'\subseteq V_1$ of vertices $v$ within some bounded distance of $\id$
that depends on the pair $G$, $\sH$.

Therefore, $\Eq$ possesses a unique solution subject to \eqref{eq:ne}
(with $V_1$ replaced by $V_1'$). All coefficients
and boundary values  in $\Eq$ and \eqref{eq:ne} are integral,
and therefore $\psi$ takes only rational values.
\end{proof}

\begin{proposition}\label{prop12}
Let  $F:V_1 \to \ZZ$ be \hdi, and non-constant on $V_1$.
There exists a \ghf\
$(h=h_F,\sH)$ such that $h$ is harmonic on $G$.
\end{proposition}

\begin{proof}
The normality of $\sH$ is used in this proof.
A vertex $v\in V$ is called a \emph{\sap} of a function $h:V \to\RR$ if $v$ has
neighbours $u$, $w$ such that $h(u)<h(v)<h(w)$. The function $h$ is said to
\emph{increase everywhere} if every vertex is a point of increase.
For $v\in V$ and a harmonic function
$h$,
\begin{equation}\label{eq:fork}
\begin{aligned}
\mbox{either: }&\mbox{ $v$ is a \sap\ of $h$,}\\
\mbox{or: }&\mbox{ $h$ is constant on $\{v\}\cup \pd v$.}
\end{aligned}
\end{equation}
An \hdi\ function $h$ on $G$ is a \ghf\ if and only if $h(\id)=0$, $h$ takes
integer values, and $h$ increases everywhere.

Let $F$ be as given, and let $\psi$ be given by Proposition \ref{102}.
Thus, $\psi:V \to\QQ$ is non-constant on $V_1$, \hdi, and harmonic on $G$.
Since $\psi$ is \hdi, we may replace it by $m\psi$ for a suitable $m\in\NN$
to obtain such a function that in addition takes integer values.
We shall work with the latter function, and thus we assume henceforth that $\psi:V \to\ZZ$.
Now, $\psi$ may not increase everywhere.
By \eqref{eq:fork}, $\psi$ has some \sap\  $w \in V$.

Let $V_1,V_2,\dots,V_M$ be the orbits of $V$ under $\sH$. Find $W$ such that
$w \in V_W$.
Since $\Ga$ acts transitively on $G$, and $\sH$ is a normal subgroup of $\Ga$ acting
quasi-transitively on $G$, there exist $\g_1,\g_2,\dots,\g_M\in \Ga$ such that
$\g_W=\id_\Ga$ and
\begin{equation*}
V_i=\g_iV_W,\qq  i=1,2,\dots,M.
\end{equation*}
Let
\begin{equation}\label{eq:psi2def}
\psi_i(v)=\psi(\g_i^{-1}v), \qq i=1,2,\dots,M,
\end{equation}
so that, in particular, $\psi_W=\psi$.
Since $w \in V_W$ is a \sap\ of $\psi$, $w_i := \g_iw$ is a \sap\
of $\psi_i$, and also $w_i \in V_i$.

\begin{lemma}\label{lem3}
For $i=1,2,\dots,M$,
\begin{letlist}
\item $\psi_i:V \to\ZZ$ is a non-constant, harmonic function, and
\item $\psi_i$ is \hdi.
\end{letlist}
\end{lemma}

\begin{proof}
(a) Since $\psi_i$ is obtained from $\psi$ by shifting the domain
according to the automorphism $\g_i$,
$\psi_i$ is non-constant and harmonic.

\noindent
(b) For $\a\in \sH$ and $u,v\in V$,
\begin{equation*}
\psi_i(\a v)-\psi_i(\a u)=\psi(\g_i^{-1}\a v)-\psi(\g_i^{-1}\a u).
\end{equation*}
Since $\sH \normal \Ga$ and $\g_i\in\Ga$,
there exists $\a_i\in\sH$ such that $\g_i^{-1}\a=\a_i\g_i^{-1}$.
Therefore,
\begin{alignat*}{2}
\psi_i(\a v)-\psi_i(\a u)&=\psi(\a_i\g_i^{-1}v)-\psi(\a_i\g_i^{-1}u)\ &&\\
&=\psi(\g_i^{-1}v)-\psi(\g_i^{-1}u) &&\text{since $\psi$ is \hdi}\\
&=\psi_i(v)-\psi_i(w) &&\text{by \eqref{eq:psi2def},}
\end{alignat*}
so that $\psi_i$ is \hdi.
\end{proof}

Let $\nu:V \to \RR$ be \hdi. For $j=1,2,\dots, M$, either every vertex in $V_j$ is a \sap\ of $\nu$,
or no vertex in $V_j$ is a \sap\ of $\nu$. We shall now
use an iterative construction in order to find a harmonic, \hdi\ function $h'$
for which every $w_i$ is a \sap.  Since the $w_i$ represent the orbits $V_i$,
the ensuing $h'$ increases everywhere.

\begin{numlist}
\item If every $w_i$ is a \sap\ of $\psi$, we set $h'=\psi$.
\item Assume otherwise, and find the smallest $j_2$ such that
$w_{j_2}$ is not a \sap\ of $\psi$.
By \eqref{eq:fork}, we may choose $c_{j_2}\in \QQ$ such that both $w$ and $w_{j_2}$ are
\saps\ of $h_2 := \psi+c_{j_2}\psi_{j_2}$. If $h_2$ increases everywhere,
we set $h' = h_2$.

\item Assume otherwise, and find the smallest $j_3$ such that
$w_{j_3}$ is not a \sap\ of $h_2$.
By \eqref{eq:fork}, we may choose $c_{j_3}\in \QQ$ such that $w$, $w_{j_2}$, and $w_{j_3}$ are
\saps\ of $h_3 := \psi+c_{j_2}\psi_{j_2}+c_{j_3}\psi_{j_3}$. If $h_3$ increases everywhere,
we set $h' = h_3$.

\item This process is iterated until we find
an \hdi, harmonic function  of the form
$$
h' =\sum_{l=1}^M c_{j_l}\psi_{j_l},
$$
with $j_1=W$, $c_W=1$, and $c_{j_l} \in \QQ$, which increases everywhere.
\end{numlist}

The function $h'-h'(\id)$ may fail to be a \ghf\ only in that
it may take rational rather than integer values.
Since the $c_{j_l}$ are rational, there exists $m\in \ZZ$ such that
$h=m(h'-h(\id))$ is a  \ghf.
\end{proof}

\begin{proof}[Proof of Theorem \ref{prop:indep}]
By Propositions \ref{prop11} and \ref{102}, there exists $\psi:V\to\QQ$ satisfying (i).
The existence of $\psi':V\to\QQ$, in (ii), follows as in Proposition \ref{prop12}.
Similarly, $\psi$, $\psi'$ may be taken to be integer-valued,
and the unimodularity holds since $\sH$ is assumed unimodular.
\end{proof}

\section{Proof of Theorem \ref{cor:nonunim}}\label{sec:nonunim}

Let $G$, $\Ga$, $\sH$ be as given. The idea is to apply Theorem \ref{prop:indep}
to a suitable triple  $G'$, $\Ga'$, $\sH'$, and
to extend the resulting \ghf\ to the original graph $G$. The required function $F$ of the theorem
will be derived from the modular function of $G$ under $\sH$.

By \cite[Thm 8.10]{LyP}, we may define a positive \emph{weight function} $M:V\to(0,\oo)$ satisfying
\begin{equation}\label{eq:modf}
\frac{M(u)}{M(v)} = \frac{|\Stab_u v|}{|\Stab_v u|}, \qq u,v\in V,
\end{equation}
where $|\cdot|$ denotes cardinality. The weight function is uniquely
defined up to a multiplicative constant, and is automorphism-invariant
up to a multiplicative constant.
Since $G$ is assumed non-unimodular, $M$ is non-constant on some orbit
of $\sH$. Without loss of generality, we assume $\id$ lies in such an orbit and that $M(\id)=1$.
See \cite[Sect.\ 8.2]{LyP} for an account of (non-)unimodularity.

Let $\sS$ be the normal subgroup of $\Ga$ generated by $\bigcup_{v\in V} \Stab_v$,
where $\Stab_v=\Stab_v^\sH$.
Let $G'$ denote the quotient graph $G/\sS$ (as in \cite[Sect.\ 2]{GL-loc}),
which we take to be simple in that
every pair of neighbours is connected by just one edge, and any loop is removed.

\begin{lemma}\label{lem:gprime}\mbox{}
\begin{letlist}
\item $\sS \normal \sH$.
\item The function $F':V/\sS\to(0,\oo)$ given by $F'(\sS v) = \log M(v)$, $v \in V$,
is well defined, in the sense that $F'$ is constant on each  orbit in $\sS$.
\item The quotient group $\Ga':=\Ga/\sS$ acts transitively on $G'$,
and $\sH':=\sH/\sS$ acts quasi-transitively on $G'$.
\item The quotient graph $G'=G/\sS$ satisfies $G' \in \sG$.
\item $\sH'$ is unimodular on $G'$.
\end{letlist}
\end{lemma}

\begin{proof}
(a)  Since $\sS\normal\Ga$ and $\sH \le \Ga$, it suffices to show that $\sS\le\sH$.
Now, $\sS$ is the set of all products of the form $(\g_1\si_1\g_1^{-1})(\g_2\si_2\g_2^{-1})
\cdots (\g_k\si_k\g_k^{-1})$ with $k \ge 0$, $\g_i\in\Ga$, $\si_i\in\Stab_{w_i}$, $w_i\in V$.
Since $\g_i\si_i\g_i^{-1} \in \Stab_{\g_iw_i}$, we have that  $\sS\le\sH$ as required.

\noindent
(b) If $u=\si v$ with $\si\in\Stab_w$, then
\begin{equation*}
\frac  {M(u)}{M(w)} = \frac{|\Stab_u w|}{|\Stab_w u|}
=\frac{|\Stab_{\si v}(\si w)|}{|\Stab_{\si w}(\si v)|}
= \frac{|\Stab_{ v} w|}{|\Stab_{ w} v|}=\frac{M(v)}{M(w)},
\end{equation*}
so that $M(u)=M(v)$. As in part (a), every element of $\sS$ is the product of members of
the stabilizer groups $\Stab_w$, and the claim follows.

\noindent
(c) Let $u,v\in V$, and find $\g\in\Ga$ such that $v=\g u$. Since $\sS\normal \Ga$,
$\sS \g (\sS u)=\sS \g u=\sS v$, so that $\sS \g: \sS u\mapsto \sS v$.
The first claim follows, and the second is similar since $\sH$ acts quasi-transitively
on $G$.

\noindent
(d) Since $M$ is non-constant on the orbit $\sH\id$, there exist $v,w\in\sH\id$ such that
$\xi:=M(w)/M(v)$ satisfies $\mu>1$. Let $\a\in\sH$ be such that $w=\a v$. By \eqref{eq:modf},
$M(\a^k v)/M(v)=\xi^k$, whence the range of $M$ is unbounded. By part (b),
$G'$ is infinite. (The non-constantness of the modular function has been used also in \cite{GHP}.)
The graph $G'$ is connected since $G$ is connected, and  is transitive
by part (c). It is locally finite since its vertex-degree is no greater than that of $G$.

\noindent
(e) It suffices for the unimodularity that, for $u\in V$ and $\ol u:= \sS u$,  we have that
$\Stab_{\ol u}:=\Stab^{\sH'}_{\ol u}$
is a single element, namely the identity element $\sS$ of $\sH'$.
Let $\a\in\sH$ be such that $\sS \a \in \Stab_{\ol u}$.
Then $\sS\a (\sS u)=\a\sS u =\sS u$. Therefore, there exists $s \in \sS$ such that $\a s(u)=u$,
so that $\a s\in\sS$. It follows that $\a\in\sS$, and hence $\sS\a=\sS$ as required.
\end{proof}

Since $M$ is non-constant on $\sH\id$, $F'$ is non-constant on the
orbit of $\sH'$ containing $\sS\id$. By Theorem \ref{prop:indep} applied
to $(G',\Ga',\sH',F')$,
$G'$ has a harmonic, \ughf\ $(\psi',\sH')$ satisfying $\psi'(\sS\id)=0$.
Let $\psi:V \to \ZZ$ be given by $\psi(v)=\psi'(\sS v)$.
We claim that $(\psi,\sH)$ is a \ghf\ on $G$.

Firstly, for $\a\in\sH$,
\begin{alignat*}{2}
\psi(\a v)-\psi(\a u) &=\psi'(\sS\a v) - \psi'(\sS\a u)\\
&=\psi'(\a \sS v) - \psi'(\a \sS u) \q&&\text{since } \sS\normal \sH\\
&=\psi'(\sS v) - \psi'(\sS u) &&\text{since $(\psi',\sH')$ is a \ghf}\\
&=\psi(v)-\psi(u),
\end{alignat*}
whence $\psi$ is \hdi.
Secondly, let $v\in V$, and find $u, w\in \pd v$ such that
$\psi'(\sS u) < \psi'(\sS v) < \psi'(\sS w)$. Then
$\psi(u)<\psi(v)<\psi(w)$, so that $v$ is a point of
increase of $\psi$. Therefore, $(\psi,\sH)$ is a \ghf\ on $G$.

Finally, we give an example in which the above recipe leads to a \ghf\
which is not harmonic. Consider the \lq grandparent graph' introduced
in \cite{Trof} (see also \cite[Example 7.1]{LyP}) and defined as follows.
Let $T$ be an infinite degree-$3$ tree, and
select an \lq end' $\om$. For each vertex $v$, we add an edge to the unique
grandparent of $v$ in the direction of $\om$. Let $\sH$ be the set of automorphisms of the
resulting graph $G$. Note that $\sH$ acts transitively on $G$,
and is non-unimodular. The above recipe yields (up to
a multiplicative constant which we take to be $1$) the \ghf\ on $T$ which
measures the (integer) height of a vertex in the direction of $\om$.
The neighbours of a vertex with height $h$ have average height
$h-\frac 78$, whence $h$ is not harmonic.

\section*{Acknowledgements}

This work was supported in part
by the Engineering and Physical Sciences Research Council under grant EP/I03372X/1.
ZL's research is supported by the Simons Foundation $\#$351813 and the
National Science Foundation DMS-1608896.
Jack Button kindly pointed out an error in an earlier version of this work,
and gave advice on certain group-theoretic aspects. The authors are grateful to
Yuval Peres for his comments on the work, and for proposing
the non-unimodular case of Theorem \ref{cor:nonunim}.

\input{cayley-2.bbl}

\end{document}

redpajama_set_name: RedPajamaArXiv


No twirly skirt for Divine, but I would like one of these in my size.
I like the subtle detail on the scattered pigs.
I love the scattered pigs! She will love it.
you are so creative! they are all just fab!
Did you design the iron on images? Love the pigs and the crayons--so stinking fun! I want circle crayons!
AND I KNOW WHERE THE STORE IS!
Love the pigs in the polk-a dots.
Just this morning I cried about leaving Dongdaemun. And you, my dear, know the perfect way to use all those things in Dongdaemun, and I THINK SO. Love the pigs.
Love it! Ava made crayons this year as well, love the iron on shirts too.


text:
Home > Marvel Comics > Sabretooth to the Supernaturals... > Stand: Hardcases Comics, The
The Stand Hardcases Premiere Hardcover Graphic Novel Stephen King Marvel Comics
price: £17.95 inc vat
The Stand Hardcases Premiere Hardcover Graphic Novel from Marvel Comics. Written by ROBERTO AGUIRRE-SACASA, Penciled by MIKE PERKINS, Cover by TOMM COKER. The Trashcan Man was spared for a reason: to join the Dark Man Randall Flagg's swelling army and to set the world on fire. It's a dream come true for the pyromaniac, but he has to reach Flagg's compound in Las Vegas first - and somehow survive crossing paths with "the Kid," a hardcase who's even crazier and more dangerous than Trashy is! The villains are front-and-center as we cross the mid-point of this epic adaptation of Stephen King's horror masterpiece! Collecting THE STAND: HARDCASES #1-5.

redpajama_set_name: RedPajamaCommonCrawl


A global best-seller, "The Book Thief" tells the story, as narrated by Death, of a German girl in World War II whose foster parents helped hide a young Jewish man. One of the novel's key messages is that everyone is made of words and stories and that without them, we cannot exist. To promote the film version of the book, and to show that it wasn't simply another movie about World War II, 20th Century Fox and Allied Integrated Marketing created a campaign that brought home the importance of words.
In a literal interpretation of the core message, 20th Century Fox ran two consecutive blank pages in The New York Times on Oct. 23, 2013, in effect taking away words from the newspaper's more than 2 million readers. A third page featured just the URL wordsarelife.com. Readers who visited that website viewed a video message from the star of "The Book Thief," Sophie Nélisse, explaining the power of words and storytelling.


text:
An Inconvenient Truth - Al Gore's film on climate change
Linux Club Via Giuseppe Libetta 15c (Ostiense, Metro Garbatella), Roma (RM)
The documentary "An Inconvenient Truth" will be screened in English followed by a discussion as part of the Eyes Wide Open Film Series presented by U.S. Citizens for Peace & Justice - Rome.
Film Synopsis:
Offers a passionate and inspirational look at one man's fervent crusade to halt global warming's deadly progress in its tracks by exposing the myths and misconceptions that surround it. That man is former Vice President Al Gore, who, in the wake of defeat in the 2000 election, re-set the course of his life to focus on a last-ditch, all-out effort to help save the planet from irrevocable change. In this eye-opening and poignant portrait of Gore and his "traveling global warming show," Gore also proves himself to be one of the most misunderstood characters in modern American public life. Here he is seen as never before in the media - funny, engaging, open and downright on fire about getting the surprisingly stirring truth about what he calls our "planetary emergency" out to ordinary citizens before it's too late.
With 2005, the worst storm season ever experienced in America just behind us, it seems we may be reaching a tipping point - and Gore pulls no punches in explaining the dire situation. Interspersed with the bracing facts and future predictions is the story of Gore's personal journey: from an idealistic college student who first saw a massive environmental crisis looming; to a young Senator facing a harrowing family tragedy that altered his perspective, to the man who almost became President but instead returned to the most important cause of his life - convinced that there is still time to make a difference.
Director: Davis Guggenheim
Dec 13, 7pm
Linux Club
Via Giuseppe Libetta 15c
(Ostiense, Metro Garbatella)
Screenings are free, donations appreciated.
For more information, see our web site: http://www.peaceandjustice.it
http://www.peaceandjustice.it/
U.S. Citizens for Peace & Justice
info@peaceandjustice.it


text:
package com.creditease.uav.cache.test;

import java.util.HashMap;
import java.util.Map;
import java.util.concurrent.TimeUnit;

import com.creditease.agent.helpers.ThreadHelper;
import com.creditease.agent.log.SystemLogger;
import com.creditease.uav.cache.api.CacheManager;

public class DoTestMain {

    public static void main(String[] args) {

        SystemLogger.init("DEBUG", true, 5);
        String ip = "localhost:6379";
        CacheManager.build(ip, 0, 0, 0);
        CacheManager cm = CacheManager.instance();
        Map<String, String> hash = new HashMap<String, String>();
        hash.put("age", "1");
        hash.put("name", "zz");
        cm.del("test", "123");
        cm.putHash("test", "123", hash, 3600, TimeUnit.SECONDS);// "1#360121199403085855#774889"

        ThreadHelper.suspend(500);

        Map<String, String> m = cm.getHashAll("test", "123");

        System.out.println(m);
    }

}

redpajama_set_name: RedPajamaGithub


text:
Army spied on lockdown critics: Sceptics, including our own Peter Hitchens, long suspected they were under surveillance. Now we've obtained official records that prove they were right all along – The Mail on Sunday
A shadowy Army unit secretly spied on British citizens who criticised the Government's Covid lockdown policies, The Mail on Sunday can reveal.
Military operatives in the UK's 'information warfare' brigade were part of a sinister operation that targeted politicians and high-profile journalists who raised doubts about the official pandemic response.
They compiled dossiers on public figures such as ex-Minister David Davis, who questioned the modelling behind alarming death toll predictions, as well as journalists such as Peter Hitchens and Toby Young. Their dissenting views were then reported back to No 10.
Documents obtained by the civil liberties group Big Brother Watch, and shared exclusively with this newspaper, exposed the work of Government cells such as the Counter Disinformation Unit, based in the Department for Digital, Culture, Media and Sport, and the Rapid Response Unit in the Cabinet Office.
https://archive.today/2023.01.29-172141/https://www.dailymail.co.uk/news/article-11687675/Army-spied-lockdown-critics-Sceptics-including-Peter-Hitchens-suspected-watched.html
Tags 77th Brigade, Big Brother Watch, British Army, Civil Liberties, Counter Disinformation Unit, COVID-19, Department for Digital Culture Media and Sport, Fourth Generation Warfare, Government, Information Warfare, Journalism, Military, Ministry of Defence, Policy, Rapid Response Unit, Surveillance, The Cabinet Office, The Mail on Sunday, UK, Warfare, Whistleblowers
NHS Doctor confirms Health & Social Care Staff were instructed to let Patients die or "euthanize" them to falsely increase the COVID Death Count while Hospitals were EMPTY – The Expose
An NHS whistleblower, who wishes to remain anonymous, has come forward with allegations that the NHS hospitals were not overwhelmed during the Covid-19 pandemic, as was reported by authorities and the mainstream media.
The whistleblower also confirmed that the little care given throughout the pandemic amounted to negligence, and that the Goverment and NHS bosses essentially instructed staff to let people die, or in some cases kill them through the 'End of Life Care' programme and falsely label the deaths as being due to Covid-19.
https://archive.today/2023.01.21-071551/https://expose-news.com/2023/01/21/nhs-staff-told-to-kill-patients-for-covid/
Tags Accident & Emergency, Care Quality Commission, Collateral Damage, COVID-19, Department of Health and Social Care, Disinformation, DNR, Do Not Attempt Cardiopulmonary Resuscitation (DNACPR), Euthanasia, Excess Deaths, Excess Mortality, Government, Hospitalisation, Human Rights, Hysteria, Midazolam, Misinformation, National Institute for Health and Care Excellence (NICE), NG163, NG191, NHS, NHS England, Policy, Propaganda, Statistics, The Expose, UK, Whistleblowers
Neil Oliver Interviews Dr. John Campbell – Dr. John Campbell
Dr. John Campbell is an internet sensation, giving facts & data to help us navigate healthcare in the present day. The interview covers John's development, work, rise to fame & what he would do to help get healthcare back in shape.
Tags Absolute Risk Reduction, Age Groups, Cause of Death, Censorship, Collateral Damage, Conflicts of Interest, COVID-19, Cycle Threshold, Dr. John Campbell, Dr. Pierre Cory, Dr. Tess Lawrie, Early Treatment Protocols, Education, Efficacy, Emergency Use Authorization, Errors, Excess Deaths, Excess Mortality, Financial Incentives, Government, Healthcare, Heart Problems, Hydroxychloroquine, Immunity, Investigation, Ivermectin, Misinformation, Moderna, mRNA, Myocarditis, NHS, Nursing, Omicron Variant, PCR, Peer-Review, Pharmaceutical Industry, Policy, Prednisolone, Publication Bias, Randomized Controlled Trials, Relative Risk Reduction, Risk, Save the NHS, Side-effects, Spike Protein, Statistics, Steroids, Testing, Training, Treatment, UK, Vaccination, Vaccine Damage, Variants
The problem with the BBC's reporting on excess deaths – The Spectator
I recall the newsroom conversations during the dark days of the pandemic only too well. They were upsetting at the time. Now, as we see a disturbing rise in excess deaths across the country, the thought of them fills me with horror and outrage.
'You do realise these lockdowns and restrictions will end up killing people too, don't you?' I would say to senior editorial colleagues with something approaching desperation in my voice. 'Sure, the virus is a serious threat to a small proportion of the population but the longer-term consequences of shutting the economy down and closing off the NHS will be deadly for huge numbers who were never at serious risk from the virus, people with years of life ahead of them. Shouldn't we be reflecting that in our coverage? Shouldn't we be considering the possibility that the government is going down the wrong path on this?'
The response of these colleagues would vary in tone, from patient but patronising good humour to open mockery. Many were influenced, I believe, by social media echo chambers (curated by pernicious algorithms). My colleagues had swallowed the myopic belief, adopted by people largely on the liberal left, that only lockdowns could 'save lives' and 'protect the NHS' from the devastation threatened by Covid-19. Anyone who demurred was, as far as they were concerned, clearly a right-wing lunatic.
Now we can all see how well that is working out. Provisional figures released this week reveal that more than 650,000 deaths were registered in the UK in 2022 – 9 per cent more than 2019. This is one of the largest excess death levels outside the pandemic in 50 years. But despite many of the causes of this being obvious, the BBC is pretending the development has come as something of a shock.
https://web.archive.org/web/20230115091426/https://www.spectator.co.uk/article/the-problem-with-the-bbcs-reporting-on-excess-deaths/
Tags BBC, Censorship, Collateral Damage, COVID-19, Disinformation, Dr. Aseem Malhotra, Government, Journalism, Lockdown, Misinformation, NHS, Policy, Save the NHS, The Spectator, UK
Is this the true, devastating cost of lockdown? – The Telegraph
Up to 100 times more may have been spent on preventing each Covid death than on preventing each non-Covid death
http://archive.today/2023.01.12-192720/https://www.telegraph.co.uk/news/2023/01/12/true-devastating-cost-lockdown/
Tags Collateral Damage, COVID-19, Economy, Government, Lockdown, National Institute for Health and Care Excellence (NICE), NHS England, Policy, Professor Sunetra Gupta, Public Health, Quality Adjusted Life Year, SAGE, The Telegraph, UK
How Stanford Failed the Academic Freedom Test – Dr. Jay Bhattacharya
Academic freedom at Stanford is clearly dying. It cannot survive if the administration fails to create an environment where good-faith discussions can occur outside of a framework of ideological rigidity and the false certainties that ideologues—and governments—wish to impose on us. Stanford missed the opportunity to sponsor COVID policy forums and it deplatformed dissenting voices. Several prominent faculty exploited this environment, engaging in actions that directly violated basic academic norms.
http://archive.today/2023.01.12-115607/https://www.tabletmag.com/sections/arts-letters/articles/stanford-failed-academic-freedom-test
Tags Academia, CDC, Censorship, COVID-19, Democracy, Dr. Jay Bhattacharya, Freedom, Government, Harvard University, Lockdown, Oxford Universtity, Policy, Press, Professor Martin Kulldorff, Professor Sunetra Gupta, SARS-CoV-2, Stanford University, Sweden, The Great Barrington Declaration, USA
Boris Johnson was secretly 'nudged' into wearing a Covid mask by government scientists – The Telegraph
Boris Johnson was secretly 'nudged' into wearing a facemask during the pandemic, according to the head of the government's Behavioural Insights Team.
http://archive.today/2023.01.05-214830/https://www.telegraph.co.uk/global-health/science-and-disease/boris-johnson-secretly-nudged-wearing-covid-mask-government/
Tags Applied Behavioural Psychology, Behavioural Insights Unit, COVID-19, Face Masks, Government, MINDSPACE, Nudge Unit, Policy, Professor David Halpern, Reckitt Global Hygiene Institute, SAGE, The Telegraph, UK
NHS managers are gaslighting the British public – The Telegraph
[T]he British people must do something that goes against the grain. We must break free of the coercive control exercised over us by the NHS. We are not to blame for putting it under pressure. We are not at fault for expecting a feverish child or a 90-year-old with a broken hip to receive prompt attention. The NHS is to blame.
https://archive.today/2023.01.04-125118/https://www.telegraph.co.uk/columnists/2023/01/03/anyone-could-do-better-job-nhss-useless-irresponsible-managers/
Tags Collateral Damage, COVID-19, Economy, Gaslighting, Government, Healthcare, Mismanagement, NHS, Policy, Propaganda, Save the NHS, Taxpayers, UK
The Shadowy Group That Brought Mask Mandates to the U.K. – Dr. Gary Sidley, The Daily Sceptic
In our supposedly liberal and transparent democratic country, it seems that it was a group of remote and unnamed scientists, outside of the formal SAGE infrastructure, who effectively imposed masks on British citizens. Were they well-meaning academics offering their expertise, or conflicted ideologues engaged in a global endeavour to control the masses? You decide.
https://dailysceptic.org/2022/12/22/the-shadowy-group-that-brought-mask-mandates-to-the-u-k/
Tags Advertising Standards Agency, COVID-19, Data Evaluation and Learning for Viral Epidemics, DELVE, Dr. Gary Sidley, Face Masks, Government, Policy, SAGE, The Daily Sceptic, The Royal Society, UK
Megacities on the Move – Plannedopolis – Forum for the Future
In a world of fossil fuels and expensive energy, the only solution is tightly planned and controlled urban transport.
This video released on the animator's channel on 23 September 2011. A full archive of the Forum for the Future article published 25 Mar 2010 can be found below:
Megacities on the move
For the first time in history more than half the world's population is living in towns and cities. We passed this milestone in 2008 and by 2040 two thirds of us are expected to live in urban environments.
Urbanisation presents us with a wealth of new opportunities and huge challenges. It has the potential to further economic development and innovation, but also threatens to exacerbate key global problems, including resource depletion, climate change, and inequality.
Megacities on the move sets out to find solutions to one of the biggest challenges – how billions of city-dwellers can access what they need without putting intolerable strains on the planet.
Our definition of mobility embraces all the ways in which people will access goods, services and information and make contact with each other. It ranges from transport to ICT solutions and innovative urban design.
The project is a multi-stakeholder futures initiative designed to help policy makers, industry and civil society organisations, such as NGOs and community groups, understand the complex challenges that lie ahead and to collaborate to find sustainable solutions.
Forum for the Future, working in partnership with Vodafone, the FIA Foundation and EMBARQ, will develop a set of four global scenarios in order to highlight cutting-edge thinking about mobility, identify future challenges and explore sustainable solutions designed for cities.
The scenarios will be piloted in spring of 2010 in workshops held in Mumbai and Istanbul, with a diverse audience of mobility thinkers and practitioners – including public policy decision-makers, planners, automotive companies, architects and technology solution providers.
We will analyse the specific implications of the scenarios for each city, and examine the challenges and opportunities which lie ahead. Our aim is to jointly develop a vision for sustainable mobility in each city, and draft strategic action plans to address key issues.
For more information read a presentation introducing Sustainable Urban Mobility 2040 or a summary of the project.
The project launch and publication is set for November 2010. Stay tuned for further details.
The Megacities on the Move team is back in town after successful city workshops! The first one took place in Mumbai on June 2-3rd, and the second in Istanbul on June 16-17. We had a rich, creative process, with the participants using the SUM2040 scenarios to identify key mobility risks and opportunities facing their cities in the future. They also developed a number of prototype solutions with the aim of creating more sustainable mobility systems in the cities. A big thank you to everyone who came out and contributed their creative thinking to this process! We will be publishing the results of the two workshops along with the rest of the project outputs.
The project was recently profiled at Sustainable City Finance, a conference hosted by the Urban Age Institute in New York in January. Sheila Watson, Director of Environment at the FIA Foundation spoke about the key objectives of the project and the upcoming workshops in Mumbai and Istanbul.
Chris Dewey, Associate
Rupert Fausset, Principal Strategic Advisor
Ivana Gazibara, Senior Strategic Advisor
James Goodman, Head of Futures
Clare Jenkinson, Strategic Advisor
Peter Madden, CEO
Nathalie Nathe, Admin and Events Assistant
Ivana Gazibara, [email protected], +44 (0)207 324 3673
Screenshot of archive.org archive 1 Nov 2010
https://web.archive.org/web/20101101095626/https://www.forumforthefuture.org/projects/megacities-on-the-move
Tags 15-minute City, Agenda 2030, Climate Change, Economy, Food, Forum for the Future, Government, Megacities, Plannedopolis, Policy, Sustainable Development Goals, Technocracy, The Great Reset, Traffic, Urban Farming, Urban Planning
Publications Videos
Catastrophic Contagion – Center for Health Security
The Johns Hopkins Center for Health Security, in partnership with WHO and the Bill & Melinda Gates Foundation, conducted Catastrophic Contagion, a pandemic tabletop exercise at the Grand Challenges Annual Meeting in Brussels, Belgium, on October 23, 2022.
The extraordinary group of participants consisted of 10 current and former Health Ministers and senior public health officials from Senegal, Rwanda, Nigeria, Angola, Liberia, Singapore, India, Germany, as well as Bill Gates, co-chair of the Bill & Melinda Gates Foundation.
The exercise simulated a series of WHO emergency health advisory board meetings addressing a fictional pandemic set in the near future. Participants grappled with how to respond to an epidemic located in one part of the world that then spread rapidly, becoming a pandemic with a higher fatality rate than COVID-19 and disproportionately affecting children and young people.
Participants were challenged to make urgent policy decisions with limited information in the face of uncertainty. Each problem and choice had serious health, economic, and social ramifications.
http://archive.today/2022.12.10-100729/https://www.centerforhealthsecurity.org/our-work/exercises/2022-catastrophic-contagion/index.html
Tags Bill & Melinda Gates Foundation, Biosecurity, Catastrophic Contagion, Center for Health Security, COVID-19, Economy, Government, Grand Challenges Annual Meeting, Johns Hopkins University, Pandemic Treaty, Pandemics, Policy, Response, SEERS, Simulations, Wargaming, WHO
Discriminatory Attitudes Against the Unvaccinated During a Global Pandemic – Nature
Post date 8 December 2022
During the COVID-19 pandemic sizeable groups of unvaccinated minorities persist even in countries with high vaccine access1. Consequently, vaccination became a controversial subject of debate and even protest2. Here, we assess whether people express discriminatory attitudes in the form of negative affect, stereotypes and exclusionary attitudes in family and political settings across groups defined by COVID-19 vaccination status. We quantify discriminatory attitudes between vaccinated and unvaccinated citizens in 21 countries, covering a diverse set of cultures across the world. Across three conjoint experimental studies (N=15,233), we demonstrate that vaccinated people express discriminatory attitudes towards the unvaccinated, as high as the discriminatory attitudes suffered by common targets like immigrant and minority populations3,4.5. In contrast, there is an absence of evidence that unvaccinated individuals display discriminatory attitudes towards vaccinated people, except for the presence of negative affect in Germany and United States. We find evidence in support of discriminatory attitudes against the unvaccinated in all countries except Hungary and Romania and find that discriminatory attitudes are more strongly expressed in cultures with stronger cooperative norms. Prior research on the psychology of cooperation has shown that individuals react negatively against perceived free-riders6,7 including in the domain of vaccinations8,9. Consistent with this, the present findings suggest that contributors to the public good of epidemic control (i.e., the vaccinated) react with discriminatory attitudes against perceived free-riders (i.e., the unvaccinated). Elites and the vaccinated general public appealed to moral obligations to increase COVID-19 vaccine uptake10,11 but the present findings suggest that discriminatory attitudes including support for the removal of fundamental rights simultaneously emerged.
http://archive.today/2022.12.09-160919/https://www.nature.com/articles/s41586-022-05607-y
Tags Collateral Damage, COVID-19, Discrimination, Government, Policy, Studies, Vaccination
Blackouts will trigger a people's revolt against the new eco-tyranny – The Telegraph
We are nearing a turning point for democratic support for environmentalism. Gordon Brown's 2008 Climate Change Act legislated to slash CO2 emissions by 80 per cent by 2050, a seismic shift pushed through with little debate but much superficial public approval. Theresa May strengthened this to 100 per cent by 2050, the "net zero" target; again, the public liked the sound of this, if not of Mrs May. China will continue to increase its emissions by more than we cut ours, but our entire ruling class has signed up to this iron-clad legal framework, with no dissent tolerated.
http://archive.today/2022.12.08-192346/https://www.telegraph.co.uk/news/2022/12/07/blackouts-will-trigger-peoples-revolt-against-new-eco-tyranny/
Tags Blackouts, Carbon Neutral, China, Climate Change, Climate Change Act, Deindustrialisation, Dictatorship, Emissions, Energy, Energy Crisis, Food, Germany, Government, Net Zero, New Normal, Policy, Sustainable Cities, Sustainable Development Goals, The Great Reset, The Netherlands, UK, Zero Carbon
The Future of Urban Consumption in a 1.5°C World – Arup
C40 is delighted to publish this pioneering piece of thought leadership, The Future of Urban Consumption in a 1.5°C World. The report demonstrates that mayors have an even bigger role and opportunity to help avert climate emergency than previously thought. But to grasp that opportunity, city leaders need to be even more entrepreneurial, creating and shaping markets and engaging in sectors that may not previously have been considered within the domain of city government, and working out how to support their citizens and businesses in achieving a radical, and rapid, shift in consumption patterns.
http://archive.today/2022.12.05-135640/https://www.arup.com/perspectives/publications/research/section/the-future-of-urban-consumption-in-a-1-5c-world
Tags Agenda 2030, Arup, Aviation Industry, C40, Climate Change, Consumption, Deadline 2020, Dietary Change, Emissions, Food, Food Industry, Global Warming, Government, Greenhouse Gas, IPCC, Low Carbon, Mayors, Meat Consumption, National Government Commitments, Policy, Private Sector, Sustainable Cities, Sustainable Development Goals, The Great Reset, Transportation, University of Leeds, Urban Planning, US Climate Action Partnership
NHS creates £1.3bn pot for Covid compensation claims – The Telegraph
The NHS has set aside £1.3 billion to cope with compensation claims arising from the pandemic this year with claims for treatment delays, cancellations and misdiagnosis expected.
An annual report from NHS Resolution, which deals with patient disputes, shows that the health service anticipates it will need to pay out more than a billion pounds this financial year to settle claims arising from poor service during Covid.
http://archive.today/2022.11.28-210431/https://www.telegraph.co.uk/news/2022/11/28/nhs-creates-13bn-pot-covid-compensation-claims/
Tags Cancellations, Cancer, Collateral Damage, Compensation, COVID-19, Dr. Charles Levinson, Government, Lockdown, Misdiagnosis, NHS, NHS Resolution, Policy, Save the NHS, The Telegraph, UK, Waiting Lists
The nationalised NHS model was doomed from the very start – The Telegraph
If demand for health care is nearly unlimited and cannot be rationed by price, it must be rationed in some other way. The NHS rations through shortages – of staff – and waiting lists. Additional rationing is achieved by those who "do not want to bother the doctor", often at cost to their health, while large numbers go private, so creating the two-tier health system Bevan hoped to avoid – though he did not use an NHS GP but rather Sir Daniel Davies, physician to George VI.
http://archive.today/2022.11.29-042954/https://www.telegraph.co.uk/news/2022/11/28/nationalised-nhs-model-doomed-start/
Tags Economics, Government, Health, History, NHS, Policy, Save the NHS, Socialism, The Telegraph, UK
The Macroeconomics of De-Cashing – IMF
The paper presents a simple framework for the analysis of the macroeconomic implications of de-cashing. Defined as replacing paper currency with convertible deposits, de-cashing would affect all key macroeconomic sectors. The overall macreconomic impact of de-cashing would depend on the balance of growth-enhancing and growth-constraining factors. Starting from a traditional saving-investment balance, the paper develops a four-sector macroeconomic framework. It is purely illustrative and is designed to provide a roadmap for a systematic evaluation of de-cashing. The framework is disaggregated into the real, fiscal, monetary, and external sectors and potential implications of de-cashing are then identified in each sector. Finally, the paper draws a balance on possible positive and negative macroeconomic implications of de-cashing, and proposes policies capable of augmenting its economic and social benefits, while reducing potential costs.
http://archive.today/2022.11.25-093919/https://www.imf.org/en/Publications/WP/Issues/2017/03/27/The-Macroeconomics-of-De-Cashing-44768
Tags Cashless Society, Economy, Government, International Monetary Fund, Macroeconomics, Policy, The Great Reset
Traffic filters will divide city into six "15 minute" neighbourhoods, agrees highways councillor – Oxford Mail
ROAD blocks stopping most motorists from driving through Oxford city centre will divide the city into six "15 minute" neighbourhoods, a county council travel chief has said.
And he insisted the controversial plan would go ahead whether people liked it or not.
http://archive.today/2022.11.25-110055/https://www.oxfordmail.co.uk/news/23073992.traffic-filters-will-divide-city-six-15-minute-neighbourhoods-agrees-highways-councillor/
Tags 15-minute City, Agenda 2030, Climate Change, Economy, Government, Oxford, Oxford Mail, Policy, Sustainable Development Goals, The Great Reset, Traffic, UK, Urban Planning
The lockdown cancer wave has only just begun – Professor Karol Sikora, UnHerd
Advocates for harsh Covid measures are finally waking up to what they have done.
The underreported story of the entire pandemic is excess deaths — not from Covid, but from other health conditions which were so brutally pushed to one side. There have been huge rises in the number of people dying from causes unrelated to the virus, accelerating throughout the year and showing no signs of slowing down.
http://archive.today/2022.11.21-081435/https://unherd.com/thepost/the-lockdown-cancer-wave-has-only-just-begun/
Tags Cancer, Collateral Damage, COVID-19, Diabetes, Equitix, Excess Deaths, Government, Lockdown, Medical Experts, NHS Digital, Policy, Professor Karol Sikora, Testing, UK, UnHerd


This diner has been open for 14 yrs. located in a busy shopping area. They are open for breakfast, lunch and early dinner. The food is home cooked quality with a very welcoming staff. The place needs a little face lift nothing a little paint can't fix. The owner is ready to retire...he said to bring all offers-priced for a quick sale. Call Pete Sheehan 516-639-2465.


Sure, with apple,when you bought insurance coverage, you are able to switch up till 2 hours earlier than the airplane departure, I think. Apple will care for all of the flights, transfers and resort arrangements for simply the 2 of you – or for your whole group, and we will put you instantly in touch with the resort to plan and ebook the marriage bundle of your alternative. We always take pleasure in ourselves when we go to Punta Cana and we were anticipating how way more fun it could be this 12 months now that Gray was old enough to play and run around and that Gem was in a position to crawl and dance.
All inclusive trip packages low cost last minute vacations and all inclusive trip packages combine pleasure spontaneity and best of all savings. Horseback riding holidays at Brush Creek Ranch provide adventure for all ages and talent levels. Apple Holidays Resort Representatives present the very best degree of personalised service to customers, and consumer satisfaction is carefully monitored via Vacation Satisfaction Questionnaires.
Horseback riding holidays at c lazy u are the final word colorado household trip our luxurious dude ranch and 5. Asheville horseback driving vacations in western north carolina at clear creek guest ranch horseback using trails within the pisgah national forest.
Unfortunately, latest protection has inaccurately characterized the extent of the problem in Mexico, unfairly dramatizing the dangers to vacationers, and disparaging the various safeguards that the Mexico tourism business. I signed as much as obtain the airline's electronic mail promotions, making me eligible for a monthly drawing to decide on the winner of a free Apple trip.
Never in one million years would I've thought to use flour tortillas to make apple dumplings. Today I talked about why it is best to e book your trip now, some nice offers on the market and offered tips on maximizing your trip. Apple has been bringing passengers to the glorious white sand beaches of Punta Cana since its tourism beginnings.


Scratched Build Skin of Alen Berg TicTac BMW, Team Tauber, from the 1991 Season. Including Driver-Suit, Helmet, painted windows, crew-board and internal glass.
Orange4000, Epistolarius, karina-moskva and 3 others like this.
Possibly in my top 5 of all time liveries. Many thanks for this.


What Constitutes a Quid Pro Quo or Token Gift for Charitable Contributions?
January 1, 2019. There's no need to tell donors to subtract out the value of return gifts to them that were merely "token" items--but how much is that?
January 19, 2018. The IRS has made changes to Form 1023-EZ application.
January 1, 2018. Here are some of the most important changes for nonprofits under the new tax law.


Well! Detachable sleeves are the best proposition. A person can enjoy two outfits in the price of one. Worth it material used to make this outfit perfect. My friend really likes the jacket when I wear, he told me he will get the same from your site but he couldn't do so because of shortage of money!


It seems Halo Wars is set to become the new Crackdown, with Microsoft enticing franchise fence sitters with goodies from Halo 3. The real-time strategy game set within Bungie's sprawling, hoop-filled universe will arrive this forthcoming February in both regular form and a Limited Collector's Edition. The latter should prove tempting even to those who prefer to see their aliens on the other side of a crosshair, as it will include early access to Halo 3's Mythic multiplayer map pack.
The three maps, named "Assembly," "Orbital" and "Sandbox," join Halo Wars: Genesis, a graphic novel by Phil Noto, Graeme Devine and Eric Nylund; a "unique" in-game Wraith vehicle; six leader cards (they "possess elite skills which can turn the tides of battle," it says here) and a "Spirit of Fire" patch, which your mom can sew onto your favorite jacket. Just be sure to pay her back the $79.99 as well.


text:
#winning
Posted by Melissa McEwan at Wednesday, November 07, 2012
WIN: President Barack Obama was reelected.
WIN: Relatedly, terrible plutocrat Mitt Romney lost.
WIN: Voters in Maryland, Maine, and Washington (projected) break a 32-state losing streak and legalize same-sex marriage by popular vote.
WIN: Voters in Minnesota vote down a proposed amendment to the state constitution that would have defined marriage as a union between one man and one woman.
WIN: The 113th Congress will have a record number of female senators: 20. Sixteen of them are Democrats; four are Republicans.
WIN: Elizabeth Warren (MA) is one of them. Elizabeth Warren won.
WIN: Mazie Hirono (HI) is also one of them. She is the first Asian American female senator!
WIN: Tammy Baldwin (WI) is also one of them, defeating popular Republican Tommy Thompson in the Wisconsin Senate race, making her both Wisconsin's first female senator and the first openly gay member of the US Senate. Said Baldwin: "I am well aware that I will have the honour to be Wisconsin's first woman US senator, and I am well aware that I will be the first openly gay member of the United States Senate, but I didn't run to make history. I ran to make a difference. But in choosing me to tackle those problems the people of Wisconsin have made history."
WIN: Claire McCaskill (MO) is also one of them, defeating the loathsome anti-choice rape apologist Todd Akin.
WIN: In Indiana, Democrat Joe Donnelly beat the loathsome anti-choice rape apologist Richard Mourdock in their race for the US Senate.
WIN: In Washington, Democrat Suzan DelBene beat the loathsome anti-choice rape apologist John Koster in their race for seat in the state congress.
WIN: In Illinois, Tammy Duckworth beat the loathsome anti-choice rape apologist Joe Walsh in their race for the US House of Representatives. Duckworth, who lost both legs serving in Iraq, is the first female war veteran with disabilities elected to the US Congress.
WIN: US voters chose women of color, women with disabilities, women who are gay, pro-choice women, and rejected men who minimize rape.
WIN: New Hampshire became the first state in the nation to have an all-female Congressional delegation: Two female senators and two female representatives. Their newly elected governor is also a woman.
WIN: Colorado and Washington voters legalized marijuana for recreational use. Massachusetts voters legalized marijuana for medical use.
WIN: A majority of US voters support legal abortion, and voted accordingly.
WIN: A majority of US voters rejected bigotry and reelected our African American President.
WIN: US voters rejected voter suppression efforts and often withstood long lines and shady disincentives to vote. vote. vote.
That is not a comprehensive list. And there were also some disappointments. Some state initiatives passed that are shitty. Indiana's new governor is Mike Pence, and we certainly aren't the only state who elected a gross conservative to a statehouse or Congress in a hard-won race.
But it was a big night. A big night for progress. A broad mandate.
I hope the President hears the roar of his progressive base. I hope he knows what a difference it made when he spoke out in favor of marriage equality. I hope he governs like a person who won because of people who expect more.
I hope we all muster the strength and sustain the will to urge him ever forward.
Labels: 2012, democrats, Marriage Equality, Obama, progressives, Whoops Your Political Philosophy Is Garbage, women