**Tema 10: Poincaré series**

A fairly good reference for principle ideas in this section (see results stated in [Section 2.5, Knieper '02] is also [Nicholls '89]. Let $X$ be a metric space (we have in mind a Hadamard manifold). 
Let $\Gamma$ be a discrete infinite subgroup of the group of isometries acting on $X$. 
In particular
\[
	d(\gamma p,\gamma q)
	=d(p,q)
	\quad\text{ for all }\gamma\in\Gamma, p,q\in X.
\]
The action of $\Gamma$ on $X$ is //properly discontinuous//, that is, for any $K\subset X$ compact the set $\{\gamma\in\Gamma\colon \gamma K\cap K\ne\emptyset\}$ is finite.

Given $p,q\in X$ and $s\in\mathbb R$, consider the //Poincaré series//
\[
	P(s,p,q)
	:= \sum_{\gamma\in\Gamma}e^{-sd(p,\gamma q)}.
\]
Given $k\in\mathbb N$, let
\[
	a_k(p,q)
	:= {\rm card}\{\gamma\in\Gamma\colon k-1\le d(p,\gamma q)<k\}
\]
and define 
\[
	\delta(p,q)
	:= \limsup_{k\to\infty}\frac1k\log a_k(p,q).
\]

**Lemma 1.** $\delta$ is independent on $p$ and $q$.

**Proof.** Observe that by the triangle inequality, it holds
\[
	d(p,\gamma q)-d(p,q)
	\le d(q,\gamma q)
	\le d(p,\gamma q)+d(p,q).
\]
Analogously,
\[\begin{split}
	d(p,\gamma q)-d(q,p)
	&= d(p,\gamma q)-d(\gamma q,\gamma p)\\
	&\le d(p,\gamma p)\\
	&\le d(p,\gamma q)+d(\gamma q,\gamma p)
	= d(p,\gamma q)+d(q,p).
\end{split}\] 
It hence follows that $\delta(p,q)$ only depends on $\Gamma$ and not on $p,q$.

The number $\delta=\delta(\Gamma)$ is called //critical exponent// for $\Gamma$. 

**Remark 1.**
The main construction principle for conformal measures in [Denker, Urbanski '91] is the simple fact (apply the root test to check for convergence of a power series) that for a sequence $(A_k)_{k\ge1}$ of real numbers the //transition parameter//
\[
	c:=\limsup_{k\to\infty}\frac1kA_k
\]	 
is uniquely defined by the fact that the series $\sum_{k\ge1}e^{A_k-ks}$ converges for $s>c$ and diverges for $s<c$. For $s=c$ the series may converge or diverge. Moreover, if $c>0$ then
\[
	c
	:= \limsup_{k\to\infty}\frac1kA_k
	= \limsup_{k\to\infty}\frac1k\sum_{\ell=0}^{k-1}A_\ell.
\]
A trivial, but sometimes useful, remark is that, in the case when $c=0$, the sequence $(B_k)_{k\ge1}$ given by $B_k:= A_k+kd$ for any $d>0$ has as critical exponent $d>0$.

**Lemmq 2.** $P(s,p,q)$ converges for $s>\delta(\Gamma)$ and diverges for $s<\delta(\Gamma)$.

**Proof.** Note that
\[
	a_k(p,q)e^{-s(k-1)} 
	\le \sum_{\gamma\in\Gamma\colon k-1\le d(p,\gamma q)<k}e^{-sd(p,\gamma q)}
	< a_k(p,q)e^{-sk}. 
\]	
Applying Remark 1 to $A_k:= \log a_k$ implies the claim. 

The group $\Gamma$ is of //divergence type// if the Poincaré series diverges at $s=\delta(\Gamma)$.

**Lemma 2.** Assume that $X/\Gamma$ is a compact manifold of nonpositive curvature. The Poincaré series is of divergence type if and only if
\[
	\int_0^\infty e^{-\delta(\Gamma)r}{\rm vol} B_r(p)\,dr
\]	
diverges for one, and hence for every, $x\in X$.

**Proof.** Let $D\subset X$ be a fundamental domain for $\Gamma$. Then, by definition, $D$ is the set which contains from any orbit $\{\gamma p\colon\gamma\in\Gamma\}$ exactly one point.  Hence
\[
	\int_D\sum_{\gamma\in\Gamma}e^{-sd(p,\gamma x)}\,d{\rm vol}(x)
	= \int_Xe^{-sd(p,x)}\,d{\rm vol}(x)
	= \int_0^\infty e^{-sr}{\rm vol} B_r(p)\,dr.
\]
This implies the claim.

The above criterium is particularly interesting taking into account a result by [Manning '79] which, assuming that $(M,g)$ is a compact Riemannian manifold and $X$ its universal covering space, states that the limit
\[
	h(g)
	:= \lim_{r\to\infty}\frac1r\log{\rm vol} B_r(p)
\] 
exists and is independent of $p\in X$, and called //volume entropy// of $(M,g)$. It is intimately related to the //topological entropy// $h_{\rm top}(\phi)$ of the geodesic flow $\phi=\{\phi^t\}_t$ of $SM$ by
\[
	h(g)
	\le h_{\rm top}(\phi).
\] 
If, moreover, $(M,g)$ has nonpositive curvature then 
\[
	h(g)
	= h_{\rm top}(\phi).
\] 
[Freire, Mañé '82] extended this result to manifolds without conjugate points.