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.