Patterson–Sullivan Measures and Growth of Relatively Hyperbolic Groups

Abstract

We prove that for a relatively hyperbolic group, there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of the group. Under natural assumptions, a similar result holds for the critical exponent of a cusp-uniform action of the group on a hyperbolic metric space. As a corollary, we obtain that the critical exponent of a torsion-free geometrically finite Kleinian group can be arbitrarily approximated by those of proper quotient groups. This resolves a question of Dal’bo–Peigné–Picaud–Sambusetti. Our approach is based on the study of Patterson–Sullivan measures on Bowditch boundary of a relatively hyperbolic group and gives a series of results on growth functions of balls and cones.

Appendices

Appendix A: Uniqueness and Ergodicity of PS-Measures

This appendix is devoted to the proof of the uniqueness and ergodicity of PS-measures (in fact, for any quasi-conformal density without atoms). The results proven here will not be used in this paper. We first state a variant of Vitali covering lemma.

Let $B \in {\mathcal {B}}$ be a family of subsets in a space X, to which an extension $E(B) \supset B$ in X is assigned. Then ${\mathcal {B}}$ is called hierarchically locally finite if there exist a positive integer $N>0$ and a disjoint decomposition ${\mathcal {B}}=\bigsqcup {\mathcal {B}}_n$ for $n\ge 0$ such that the following two properties hold:

• (1) Given $B\in {\mathcal {B}}_n$ for $n\ge 0$, the set ${\mathcal {N}}(B)$ of $B'\in {\mathcal {B}}_{n-1} \cup {\mathcal {B}}_n\cup {\mathcal {B}}_{n+1}$ with $B'\cap B\ne \emptyset$ is of cardinality at most N,  where ${\mathcal {B}}_{-1}{{:}{=}}\emptyset$.

• (2) Given $n-m\ge 2$, if $B'\in {\mathcal {B}}_n$ intersects $B\in \mathcal B_m$, then $B'$ is contained in the extension E(B).

By definition, any subfamily of ${\mathcal {B}}$ is also hierarchically locally finite.

Lemma A.1

Let ${\mathcal {B}}$ be a hierarchically locally finite family of subsets in a space X. Then there exists a sub-family ${\mathcal {B}}' \subset {\mathcal {B}}$ of pairwise disjoint subsets such that the following holds

$$\begin{aligned} \bigcup \limits _{B \in {\mathcal {B}}} B \subset \bigcup _{A \in \mathcal B'} \left( \bigcup _{B\in {\mathcal {N}}(A)}E(B) \right) . \end{aligned}$$

(A.1)

Proof

A sequence ${\mathcal {A}}_n$, with $A_n \subset {\mathcal {B}}_n$, is defined inductively as follows. First, choose a maximal disjoint sub-family ${\mathcal {A}}_0$ of ${\mathcal {B}}_0$. Assuming that ${\mathcal {A}}_{n-1}$ is defined, let ${\mathcal {A}}_{n}$ be a maximal disjoint sub-family of ${\mathcal {B}}_n$ such that for each $B\in {\mathcal {A}}_{n}$, we have

$$\begin{aligned} B\cap \left( \bigcup _{0\le m< n} \{A: A\in {\mathcal {A}}_m \}\right) =\emptyset . \end{aligned}$$

(A.2)

We claim that ${\mathcal {B}}' {{:}{=}} \bigcup _{n=0}^\infty {\mathcal {A}}_n$ satisfies the conclusion. Indeed, consider $B\in {\mathcal {B}}$, and say $B\in {\mathcal {B}}_n$, for definiteness. If the property (A.2) holds for $B\in {\mathcal {B}}_n$, then by the maximality of ${\mathcal {A}}_{n}$, there exists some $A\in \mathcal A_n$ such that $B\cap A\ne \emptyset$ so $B\in {\mathcal {N}}(A)$. Thus, B is contained in the right-hand union of (A.1).

Otherwise, assume that (A.2) is false: there exists some $A\in {\mathcal {A}}_m$ for $m<n$ such that $B\cap A\ne \emptyset$. So the second property of the above definition implies $B\subset E(A)$, where $A\in {\mathcal {B}}'$. In this case, the inclusion (A.1) holds. The proof is thus complete. $\square$

Construction of Hierarchically Locally Finite Family

We fix $r_0 = \varphi (\kappa /2)$ and $r_1 = \varphi (\kappa /4)>r_0$, where $\kappa =\kappa (\epsilon , R)>0$ is given by Lemma 2.15 and $\varphi$ given by Lemma 2.12. Let $\lambda >1$ be given by Theorem 2.13.

For each conical point $\xi \in \partial ^c{G}$, we make a choice of a geodesic $[1, \xi ]$. For an $(\epsilon , R)$-transition point v on $[1, \xi ]$, we have $\rho _v(1, \xi ) \ge \kappa$ by Lemma 2.15. Let $B_v(\xi ){{:}{=}}\{\eta \in \partial ^c{G}: \rho _v(\xi , \eta )< \kappa /2\}$ be the open ball centered at $\xi \in \partial ^c{G}$. So $\rho _v(1, \eta )\ge \kappa /2$ and by Lemma 2.12, the shadow $\Pi _{r_1}(v)$ contains $B_v(\xi )$, which is defined to be the extension of $B_v(\xi )$. By Lemma 4.11, we have

$$\begin{aligned} \mu _1(\Pi _{r_1}(v)) \asymp \mu _1(B_v(\xi )) \asymp \exp (-\delta _{G} d(1,v)). \end{aligned}$$

(A.3)

Denote by ${\mathcal {B}}$ the family of all such $B_v(\xi )$ where $\xi \in \partial ^c{G}$ and v is an $(\epsilon , R)$-transition point on $[1, \xi ]$. For a fixed $L>0$, a sequence of sub-families ${\mathcal {B}}_n$ for $n\ge 0$ is defined as follows,

$$\begin{aligned} {\mathcal {B}}_n=\{B_v(\xi )\in {\mathcal {B}}: \xi \in \partial ^c{G};\; nL\le d(1, v)< (n+1)L\}. \end{aligned}$$

Lemma A.2

There exists a uniform constant $L>0$ such that the so-defined ${\mathcal {B}}$ as above is a hierarchically locally finite family over $\partial {G}$.

Proof

We prepare some relations. Consider $B_v(\xi )\in {\mathcal {B}}_n$ for $n\ge 1$ and an arbitrary $B_w(\zeta ) \in {\mathcal {B}}$. By definition, v and w are $(\epsilon , R)$-transition points respectively on $\alpha {{:}{=}}[1, \xi ]$ and $\beta {{:}{=}}[1, \zeta ]$ so $\rho _v(1, \xi ) \ge \kappa$ and $\rho _w(1, \zeta ) \ge \kappa$.

Fix a point $\eta \in B_v(\xi )\cap B_w(\zeta )$ so by definition, $\rho _v(\xi , \eta ),\; \rho _w(\zeta , \eta )\le \kappa /2$. For a fixed geodesic $\gamma {{:}{=}}[1, \eta ]$, there exist $v', w'\in \gamma$ such that $d(1, w)=d(1, w')$ and $d(1, v)=d(1, v')$ and $d(v, v'), d(w, w')\le 2r_0,$ where $r_0 = \varphi (\kappa /2)$.

Now we verify the first condition. If $B_w(\zeta ) \in \mathcal B_{n-1}\cup {\mathcal {B}}_n\cup {\mathcal {B}}_{n+1}$ intersects $B_v(\xi )$, we have $|d(1, w) -d(1, v)|\le 2L$. It follows that $d(v, w)\le d(v, v') +d(v', w')+d(w', w)\le 2L+4r_0$. Set

$$\begin{aligned} N{{:}{=}}\;\sharp B(1, 2L+4r_0) \end{aligned}$$

so there are at most N possibilities of $B_w(\zeta )$ such that $B_v(\xi )\cap B_w(\zeta )\ne \emptyset .$ So the first condition of ${\mathcal {B}}$ is verified.

So next, assume that $B_w(\zeta ) \in {\mathcal {B}}_m$ for $m<n$ so $d(v, w)=d(1, v)- d(1, w)\ge L$. Choosing L big enough, we have

Claim

If $B_v(\xi )\cap B_w(\zeta )\ne \emptyset$, then $B_v(\xi )\subset \Pi _{r_1}(w)$.

Proof of Claim

Since $\gamma$ is a $\rho _1$-Floyd geodesic, we have

$$\begin{aligned} \rho _{1}(v', \eta )=\rho _{1}([v', \eta ]_\gamma )\le \frac{\lambda ^{-d(1,v)}}{1-\lambda ^{-1}}. \end{aligned}$$

By the inequality (2.1), we have

$$\begin{aligned} \rho _w(v', \eta )\le \rho _1(v', \eta )\cdot \lambda ^{d(1, w)} \le \frac{\lambda ^{-L}}{1-\lambda ^{-1}}. \end{aligned}$$

Consider an arbitrary point $\xi ' \in B_v(\xi )$ so we have $d(v, \gamma ')\le r_0$ for any geodesic $\gamma '{{:}{=}}[1,\xi ']$. Choose $v''\in \gamma '$ such that $d(1, v)=d(1, v'')$ and then $d(v, v'')\le 2r_0$. The same argument as above gives

$$\begin{aligned} \rho _w(v'', \xi ')\le \frac{\lambda ^{-L}}{1-\lambda ^{-1}}. \end{aligned}$$

On the other hand, we calculate $\rho _w(v', v'')$ which is bounded by the Floyd length of a geodesic $[v, v']$ with length at most $d(v', v'')\le 4r_0$ and $d(w, v')\ge d(w, v) -d(v, v')\ge L-4r_0$. A direct computation shows $\rho _w(v', v'')\le {4r_0} \cdot {\lambda ^{-L+8r_0}}.$

Note that $\rho _w(1, \eta )\ge \rho _w(1, \zeta )-\rho _w(\zeta , \eta ) \ge \kappa /2$ for $\eta \in B_w(\zeta )$. So $\rho _w(1, \xi ')\ge \rho _w(1, \eta )-\rho _v(\eta , v')-\rho _v(v', v'')-\rho _u(v'', \xi ')\ge \kappa /4$, where we choose $L>0$ large enough such that

$$\begin{aligned} \frac{2\lambda ^{-L}}{1-\lambda ^{-1}}- {4r_0}\cdot {\lambda ^{-L+8r_0}} \le \kappa /4. \end{aligned}$$

(A.4)

This implies that for any $\xi '\in B_v(\xi )$, we have $\xi \in \Pi _{r_1}(w)$. The claim is proved. $\square$

Note here that $\Pi _{r_1}(w)$ is the extension of $B_w(\zeta )$. So the second condition follows by the claim. Both conditions are verified and the lemma is proved. $\square$

Note that ${\mathcal {B}}=\bigcup _{n\ge 0} {\mathcal {B}}_n$. By Lemma 4.11, we see that the $\rho _1$-diameter of each $B\in {\mathcal {B}}_n$ tends to 0 as $n\rightarrow \infty$. Recall that for $\xi \in \partial ^c{G}$, there are infinitely many $(\epsilon , R)$-transition points v on $[1, \xi ]$. Given $\varepsilon >0$, there exists $n_0>0$ such that

$$\begin{aligned} \partial ^c{G}\subset \bigcup \bigg\{B: B\in \bigcup _{n\ge n_0}{\mathcal {B}}_n\bigg\}. \end{aligned}$$

Since $\partial ^c{G}$ is dense in $\partial {G}$, we have that the open set $\bigcup _{n\ge n_0} {\mathcal {B}}_n$ covers $\partial {G}$. So, for any subset $A \subset \partial {G}$, there exists a countable subfamily of ${\mathcal {B}}'\subset {\mathcal {B}}$ which is an $\epsilon$-covering of A. This follows from that $\partial {G}$ is Lindelöf.

So, we could consider a Hausdorff–Caratheodory measure $H_\sigma (, {\mathcal {B}})$ relative to ${\mathcal {B}}$ and the gauge function $r^\delta$. The Hausdorff measure in the usual sense is to take ${\mathcal {B}}$ as the set of all metric balls in $\partial {G}$.

Lemma A.3

Let $\{\mu _v\}_{v\in G}$ be a $\sigma$-dimensional quasi-conformal density without atoms on $\partial {G}$. Then we have

$$\begin{aligned} H_{\sigma }(A, {\mathcal {B}}) \asymp \mu _1(A) \end{aligned}$$

for any subset $A \subset \partial {G}$. In particular, $\mu$ is unique in the following sense: if $\mu , \mu '$ are two such quasi-conformal densities, then the Radon–Nikodym derivative $\text{d}\mu /\text{d}\mu '$ is bounded from up and below.

Proof

Take an $\epsilon$-covering ${\mathcal {B}}' \subset {\mathcal {B}}$ of A, so that $\mu _1(A) \le \sum _{B \in {\mathcal {B}}'} \mu _1(B)$. Let $\epsilon \rightarrow 0$. By (A.3), we obtain that $\mu _1(A) \prec H_{\sigma }(A, {\mathcal {B}})$.

To establish the other inequality, for any $0< \epsilon < \epsilon _0$, let ${\mathcal {B}}_1 \subset {\mathcal {B}}$ be an $\epsilon$-covering of K. By Lemma A.1, there exist an integer $N>1$ and a disjoint sub-family ${\mathcal {B}}_2$ of ${\mathcal {B}}_1$ such that (A.1) holds. So

$$\begin{aligned} H_\sigma (K, {\mathcal {B}}) \prec \sum _{B \in {\mathcal {B}}_2} N \cdot {\Vert B\Vert }^\sigma \prec \mu _1(U), \end{aligned}$$

by (A.3). Letting $\epsilon \rightarrow 0$ yields $H_\sigma (A, {\mathcal {B}}) \prec \mu _1(A)$. $\square$

Lemma A.4

Let $\{\mu _v\}_{v\in G}$ be a $\sigma$-dimensional quasi-conformal density without atoms on $\partial {G}$. Then $\{\mu _v\}_{v\in G}$ is ergodic with respect to the action of G on $\partial {G}$.

Proof

Let A be a G-invariant Borel subset in $\partial {G}$ such that $\mu _1(A) >0$. Restricting $\mu$ on A gives rise to a $\delta _{G}$-dimensional conformal density without atoms on A.

By Lemma A.3, there exists $C>0$ such that for any subset $X \subset \partial {G}$, we have $\mu _1(X) \le C \cdot H_\sigma (A \cap X, {\mathcal {B}})$. Thus, $\mu _1(\partial {G}\setminus A)=0$. $\square$

By Proposition 4.13, we have the following proposition.

Proposition A.5

The PS-measure $\{\mu _v\}_{v \in G}$ is a $\delta _{G}$-dimensional quasi-conformal density without atoms. Moreover, it is unique and ergodic.

Appendix B: Finiteness of Partial Cone Types

Recall that the finiteness of cone types is established by Cannon in [9] for a hyperbolic group. We shall state an analogous “partial” result in the relative setting.

For fixed $\epsilon , R>0$, we define two partial cones $\Omega _{\epsilon , R}(g), \Omega _{\epsilon , R}(g')$ to be of same type if there exists $t \in G$ such that $t\cdot \Omega _{\epsilon , R}(g)= \Omega _{\epsilon , R}(g')$.

Lemma B.1

(Finiteness of partial cone types) There exist $\epsilon , R_0>0$ such that for any $R > R_0$, there are at most $N=N(\epsilon , R)$ types of all $(\epsilon , R)$-partial cones $\{\Omega _{\epsilon , R}(g): \forall g \in G\}$.

Proof

Let $\epsilon , R_0>0$ be given by Lemma 2.15. For any $R > R_0$, let $\Omega _{\epsilon , R}(g)$ be a partial cone and h an element in $\Omega _{\epsilon , R}(g)$. We first make the following claim.

Claim

Assume that $d(h, g) \ge 2R+1$ for $h\in \Omega _{\epsilon , R}(g)$. Then there exists a constant $\kappa >0$ such that the following holds

$$\begin{aligned} \rho _{g}(1, h) > \kappa . \end{aligned}$$

Proof of Claim

By definition of the partial cone, there exists some geodesic $\gamma =[1, h]$ with $g \in \gamma$ such that $\gamma$ contains an $(\epsilon , R)$-transition point v in B(g, 2R).

Since v is an $(\epsilon , R)$-transition point in $\gamma$, there exists $\kappa =\kappa (\epsilon , R)$ given by Lemma 2.15 such that $\rho _v(1, h) > \kappa$. Note that $d(v, g) \le 2R$. By the inequality (2.1) for Floyd metrics with different basepoints, we can assume that $\rho _{g}(1, h) > \kappa$, up to divide $\kappa$ by a constant depending on 2R. $\square$

Define $C =\varphi (\kappa )$, where $\varphi$ is given by Lemma 2.12. Let $F_g$ be the subset of elements $f\in B(g, 2C+1)$ such that $d(1, gf) \le d(1, g)$. Consider

$$\begin{aligned} B_g{{:}{=}}B(g, 3R+1) \cap \big (\Omega _{\epsilon , R}(g)\cup \mathcal C(g)\big ), \end{aligned}$$

where ${\mathcal {C}}(g)$ is the union of all possible geodesics [1, g] between 1 and g.

Claim

The sets $F_g$ and $B_g$ together determine the type of the partial cone $\Omega _{\epsilon , R}(g)$.

Proof of Claim

Let $g, g'\in G$ such that $F_g=F_{g'}$ and $B_g=B_{g'}$. For any $h \in \Omega _{\epsilon , R}(g)$ and $w {{:}{=}} g^{-1}h$, we shall prove that $g'w \in \Omega _{\epsilon , R}(g')$ by induction on $n=d(1, w)\ge 0$. The base case “$n\le 3R+1$” is true by $B_g=B_{g'}$.

Assume $gw \in \Omega _{\epsilon , R}(g)$ and $g'w \in \Omega _{\epsilon , R}(g')$. Let $s \in S$ be a generator such that $gws \in \Omega _{\epsilon , R}(g)$. So the following holds

$$\begin{aligned} d(1, gws) =d(1, g) +d(1, w) +1. \end{aligned}$$

(B.1)

We shall show that $g'ws \in \Omega _{\epsilon , R}(g')$ by verifying the following two properties:

• the concatenation of some geodesics $[1, g'], [g', g'w]$ and $[g'w, g'ws]$ is a geodesic, and

• it contains an $(\epsilon , R)$-transition point in $B(g', 2R)$.

By assumption, the subpath $[1, g][g, gw]\cap B(g, 3R+1)$ contains an $(\epsilon , R)$-transition point. Note that the transition point is invariant by translation. Since $B_g=B_{g'}$, the two segments $[1, g][g, gw]\cap B(g, 3R+1)$ and $[1, g'][g, g'w]\cap B(g, 3R+1)$ have the same label, and so $[1, g'][g', g'w]$ does contain an $(\epsilon , R)$-transition point in $B(g', 3R+1)$. The second property follows. So, it suffices to verify that the concatenation path is a geodesic. Suppose, by way of contradiction, that

$$\begin{aligned} d(1, g'ws) \le d(1, g') + d(1, w). \end{aligned}$$

(B.2)

Consider a geodesic $\gamma$ between 1 and $g'ws$. By Induction Assumption, $[1, g'][g', g'w]$ is a geodesic, and then we get $d(g', \gamma ) \le C$ by the previous claim. Thus, there exists $u \in \gamma$ such that $d(g', u) \le 2C$ and $d(1, u) = d(1, g')$. By (B.2), it follows that $d(u, \gamma _+) \le d(1, w)$.

Setting $f {{:}{=}} g'^{-1}u$, we have $f\in F_{g'}=F_g$ and so $d(1, gf) \le d(1, g)$. Denote by k the element represented by the geodesic segment $[u, \gamma _+]_\gamma$. Then $f k = ws$. Hence,

$$\begin{aligned} d(1, gws) =d(1, gf k) \le d(1, gf) + d(u, \gamma _+) \le d(1, g) + d(1, w). \end{aligned}$$

This contradicts with (B.1). The first property holds: $g'ws \in \Omega (g')$. The claim is proved. $\square$

The finiteness of partial cone types now follows from the second claim. $\square$