A note on growth of hyperbolic groups

TL;DR

This paper provides an alternative proof that the growth function of a non-elementary hyperbolic group is exponential, with explicit bounds, confirming the exponential growth rate of such groups.

Contribution

It offers a new proof of the exponential growth rate of hyperbolic groups, establishing explicit bounds for the growth function.

Findings

Growth of hyperbolic groups is exponential.

Explicit constants for growth bounds are provided.

Alternative proof method for known growth properties.

Abstract

The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $\lambda,D > 1$ such that \[ D^{-1}\lambda^n \leq |B_{G,X}(n)| \leq D \lambda^n \] for all $n \geq 0$, where $B_{G,X}(n)$ is the ball of radius $n$ in the Cayley graph $\Gamma(G,X)$.

Equations

$$D^{-1}{\lambda }^n\le |B_{G,X}(n)|\le D{\lambda }^n$$

$$D^{-1}{\lambda }^n\le {\mathfrak{B}}_{G,X}(n)\le D{\lambda }^n$$

$$C{n}^{\alpha }{\lambda }^n\le {\mathfrak{S}}_{G,X}(n)\le D{n}^{\alpha }{\lambda }^n$$

$$\mu :S(n)\times S(m)$$

$$(h,k)$$

$$S(n)\times S(m)=\underset{\ell =|n-m|}{\overset{n+m}{⨆}}{\mu }^{-1}(S(\ell )).$$

$$|{\mu }^{-1}(g)|=|\{h\in S(n)\mid h^{-1}g\in S(m)\}|\le |\{h\in S(n)\mid c^{-1}h\in B(N)\}|\le |cB(N)|=\mathfrak{B}(N),$$

$$C{n}^{\alpha }{\lambda }^n<\mathfrak{B}(n)<\frac{D\lambda }{\lambda -1}{n}^{\alpha }{\lambda }^n$$

$$C^2{n}^{2\alpha }{\lambda }^{2n}\le \mathfrak{S}(n{)}^2\le \sum _{\ell =0}^{2n}\mathfrak{S}(\ell )\mathfrak{B}(\delta +\lceil n-\ell /2\rceil )<\frac{D^2\lambda }{\lambda -1}\sum _{\ell =0}^{2n}{\ell }^{\alpha }(\delta +\lceil n-\ell /2\rceil {)}^{\alpha }{\lambda }^{\delta +n+\lceil \ell /2\rceil }.$$

$$\frac{C^2(\lambda -1)}{2^{\alpha }{D}^2\lambda }<\sum _{\ell =0}^{2n}\underset{\le 1}{\underbrace{{\left(\frac{\ell }{2n}\right)}^{\alpha }}}\underset{\le (1+\delta /n{)}^{\alpha }}{\underbrace{{\left(\frac{\delta +\lceil n-\ell /2\rceil }{n}\right)}^{\alpha }}}{\lambda }^{\delta +\lceil \ell /2\rceil -n}.$$

$$\sum _{\ell =0}^{2n-2\epsilon (n)}{\left(\frac{\ell }{2n}\right)}^{\alpha }{\left(\frac{\delta +\lceil n-\ell /2\rceil }{n}\right)}^{\alpha }{\lambda }^{\delta +\lceil \ell /2\rceil -n}$$

$$<2n{\left(1+\frac{\delta }{n}\right)}^{\alpha }{\lambda }^{\delta -\epsilon (n)}$$

$$\sum _{\ell =2n-2\epsilon (n)+1}^{2n}{\left(\frac{\ell }{2n}\right)}^{\alpha }{\left(\frac{\delta +\lceil n-\ell /2\rceil }{n}\right)}^{\alpha }{\lambda }^{\delta +\lceil \ell /2\rceil -n}$$

$$=2{\lambda }^{\delta }\epsilon (n){\left(\frac{\delta +\epsilon (n)}{n}\right)}^{\alpha }.$$

Full text

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for all $n\geq 0$, where $B_{G,X}(n)$ is the ball of radius $n$ in the Cayley graph $\Gamma(G,X)$.

Given a group $G$ with a finite generating set $X$ and an integer $n\geq 0$, we denote by $S_{G,X}(n)\subseteq G$ the sphere of radius $n$ in $G$ with respect to $X$: that is, the set of elements $g\in G$ that can be represented by words of $n$ (but not $n-1$) letters in $X^{\pm 1}$. We denote $|S_{G,X}(n)|$ by $\mathfrak{S}_{G,X}(n)$. Similarly, we denote by $B_{G,X}(n)\subseteq G$ the ball of radius $n$ in $G$ with respect to $X$ – that is, $B_{G,X}(n)=\bigcup_{i=0}^{n}S_{G,X}(i)$ – and we write $\mathfrak{B}_{G,X}(n)$ for $|B_{G,X}(n)|$. We write $S(n)$, $\mathfrak{S}(n)$, $B(n)$ and $\mathfrak{B}(n)$ for $S_{G,X}(n)$, $\mathfrak{S}_{G,X}(n)$, $B_{G,X}(n)$ and $\mathfrak{B}_{G,X}(n)$ (respectively) if $G$ and $X$ are clear.

Here we give a short alternative proof of the following result.

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for all $n\geq 0$.

The proof of this theorem given in [Coo93] is based on the theory of Patterson-Sullivan measures and dynamical systems; using similar methods, Yang has shown an analogous result for relatively hyperbolic groups [Yan13, Theorem 1.9]. Here we give a Patterson-Sullivan measure-free proof. It is worth noting that an analogous result also holds when $G$ is a right-angled Artin/Coxeter group that does not split as a direct product and $X$ is the standard generating set [GTT17, Theorem 2.2].

We define the (spherical) growth function of $G$ with respect to $X$ to be the formal power series $s_{G,X}(t)=\sum_{n=0}^{\infty}\mathfrak{S}_{G,X}(n)t^{n}$. Similarly, we may define the volume growth function as $b_{G,X}(t)=\sum_{n=0}^{\infty}\mathfrak{B}_{G,X}(n)t^{n}$; note that we have $s_{G,X}(t)=(1-t)b_{G,X}(t)$. We say that $G$ has rational growth with respect to $X$ if $s_{G,X}(t)$ (equivalently, $b_{G,X}(t)$) is a rational function: that is, a ratio of two polynomials. A classical result due to Cannon says that this is always the case for hyperbolic groups.

Our proof relies on the following theorem of the author.

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for all $n\geq 1$.

For the rest of the note, fix a non-elementary word-hyperbolic group $G$ and a finite generating set $X$ for $G$. Let $\delta\in\mathbb{Z}_{\geq 1}$ be a constant such that triangles in the Cayley graph $\Gamma(G,X)$ are $\delta$-thin, and let $d$ be the combinatorial metric on $\Gamma(G,X)$. We use an auxiliary lemma.

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allows us to define a partition

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It is therefore enough to show that $|\mu^{-1}(g)|\leq\mathfrak{B}\left(\delta+\left\lceil\frac{n+m-\ell}{2}\right\rceil\right)$ for all $\ell$ and all $g\in S(\ell)$.

Let $\ell\in\{|n-m|,\ldots,n+m\}$ and let $g\in S(\ell)$. Let $\gamma_{g}$ be a geodesic in $\Gamma(G,X)$ joining $1_{G}$ and $g$, and let $p_{g}$ be the point on $\gamma_{g}$ such that $d(p_{g},1_{G})=\frac{\ell+n-m}{2}$. Let $c\in G$ be a vertex of $\Gamma(G,X)$ such that $d(c,p_{g})\leq\frac{1}{2}$, and note that $c=p_{g}$ if $n+m-\ell$ is even. Let $N=\delta+\left\lceil\frac{n+m-\ell}{2}\right\rceil$.

1_{G}$$p_{g}$$\gamma_{g}$$g$$\gamma_{h}$$p_{h}\ \ \$$h$$\gamma_{k}

Let $(h,k)\in\mu^{-1}(g)$, so that $h\in S(n)$ and $k=h^{-1}g\in S(m)$. Let $\gamma_{h}$, $\gamma_{k}$ be geodesics in $\Gamma(G,X)$ joining $1_{G}$ and $h$, $h$ and $g$, respectively, and let $\Delta$ be the geodesic triangle with edges $\gamma_{g}$, $\gamma_{h}$ and $\gamma_{k}$. Let $p_{h}$ be the point on $\gamma_{h}$ such that $d(p_{h},1_{G})=\frac{\ell+n-m}{2}$. Since $\Delta$ is $\delta$-thin, we have $d(p_{g},p_{h})\leq\delta$. By construction, $d(p_{h},h)=\frac{n+m-\ell}{2}$, and so it follows by the triangle inequality that $d(1_{G},c^{-1}h)=d(c,h)\leq N$. In particular,

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as required. ∎

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for all $n\geq 1$. It is therefore enough to show that $\alpha=0$.

Pick an integer $n\geq 1$. It follows from Theorem 3, Lemma 4 and (1) that

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We may rearrange this to obtain

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Now let $\varepsilon:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 2\delta}$ be a function such that $n\left(1+\frac{\delta}{n}\right)^{\alpha}\lambda^{-\varepsilon(n)}\to 0$ and $\frac{\varepsilon(n)^{2}}{n}\to 0$ as $n\to\infty$: for instance, $\varepsilon(n)=2\delta+\lfloor\sqrt[3]{n}\rfloor$. We may split the right hand side of (2) into terms with $\ell\leq 2n-2\varepsilon(n)$ and terms with $\ell>2n-2\varepsilon(n)$. In particular, we get

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and

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Note that the left hand side of (2) is a strictly positive constant, and the right hand side of (3) tends to zero as $n\to\infty$ by the choice of $\varepsilon$. Furthermore, by the choice of $\varepsilon$, if $\alpha\geq 1$ then the right hand side of (4) tends to zero as $n\to\infty$. But by (2) this cannot happen, and so $\alpha=0$, as required. ∎