Growth of conjugacy classes in Gromov hyperbolic groups.
Consider a group \(\Gamma\) acting by isometries on a metric space \(X\). Given an element \(\gamma\in\Gamma\), the number \(\ell(\gamma)=\inf_{x\in X}d(x,\gamma x)\) only depends on the conjugacy class \([\gamma]\) of \(\gamma\) in \(\Gamma\). \(\ell(\gamma)\) is called the length of the conjugacy class \([\gamma]\). For all \(t\geq 0\), let \[ P(t)=\text{card}\{[\gamma]\mid\gamma\in\Gamma\text{ is primitive and }\ell(\gamma)\leq t\} \] denote the number of conjugacy classes of primitive elements in \(\Gamma\) having length at most \(t\). An element \(\gamma\in\Gamma\) is said to be primitive if it cannot be written as a power \(\gamma=\alpha^n\), where \(\alpha\in\Gamma\) and \(n\geq 2\). An asymptotic estimate for \(P(t)\), as \(t\) tends to infinity, has been given by \textit{G. A. Margulis} [Funct. Anal. Appl. 3, 335-336 (1969); translation from Funkts. Anal. Prilozh. 3, No. 4, 89-90 (1969; Zbl 0207.20305)] in the case when \(X\) is the universal cover of a closed Riemannian manifold \(M\) with negative sectional curvature and \(\Gamma=\pi_1M\). Margulis obtained the estimate \[ P(t)\sim\tfrac{e^{ht}}{ht} \] as \(t\to\infty\), where \(h>0\) is a constant depending only on \(X\) (the notation \(f(t)\sim g(t)\) means that \(f(t)/g(t)\to 1\)). The constant \(h\) is equal to the topological entropy of the geodesic flow on the unit tangent bundle of \(M\). It is also equal to the `volume entropy' of \(X\), i.e. one has, for all \(x\in X\), \[ h=\lim_{R\to\infty}\tfrac{\log\text{vol\;}B(x,R)}{R}=\lim_{r\to \infty}\tfrac{\log\text{card\;}\Gamma_R(x)}R \] where \(B(x,R)\) denotes the closed ball of radius \(R\) centered at \(x\) and \[ \Gamma_R(x)=\{\gamma\in\Gamma\mid d(x,\gamma x)\leq R\}. \] A generalization of Margulis' result for rank 1 manifolds has been obtained by the second author of the paper under review [Geom. Funct. Anal. 7, No. 4, 755-782 (1997; Zbl 0896.53033)] as following. Let \(X\) be the universal cover of the closed rank 1 manifold \(M\) and let \(\Gamma=\pi_1M\). Then there exist positive constants \(A,B,t_0\) and \(h\) such that \[ A\tfrac{e^{ht}}t\leq P(t)\leq Be^{ht} \] for all \(t\geq t_0\). The constant \(h\) is equal to the topological entropy of the geodesic flow on the unit tangent bundle of \(M\) and to the volume entropy of \(X\). In the paper under review an analogous generalization of Margulis' result for discrete cocompact group actions on Gromov hyperbolic metric spaces is established: Theorem 1.1. Let \(\Gamma\) be a group acting properly and cocompactly by isometries on a proper geodesic \(\delta\)-hyperbolic metric space \(X\). Assume that the Gromov boundary \(\partial X\) of \(X\) contains more than two points. Then there exist positive constants \(A,B,t_0\) and \(h\), where \(h=h(X)\) depends only on \(X\), such that \[ A\tfrac{e^{ht}}t\leq P(t)\leq Be^{ht} \] for all \(t\geq t_0\). Moreover, one has \[ h=\lim_{r\to\infty}\tfrac{\log\text{card\;}\Gamma_R(x)}R, \] where \[ \Gamma_R(x)=\{\gamma\in\Gamma\mid d(x,\gamma x)\leq R\}, \] for all \(x\in X\). The authors give an application of Theorem 1.1 to combinatorial group theory which is obtained as follows. Let \(\Gamma\) be a finitely generated group and let \(S\) be a finite generating set of \(\Gamma\). For all \(\gamma\in\Gamma\), it is denoted by \(|\gamma|_S\) the `word length' of \(\gamma\) with respect to \(S\), i.e. the length of a shortest word on the alphabet \(S\cup S^{-1}\) representing \(\gamma\). Define the `length' of the conjugacy class \([\gamma]\) of \(\gamma\) in \(\Gamma\) as being the integer \[ \ell_S([\gamma])=\min_{\alpha\in[\gamma]}|\alpha|_S. \] When \(\Gamma\) is a non-elementary word hyperbolic group, one can apply Theorem 1.1 to the natural action of \(\Gamma\) on its Cayley graph with respect to \(S\). Corollary 1.2. Let \(\Gamma\) be a non-elementary word hyperbolic group and let \(S\) be a finite generating set of \(\Gamma\). For all integers \(n\geq 1\), let \[ p_n=\text{card}\{[\gamma]\mid\gamma\in\Gamma\text{ is primitive and }\ell_S([\gamma])\leq n\}. \] Then there exist positive constants \(A,B,n_0\) and \(h\), depending only on \(\Gamma\) and \(S\), such that \[ A\tfrac{e^{hn}}n\leq p_n\leq Be^{hn} \] for all \(n\geq n_0\). Moreover, one has \[ h=\lim_{n\to\infty}\tfrac{\log b_n}n, \] where \(b_n\) denotes the number of elements in \(\Gamma\) having word length at most \(n\). The lower bound for \(p_n\) in the preceding corollary shows the exponential growth of conjugacy classes of primitive elements in non-elementary word hyperbolic groups. It was previously known (see 5.2.B of [\textit{M. Gromov}, Hyperbolic groups, in ``Essays in group theory'', Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]) that the number of conjugacy classes of primitive elements in a non-elementary word hyperbolic group is infinite.
scientific article; zbMATH DE number 1890450