On Growth of Double Cosets in Hyperbolic Groups

2018/08/02 by Rita Gitik, Gitik, Rita, Eliyahu Rips +1
Mathematics · #20B07 #20F65 #20F67 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20B07 #msc:20F65 #msc:20F67

paper · pdf · doi:10.48550/arxiv.1808.00802

arxiv created 2019/08/01 · arxiv updated 2019/08/05

Abstract

Let H be a hyperbolic group, A and B be subgroups of H, and gr(H,A,B) be the growth function of the double cosets AhB, h ∈ H. We prove that the behavior of gr(H,A,B) splits into two different cases. If A and B are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as gr(H,A,B). We can even take A=B. In contrast, for quasiconvex subgroups A and B of infinite index, gr(H,A,B) is exponential. Moreover, there exists a constant λ> 0, such that gr(H,A,B)(r) >λf_H(r) for all big enough r, where f_H(r) is the growth function of the group H. So, we have a clear dychotomy between the quasiconvex and non-quasiconvex case.

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