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Counting double cosets with application to generic 3-manifolds

Suzhen Han, Wenyuan Yang, Yanqing Zou

Abstract

We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. As an application, we confirm a conjecture of Maher that hyperbolic 3-manifolds are exponentially generic in the set of 3-manifolds built from Heegaard splitting using complexity in Teichmüller metric.

Counting double cosets with application to generic 3-manifolds

Abstract

We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. As an application, we confirm a conjecture of Maher that hyperbolic 3-manifolds are exponentially generic in the set of 3-manifolds built from Heegaard splitting using complexity in Teichmüller metric.
Paper Structure (18 sections, 32 theorems, 58 equations)

This paper contains 18 sections, 32 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Double coset growth for Morse subgroups
  3. Double coset growth for subgroups of second type
  4. Applications to generic 3-manifolds
  5. Counting common perpendiculars and further questions
  6. Preliminaries
  7. Contracting element
  8. Counting double cosets
  9. Statistically convex-cocompact actions
  10. Horofunction boundary
  11. Finite difference relation.
  12. Convergence boundary
  13. Double coset growth for Morse subgroups
  14. Double coset growth for subgroups of second type
  15. Generic 3-manifolds are hyperbolic: Theorem \ref{['Thm:HypExpGen']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that a non-elementary group $G$ acts properly on a proper geodesic metric space with a contracting element. Assume that $H$ and $K$ are Morse subgroups of infinite index. Then there exist $\delta, r_0>0$ so that for any $r>r_0$, In particular, if $G$ has purely exponential growth, then $\mathrm{gr}_{H,K}(r)\asymp\mathrm{gr}_G(r) \asymp\omega_G^r$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 52 more