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The Martin boundary of an extension by a hyperbolic group

Sara Ruth Pires Bispo, Manuel Stadlbauer

Abstract

We prove uniform Ancona-Gouëzel-Lalley inequalities for an extension by a hyperbolic group $G$ of a Markov map which allows to deduce that the visual boundary of the group and the Martin boundary are Hölder equivalent. As application, we identify the set of minimal conformal measures of a regular cover of a convex-cocompact CAT(-1)-manifold with the visual boundary of the covering group, provided that this group is hyperbolic.

The Martin boundary of an extension by a hyperbolic group

Abstract

We prove uniform Ancona-Gouëzel-Lalley inequalities for an extension by a hyperbolic group of a Markov map which allows to deduce that the visual boundary of the group and the Martin boundary are Hölder equivalent. As application, we identify the set of minimal conformal measures of a regular cover of a convex-cocompact CAT(-1)-manifold with the visual boundary of the covering group, provided that this group is hyperbolic.

Paper Structure

This paper contains 7 sections, 25 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction and statement of main results
  2. Group extensions of Markov maps
  3. The Green operator
  4. Ancona-Gouëzel inequalities for extensions by word hyperbolic groups
  5. Geometry of the Martin boundary
  6. An application to regular covers of convex-cocompact CAT(-1) metric spaces
  7. Appendix: Reduced measures and the domination principle

Key Result

Theorem A

Assume that $G$ is a non-elementary and word hyperbolic group, that $T$ is a topologically transitive and that $\mathbb{G}_R(\mathbf{1}_{\Sigma \times\{g \} })(\,\cdot\,,\mathop{\mathrm{id}}\nolimits) \asymp \mathbb{G}_R(\mathbf{1}_{\Sigma \times\{\mathop{\mathrm{id}}\nolimits \} })(\,\cdot\,,g)$. T

Figures (4)

  • Figure 2: The construction of $B_{i+1}$
  • Figure 3: Typical orbits for $D_3$ (slashed) and $E_4$ (dotted)
  • Figure 4: The positions of $\sigma,\tilde{\sigma}$ and $h,\tilde{h}$.
  • Figure 5: A configuration of $g, \tilde{g}$ and $h,\mathop{\mathrm{id}}\nolimits$ with $\xi_1 \neq \xi_2$

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • ...and 37 more