Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuity of asymptotic entropy on wreath products

Eduardo Silva

TL;DR

This work addresses the continuity of the asymptotic entropy $h(\mu)$ for random walks on countable groups, focusing on wreath products $A\wr B$ where $B$ is hyper-FC-central and contains a cubic-growth subgroup. The authors develop a general continuity criterion for $h(\mu)$ via a coarse-trajectory and lamp-configuration entropy analysis, and they connect this to the (weak) continuity of harmonic measures on Poisson boundaries to transfer results to broad group classes. A key technical achievement is proving the continuity of the base-escape probability $p_{esc}(\mu)$ under convergence of step distributions and finite-entropy assumptions, which feeds into the wreath-product estimates. The paper unifies and extends known continuity results for hyperbolic and acylindrically hyperbolic groups and applies to new families including linear groups and groups acting on CAT(0) spaces, thereby broadening the scope of entropy continuity in geometric group theory.

Abstract

We prove the continuity of asymptotic entropy as a function of the step distribution for non-degenerate probability measures with finite entropy on wreath products $ A \wr B = \bigoplus_B A \rtimes B $, where $A$ is any countable group and $B$ is a countable hyper-FC-central group that contains a finitely generated subgroup of at least cubic growth. As one step in proving the above, we show that on any countable group $G$ the probability that the $μ$-random walk on $G$ never returns to the identity is continuous in $μ$, for measures $μ$ such that the semigroup generated by the support of $μ$ contains a finitely generated subgroup of at least cubic growth. Finally, we show that among random walks on a group $G$ that admit a separable completely metrizable space $X$ as a model for their Poisson boundary, the weak continuity of the associated harmonic measures on $X$ implies the continuity of the asymptotic entropy. This result recovers the continuity of asymptotic entropy on known cases, such as Gromov hyperbolic groups and acylindrically hyperbolic groups, and extends it to new classes of groups, including linear groups and groups acting on $\mathrm{CAT}(0)$ spaces.

Continuity of asymptotic entropy on wreath products

TL;DR

This work addresses the continuity of the asymptotic entropy for random walks on countable groups, focusing on wreath products where is hyper-FC-central and contains a cubic-growth subgroup. The authors develop a general continuity criterion for via a coarse-trajectory and lamp-configuration entropy analysis, and they connect this to the (weak) continuity of harmonic measures on Poisson boundaries to transfer results to broad group classes. A key technical achievement is proving the continuity of the base-escape probability under convergence of step distributions and finite-entropy assumptions, which feeds into the wreath-product estimates. The paper unifies and extends known continuity results for hyperbolic and acylindrically hyperbolic groups and applies to new families including linear groups and groups acting on CAT(0) spaces, thereby broadening the scope of entropy continuity in geometric group theory.

Abstract

We prove the continuity of asymptotic entropy as a function of the step distribution for non-degenerate probability measures with finite entropy on wreath products , where is any countable group and is a countable hyper-FC-central group that contains a finitely generated subgroup of at least cubic growth. As one step in proving the above, we show that on any countable group the probability that the -random walk on never returns to the identity is continuous in , for measures such that the semigroup generated by the support of contains a finitely generated subgroup of at least cubic growth. Finally, we show that among random walks on a group that admit a separable completely metrizable space as a model for their Poisson boundary, the weak continuity of the associated harmonic measures on implies the continuity of the asymptotic entropy. This result recovers the continuity of asymptotic entropy on known cases, such as Gromov hyperbolic groups and acylindrically hyperbolic groups, and extends it to new classes of groups, including linear groups and groups acting on spaces.
Paper Structure (32 sections, 40 theorems, 158 equations)

This paper contains 32 sections, 40 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Continuity of entropy on wreath products
  3. Connection between the continuity of asymptotic entropy and the weak continuity of harmonic measures
  4. Sketch of the proof of Theorem \ref{['thm: main corollary continuity of asymptotic entropy over Zd']}
  5. Organization
  6. Acknowledgements
  7. Preliminaries
  8. Random walks on groups, entropy and the Poisson boundary
  9. Escape probability and the asymptotic range
  10. Wreath products
  11. Growth of groups
  12. Convergence of probability measures
  13. Continuity of return probability to the origin
  14. The comparison lemma of Coulhon and Saloff-Coste
  15. The proof of Theorem \ref{['thm: continuity of range']}
  16. ...and 17 more sections

Key Result

Theorem 1.2

Let $A$ be a countable group and let $B$ be a countable hyper-FC-central group that contains a finitely generated subgroup of at least cubic growth. Let $\mu$ be a non-degenerate probability measure $\mu$ on $A\wr B$ with $H(\mu)<\infty$. Consider a sequence $\{\mu_k\}_{k\ge 1}$ of probability measu

Theorems & Definitions (102)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Derrienic1980, KaimanovcihVershik1983
  • Remark 2.4
  • ...and 92 more