Free Semigroups of Large Critical Exponent

Fuente: arXiv
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Main Author: Skenderi, Aleksander
Format: Preprint
Published: 2025
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author Skenderi, Aleksander
author_facet Skenderi, Aleksander
contents For a convergence group equipped with an expanding coarse-cocycle, we construct finitely generated free subsemigroups, which we call $\textit{Bishop--Jones}$ $\textit{semigroups}$, of critical exponent arbitrarily close to but strictly less than the critical exponent of the ambient group. As an application, we show that for any non-elementary transverse subgroup $Γ$ of a semisimple Lie group $G$, there exist finitely generated free Anosov subsemigroups in the sense of Kassel--Potrie of critical exponent arbitrarily close to but strictly less than that of the ambient transverse group. Furthermore, we show that these semigroups admit $\mathcal{C}$-regular quasi-isometric embeddings into the symmetric space $X$ of $G$, in the sense of Kapovich--Leeb--Porti.
format Preprint
id arxiv_https___arxiv_org_abs_2502_02003
institution arXiv
publishDate 2025
record_format arxiv
spellingShingle Free Semigroups of Large Critical Exponent
Skenderi, Aleksander
Group Theory
Dynamical Systems
Geometric Topology
For a convergence group equipped with an expanding coarse-cocycle, we construct finitely generated free subsemigroups, which we call $\textit{Bishop--Jones}$ $\textit{semigroups}$, of critical exponent arbitrarily close to but strictly less than the critical exponent of the ambient group. As an application, we show that for any non-elementary transverse subgroup $Γ$ of a semisimple Lie group $G$, there exist finitely generated free Anosov subsemigroups in the sense of Kassel--Potrie of critical exponent arbitrarily close to but strictly less than that of the ambient transverse group. Furthermore, we show that these semigroups admit $\mathcal{C}$-regular quasi-isometric embeddings into the symmetric space $X$ of $G$, in the sense of Kapovich--Leeb--Porti.
title Free Semigroups of Large Critical Exponent
topic Group Theory
Dynamical Systems
Geometric Topology
url https://arxiv.org/abs/2502.02003