Free Semigroups of Large Critical Exponent

TL;DR

This paper constructs free subsemigroups with critical exponents close to the ambient group in certain geometric contexts, revealing new structures within convergence groups and Lie groups.

Contribution

It introduces Bishop–Jones semigroups with critical exponents near the ambient group's, expanding understanding of free subsemigroups in geometric group theory.

Findings

Existence of free subsemigroups with critical exponent arbitrarily close to the ambient group.

Construction of free Anosov subsemigroups in semisimple Lie groups.

These semigroups admit regular quasi-isometric embeddings into symmetric spaces.

Abstract

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 $\Gamma$ 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.