Free Semigroups of Large Critical Exponent
Aleksander Skenderi
TL;DR
This work constructs Bishop–Jones semigroups: free finitely generated subsemigroups inside convergence groups equipped with expanding coarse-cocycles, achieving critical exponents δσ(𝒯) arbitrarily close to the ambient δσ(Γ) while remaining strictly smaller. The authors develop a general, equivariant construction, prove sharp control of sigma-magnitudes in terms of word length, and show a finite-generation free structure with a robust divergence-type behavior. They apply this framework to transverse groups in semisimple Lie groups, producing free Pθ-Anosov subsemigroups with δφ(Γn)→δφ(Γ) from below, and prove that these subsemigroups admit coarse, (C,regular) embeddings into the symmetric space X. Additionally, they demonstrate a Rank-1 gap phenomenon failure for subsemigroups of Sp(n,1) and F4^{-20}, and establish a coarse triangle inequality for Bishop–Jones semigroups via a θ-admissible cone, connecting geometric growth with higher-rank Lie-theoretic data.
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 $Γ$ 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.