Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups

TL;DR

The paper studies critical exponents of discrete, Zariski dense subgroups $\Gamma$ of a connected semisimple real Lie group $G$ and proves that one can extract free, finitely generated subsemigroups $\Omega\subset\Gamma$ with $\delta(\Omega)$ arbitrarily close to $δ(Γ)$ while maintaining Zariski density and a $P$-Anosov property. The construction relies on ε-contracting loxodromic elements, north–south dynamics on flag varieties, and Quint’s growth indicator to identify large, well-positioned generating sets with disjoint shadows; freeness is ensured via shadow-control. This yields that there is no gap phenomenon for the critical exponents of discrete subsemigroups, in contrast to Leuzinger’s gap theorem for lattices in groups with property (T). As applications, the authors prove a lower semicontinuity result for the critical exponent in the Chabauty topology (under Zariski-dense limits) and provide a new, streamlined proof of related convergence results for infinite covolume subgroups in property (T) groups.

Abstract

Let $G$ be a connected algebraic semisimple real Lie group with finite center and no compact factors, and let $Γ$ be a Zariski dense discrete subgroup of $G$. We show that $Γ$ contains free, finitely generated subsemigroups whose critical exponents are arbitrarily close to that of $Γ$. Furthermore, these subsemigroups are Zariski dense in $G$ and $P$-Anosov in the sense of Kassel--Potrie. This shows that no gap phenomenon holds for critical exponents of discrete subsemigroups of Lie groups, which is in contrast with Leuzinger's critical exponent gap theorem for infinite covolume discrete subgroups of Lie groups with Kazhdan's property (T), proven in 2003. As an application, we prove that the critical exponent is lower semicontinuous in the Chabauty topology, in the following sense: if a sequence of Zariski dense discrete subgroups $\{Γ_{n}\}$ of $G$ converges in the Chabauty topology to a Zariski dense discrete subgroup $Γ$, then $\liminf_{n \to \infty} δ(Γ_{n}) \geq δ(Γ)$.

Theorems & Definitions (75)

• Theorem 1.1: Corlette, Theorem 4.4 of Cor
• Theorem 1.2: Leuzinger, Main Theorem (Dichotomy) of Leu
• Theorem 1.3: Theorem \ref{['MainTheorem2']}
• Theorem 1.4: Theorem \ref{['lowersemicontinuity2']}
• Lemma 2.1: Lemma 2.3 of Ka
• Definition 2.2
• Lemma 2.3: Lemma 6.39 in BQ1
• Lemma 2.4: Lemma 6.5 of Q2
• Definition 2.5
• Proposition 2.6: Proposition $2.3$ of CZZ2