Tent property of the growth indicator functions and applications

TL;DR

This paper establishes bounds on the growth indicator functions of Zariski dense subgroups in semisimple groups, revealing precise conditions for equality and applications to convex cocompact and Hitchin subgroups.

Contribution

It provides a new pointwise bound for the growth indicator function of Zariski dense subgroups, with characterizations for when equality holds, and explores applications to specific subgroup classes.

Findings

Derived a pointwise upper bound for the growth indicator function.

Identified conditions under which equality is achieved for Anosov subgroups.

Applied results to convex cocompact and Hitchin subgroups, deriving explicit bounds.

Abstract

Let $\Gamma$ be a Zariski dense discrete subgroup of a connected semisimple real algebraic group $G$. Let $k=\operatorname{rank} G$. Let $\psi_\Gamma:\mathfrak{a} \to \mathbb{R}\cup \{-\infty\}$ be the growth indicator function of $\Gamma$, first introduced by Quint. In this paper, we obtain the following pointwise bound of $\psi_\Gamma$: for all $v\in \mathfrak{a}$, $$ \psi_\Gamma(v) \le \min_{1\le i\le k} \delta_{\alpha_i} \alpha_i(v) $$ where $\Delta=\{\alpha_1, \cdots, \alpha_k\}$ is the set of all simple roots of $(\mathfrak{g},\mathfrak{a})$ and $0<\delta_{\alpha_i}\le \infty$ is the critical exponent of $\Gamma$ associated to $\alpha_i$. When $\Gamma$ is $\Delta$-Anosov, there are precisely $k$-number of directions where the equality is achieved, and the following strict inequality holds for $k\ge 2$: for all $v\in \mathfrak{a}-\{0\}$, $$\psi_\Gamma(v) <\frac{1}{k}\sum_{i=1}^k \delta_{\alpha_i} \alpha_i (v).$$ We discuss applications for self-joinings of convex cocompact subgroups in $\prod_{i=1}^k \operatorname{SO}(n_i,1)$ and Hitchin subgroups of $\operatorname{PSL}(d, \mathbb{R})$. In particular, for a Zariski dense Hitchin subgroup $\Gamma<\text{PSL}(d, \mathbb{R})$, we obtain that for any $ v=\operatorname{diag}(t_1, \cdots, t_d)\in \mathfrak{a}^+$, $$\psi_\Gamma (v) \le \min_{1\le i\le d-1} (t_i -t_{i+1}). $$