Tent property of the growth indicator functions and applications

Abstract

Let $\Gamma$ be a Zariski dense discrete subgroup of a connected semisimple real algebraic group G. Let $k={\text {rank}} G$. Let $\psi _\Gamma :\mathfrak {a} \rightarrow \mathbb {R}\cup \{-\infty \}$ 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 $v\in \mathfrak {a}$,

$$\begin{aligned} \psi _\Gamma (v) \le \min _{1\le i\le k} \delta _{\alpha _i} \alpha _i(v) \end{aligned}$$

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 $v\in \mathfrak {a}-\{0\}$,

$$\begin{aligned} \psi _\Gamma (v) <\frac{1}{k}\sum _{i=1}^k \delta _{\alpha _i} \alpha _i (v). \end{aligned}$$

We discuss applications for self-joinings of convex cocompact subgroups in $\prod _{i=1}^k {\text {SO}}(n_i,1)$ and Hitchin subgroups of ${\text {PSL}}(d, \mathbb {R})$. In particular, for a Zariski dense Hitchin subgroup $\Gamma <\text {PSL}(d, \mathbb {R})$, we obtain that for any $v={\text {diag}}(t_1, \cdots , t_d)\in \mathfrak {a}^+$,

$$\begin{aligned} \psi _\Gamma (v) \le \min _{1\le i\le d-1} (t_i -t_{i+1}). \end{aligned}$$