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 $k=\mathrm{rank}G$. 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 $v\in \mathfrak{a}$, $${\psi }_{\Gamma }(v)\le \underset{1\le i\le k}{\min }{\delta }_{{\alpha }_i}{\alpha }_i(v)$$ 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 ${\prod }_{i=1}^k\mathrm{SO}(n_i,1)$ and Hitchin subgroups of $\mathrm{PSL}(d,\mathbb{R})$. In particular, for a Zariski dense Hitchin subgroup $\Gamma <\text{PSL}(d,\mathbb{R})$, we obtain that for any $v=\mathrm{diag}(t_1,\cdots ,t_d)\in {\mathfrak{a}}^{+}$, $${\psi }_{\Gamma }(v)\le \underset{1\le i\le d-1}{\min }(t_i-t_{i+1}).$$

arXiv comment19 pages, 3 figures, Final version, To appear in Geometriae Dedicata