Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds

Abstract

The classical result of Patterson and Sullivan says that for a non-elementary convex cocompact subgroup $\Gamma <{\text{SO}}^{\circ }(n,1)$, $n\ge 2$, the Hausdorff dimension of the limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let $\Delta$ be a finitely generated group and ${\rho }_i:\Delta \to {\text{SO}}^{\circ }(n_i,1)$ be a convex cocompact faithful representation of $\Delta$ for $1\le i\le k$. Associated to $\rho =({\rho }_1,\cdots ,{\rho }_k)$, we consider the following self-joining subgroup of ${\prod }_{i=1}^k\text{SO}(n_i,1)$: $$\Gamma =\left(\prod _{i=1}^k{\rho }_i\right)(\Delta )=\{({\rho }_1(g),\cdots ,{\rho }_k(g)):g\in \Delta \}.$$ (1). Denoting by $\Lambda \subset {\prod }_{i=1}^k{\mathbb{S}}^{n_i-1}$ the limit set of $\Gamma$, we first prove that $${\text{dim}}_H\Lambda =\underset{1\le i\le k}{\max }{\delta }_{{\rho }_i}$$ where ${\delta }_{{\rho }_i}$ is the critical exponent of the subgroup ${\rho }_i(\Delta )$. (2). Denoting by ${\Lambda }_u\subset \Lambda$ the $u$-directional limit set for each $u=(u_1,\cdots ,u_k)$ in the interior of the limit cone of $\Gamma$, we obtain that for $k\le 3$, $$\frac{{\psi }_{\Gamma }(u)}{{\max }_i{u}_i}\le {\text{dim}}_H{\Lambda }_u\le \frac{{\psi }_{\Gamma }(u)}{{\min }_i{u}_i}$$ where ${\psi }_{\Gamma }:{\mathbb{R}}^k\to \mathbb{R}\cup \{-\infty \}$ is the growth indicator function of $\Gamma$.

To appear in Journal of Modern Dynamics. arXiv admin note: substantial text overlap with arXiv:2112.00877