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 $Γ<\text{SO}^\circ (n,1)$, $n\ge 2$, the Hausdorff dimension of the limit set of $Γ$ is equal to the critical exponent of $Γ$. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let $Δ$ be a finitely generated group and $ρ_i:Δ\to \text{SO}^\circ(n_i,1)$ be a convex cocompact faithful representation of $Δ$ for $1\le i\le k$. Associated to $ρ=(ρ_1, \cdots, ρ_k)$, we consider the following self-joining subgroup of $\prod_{i=1}^k \text{SO}(n_i,1)$: $$Γ=\left(\prod_{i=1}^kρ_i\right)(Δ)=\{(ρ_1(g), \cdots, ρ_k(g)):g\in Δ\} .$$ (1). Denoting by $Λ\subset \prod_{i=1}^k \mathbb{S}^{n_i-1}$ the limit set of $Γ$, we first prove that $$\text{dim}_H Λ=\max_{1\le i\le k} δ_{ρ_i}$$ where $δ_{ρ_i}$ is the critical exponent of the subgroup $ρ_{i}(Δ)$. (2). Denoting by $Λ_u\subset Λ$ the $u$-directional limit set for each $u=(u_1, \cdots, u_k)$ in the interior of the limit cone of $Γ$, we obtain that for $k\le 3$, $$ \frac{ψ_Γ(u)}{\max_i u_i }\le \text{dim}_H Λ_u \le \frac{ψ_Γ(u)}{\min_i u_i }$$ where $ψ_Γ:\mathbb{R}^k\to \mathbb{R}\cup\{-\infty\}$ is the growth indicator function of $Γ$.