Hausdorff dimension of divergent diagonal geodesics on product of finite-volume hyperbolic spaces

Abstract

In this paper, we consider the product space of several non-compact finite-volume hyperbolic spaces, $V_{1},V_{2},\ldots ,V_{k}$ of dimension $n$. Let $\text{T}^{1}(V_{i})$ denote the unit tangent bundle of $V_{i}$ and $g_{t}$ denote the geodesic flow on $\text{T}^{1}(V_{i})$ for each $i=1,\ldots ,k$. We define $$\begin{eqnarray}{\mathcal{D}}_{k}:=\{(v_{1},\ldots ,v_{k})\,\in \,\text{T}^{1}(V_{1})\times \cdots \times \text{T}^{1}(V_{k})\,:\,(g_{t}(v_{1}),\ldots ,g_{t}(v_{k}))\text{ diverges as }t\rightarrow \infty \}.\end{eqnarray}$$ We will prove that the Hausdorff dimension of ${\mathcal{D}}_{k}$ is equal to $k(2n-1)-((n-1)/2)$. This extends a result of Cheung.