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

Abstract

In this article, we consider the product space of several non-compact finite volume hyperbolic spaces, $V_1,V_2,\dots ,V_k$ of dimension $n$. Let ${\mathrm{T}}^1(V_i)$ denote the unit tangent bundle of $V_i$ for each $i=1,\dots ,k$, then for every $(v_1,\dots ,v_k)\in {\mathrm{T}}^1(V_1)\times \cdots \times {\mathrm{T}}^1(V_k)$, the diagonal geodesic flow $g_t$ is defined by $g_t(v_1,\dots ,v_k)=(g_t{v}_1,\dots ,g_t{v}_k)$. And we define $${\mathfrak{D}}_k=\left\{(v_1,\dots ,v_k)\in {\mathrm{T}}^1(V_1)\times \cdots \times {\mathrm{T}}^1(V_k):g_t(v_1,\dots ,v_k)\text{ divergent, as }t\to \infty \right\}.$$ We will prove that the Hausdorff dimension of ${\mathfrak{D}}_k$ is equal to $k(2n-1)-\frac{n-1}{2}$. This extends the result of Yitwah Cheung ~\cite{Cheung1}.

29 pages. The paper is rewritten according to referees' suggestions