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

TL;DR

This paper determines the Hausdorff dimension of divergent diagonal geodesics in the product of finite volume hyperbolic spaces, extending previous results to higher dimensions and multiple spaces.

Contribution

It provides a precise calculation of the Hausdorff dimension for divergent geodesics in product hyperbolic spaces, generalizing earlier work by Cheung.

Findings

Hausdorff dimension of divergent geodesics is k(2n-1) - (n-1)/2

Extends Cheung's results to multiple hyperbolic spaces

Provides a formula applicable to various dimensions and products

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 $$\mathfrak{D}_k =\left\{ (v_1, \dots, v_k) \in \mathrm{T}^1 (V_1) \times \cdots \times \mathrm{T}^1 (V_k): g_t(v_1, \dots, v_k) \text{ divergent, as } t\rightarrow \infty\right\}. $$ 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}.