Bounded Geodesics on Locally Symmetric Spaces

TL;DR

The paper investigates bounded geodesics on locally symmetric spaces of non-compact type with finite volume, focusing on the set of directions at a point whose geodesic rays remain bounded. It proves that for the specific case $Y=SO_3(\ ext{R})\backslash SL_3(\ ext{R})/\Gamma$ (with $\Gamma$ commensurable to $SL_3(\mathbb{Z})$), the direction set $E^+(y,\infty)$ is hyperplane absolute winning (HAW) on the unit tangent sphere, a strengthening of previous thickness results and implying robust intersection properties. The core method combines homogeneous dynamics with Schmidt games, recasting the boundedness problem as a dynamical-recurrence property on $X=G/\Gamma$ and exploiting the hyperplane absolute game (HAW) framework to obtain thickness under countable intersections and diffeomorphisms. A key technical pillar is Theorem ru, which handles a fibered Diophantine-approximation argument across parameter slices, supported by two technical lemmas (Lmain1 and Lmain2) that control multiscale geometry via hyperplane constraints; together these yield the main theorems and their corollaries, including extending results to products of $SL_2$ and consequences for thickness in the associated tangent spaces.

Abstract

Let $Γ$ be a torsion-free subgroup of $SL_3(R)$ commensurable with $SL_3(Z)$, and $Y=SO_3(R)\backslash SL_3(R)/Γ$ be endowed with the natural locally symmetric space structure. We prove that for any point y in Y, the set of directions in which the geodesic ray starting from y is bounded in Y, is hyperplane absolute winning.

Theorems & Definitions (40)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7
• Corollary 1.8
• Remark 1.9
• Lemma 2.1: BFKRW