Coarse models of homogeneous spaces and translations-like actions

TL;DR

This paper explores translation-like actions of lattices in semisimple Lie groups, showing they can act on each other in bounded, free ways, revealing new geometric relationships among these groups.

Contribution

It extends previous work by demonstrating that cocompact lattices in certain Lie groups admit translation-like actions by , and establishes new quasi-isometric relationships between lattice quotients and homogeneous spaces.

Findings

Cocompact lattices in most semisimple Lie groups admit  translation-like actions.

Lattices in the unipotent radical act translation-like on other cocompact lattices.

Quasi-isometry between lattice quotients and homogeneous spaces is established.

Abstract

For finitely generated groups $G$ and $H$ equipped with word metrics, a translation-like action of $H$ on $G$ is a free action where each element of $H$ moves elements of $G$ a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group $\mathbf{G}$ that is not isogenous to $\mathrm{SL}(2,\mathbb{R})$ admit translation-like actions by $\mathbb{Z}^2$. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical $\mathbf{N}$ of the Borel subgroup $\mathbf{AN}$ of $\mathbf{G}$ acts translation-like on any cocompact lattice in $\mathbf{G}$. We also prove that for noncompact simple Lie groups $G,H$ with $H<G$ and lattices $\Gamma < G$ and $\Delta < H$, that $\Gamma/\Delta$ is quasi-isometric to $G/H$ where $\Gamma/\Delta$ is the quotient via a translation-like action of $\Delta$ on $\Gamma$.