Infinite volume ends of quotient graphs and homogeneous spaces

TL;DR

This paper introduces a new concept of infinite volume ends for lcsc spaces, demonstrating that under certain conditions, quotients by specific groups have exactly one infinite volume end, with applications to symmetric spaces and graphs.

Contribution

It defines the space of infinite volume ends for lcsc spaces and proves that certain quotients have exactly one infinite volume end, extending classical end theory.

Findings

Quotients of path-connected unimodular lcsc groups with property (T) have one infinite volume end.

Locally finite graphs with transitive group actions have exactly one end.

Infinite volume ends of quotients of symmetric spaces of noncompact type are uniquely determined.

Abstract

We introduce the space of infinite volume ends of a locally compact second countable (lcsc) space that admits a Radon measure. In certain cases, this coincides with the classical space of ends. Consider a discrete subgroup $\Gamma$ of a unimodular lcsc group $G$ that is not coamenable. Assume that $G$ has property (T) and the associated homogeneous space $G/\Gamma$ is equipped with the Haar measure. We demonstrate that if $G$ is path connected, then $G/\Gamma$ has exactly one infinite volume end. In a related vein, if $G$ acts transitively on a locally finite connected graph $X$ with compact open vertex stabilizers and the action of the subgroup $\Gamma$ is free, we show that $X/\Gamma$ has exactly one end. We also obtain identical results for certain discrete subgroups $\Gamma$ of nonamenable product groups $G$. These results can be applied to understand ends of Schreier graphs and infinite volume ends of quotients of symmetric spaces of noncompact type. For instance, for symmetric spaces $X$ of noncompact type without real or complex hyperbolic factors, every infinite-covolume quotient $\Gamma\backslash X$ has exactly one end of infinite Riemannian volume.