Infinite volume ends of quotient graphs and homogeneous spaces

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.