Ends of unimodular random manifolds

TL;DR

This paper investigates the structure of ends in unimodular random manifolds, providing detailed descriptions for surfaces, foliations, and hyperbolic quotients under invariant measures.

Contribution

It introduces a general framework for analyzing ends of unimodular random manifolds and applies it to specific cases like surfaces and hyperbolic quotients.

Findings

Characterization of ends for generic manifolds under unimodular measures

Description of generic leaves in foliations with invariant measures

Analysis of quotients of hyperbolic plane by invariant random subgroups

Abstract

We study the ends of a generic manifold, with respect to a unimodular measure on the space of pointed Riemannian manifolds with bounded curvatures. We apply our general result to the case of surfaces and obtain as corollaries a very precise description of generic leaves for foliations with invariant measures and of quotients of the hyperbolic plane by invariant random subgroups of the isometry group.