Unimodular measures on the space of all Riemannian manifolds

TL;DR

This paper investigates unimodular measures on the space of all pointed Riemannian manifolds, developing a structure theory, exploring their limits, and applying these concepts to understand the geometry of finite volume manifolds.

Contribution

It introduces a comprehensive structure theory for unimodular measures, linking them to invariance properties and transverse measures, and applies these ideas to compactness and hyperbolic 3-manifolds.

Findings

Unimodular measures are preserved under weak* limits.

They can be used to compactify sets of finite volume manifolds.

A geometric proof of compactness for negatively curved manifolds is provided.

Abstract

We study unimodular measures on the space $\mathcal M^d$ of all pointed Riemannian $d$-manifolds. Examples can be constructed from finite volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak* limits, and under certain geometric constraints (e.g. bounded geometry) unimodular measures can be used to compactify sets of finite volume manifolds. One can then understand the geometry of manifolds $M$ with large, finite volume by passing to unimodular limits. We develop a structure theory for unimodular measures on $\mathcal M^d$, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated `desingularization' of $\mathcal M^d$. We also give a geometric proof of a compactness theorem for unimodular measures on the space of pointed manifolds with pinched negative curvature, and characterize unimodular measures supported on hyperbolic $3$-manifolds with finitely generated fundamental group.