Unimodular measures on the space of all Riemannian manifolds

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${}^{\ast }$ limits, and under certain geometric constraints (eg 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.