Unimodular measures on the space of all Riemannian manifolds

We study unimodular measures on the space Md of all pointed Riemannian d– manifolds. Examples can be constructed from finite-volume manifolds, from mea-sured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak* 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 Md, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated “desingularization” of Md. 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. © 2022, Mathematical Sciences Publishers. All rights reserved.