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* 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.

81 pages, updated with some minor changes and corrections

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Cited by in corpus (4)

• Eigenfunctions and Random Waves in the Benjamini-Schramm limit
• Metrizing the Chabauty topology
• Genericity of chaos for colored graphs
• On the Chabauty space of ${\text{PSL}}_2(\mathbb{R})$, I: lattices and grafting