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Unimodular measures on the space of all Riemannian manifolds

### Miklós Abért and Ian Biringer

Geometry & Topology 26 (2022) 2295–2404
##### Abstract

We study unimodular measures on the space ${\mathsc{ℳ}}^{d}$ 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 ${\mathsc{ℳ}}^{d}$, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated “desingularization” of ${\mathsc{ℳ}}^{d}$. 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.

##### Keywords
unimodular measures, Benjamini–Schramm convergence, invariant random subgroup, hyperbolic geometry, Lie group
##### Mathematical Subject Classification
Primary: 28D99, 53C12, 57K32
##### Publication
Revised: 31 May 2021
Accepted: 6 July 2021
Published: 12 December 2022