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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 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 (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 d, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated “desingularization” of 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
References
Publication
Received: 16 November 2020
Revised: 31 May 2021
Accepted: 6 July 2021
Published: 12 December 2022
Proposed: Mladen Bestvina
Seconded: David M Fisher, Dmitri Burago
Authors
Miklós Abért
Alfréd Rényi Institute of Mathematics
Budapest
Hungary
Ian Biringer
Department of Mathematics
Boston College
Chestnut Hill, MA
United States