Unimodular measures on the space of all Riemannian manifolds

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

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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

Volume 26, issue 5 (2022)

Miklós Abért and Ian Biringer

Abstract