Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Unimodular measures on the space of all Riemannian manifolds

Miklós Abért and Ian Biringer

Geometry & Topology 26 (2022) 2295–2404

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.

unimodular measures, Benjamini–Schramm convergence, invariant random subgroup, hyperbolic geometry, Lie group
Mathematical Subject Classification
Primary: 28D99, 53C12, 57K32
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
Miklós Abért
Alfréd Rényi Institute of Mathematics
Ian Biringer
Department of Mathematics
Boston College
Chestnut Hill, MA
United States