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28 March 2024
 
  » arxiv » 1606.3360

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Unimodular measures on the space of all Riemannian manifolds
Ian Biringer ; Miklos Abert ;
Date 10 Jun 2016
AbstractWe 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.
As motivation, we explain how unimodular measures can play the role of invariant random subgroups in the proof of the uniform L"uck approximation theorem of ABBGNRS (this https URL). We also show how a rank rigidity result of Adams for measured foliations can be translated using our machinery into a theorem for finite volume manifolds $M$ that only requires geometric control over a large proportion of $M$.
Source arXiv, 1606.3360
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