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Unimodular measures on the space of all Riemannian manifolds | Ian Biringer
; Miklos Abert
; | Date: |
10 Jun 2016 | Abstract: | We 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 | Services: | Forum | Review | PDF | Favorites |
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