A Spectral Strong Approximation Theorem for Measure Preserving Actions

Let $\Gamma$ be a finitely generated group acting by probability measure preserving maps on the standard Borel space $(X,\mu)$. We show that if $H\leq\Gamma$ is a subgroup with relative spectral radius greater than the global spectral radius of the action, then $H$ acts with finitely many ergodic components and spectral gap on $(X,\mu)$. This answers a question of Shalom who proved this for normal subgroups.Comment: 17 page

Unimodular measures on the space of all Riemannian manifolds

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.Comment: 81 page