382 research outputs found
Unimodularity of Invariant Random Subgroups
An invariant random subgroup is a random closed subgroup whose law
is invariant to conjugation by all elements of . When is locally compact
and second countable, we show that for every invariant random subgroup there almost surely exists an invariant measure on . Equivalently, the
modular function of is almost surely equal to the modular function of ,
restricted to .
We use this result to construct invariant measures on orbit equivalence
relations of measure preserving actions. Additionally, we prove a mass
transport principle for discrete or compact invariant random subgroups.Comment: 23 pages, one figur
Thick hyperbolic 3-manifolds with bounded rank
We construct a geometric decomposition for the convex core of a thick
hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in
terms of rank and injectivity radius on the Heegaard genus of M and on the
radius of any embedded ball in the convex core of M.Comment: 170 pages, 17 figure
Unimodular measures on the space of all Riemannian manifolds
We study unimodular measures on the space of all pointed
Riemannian -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 with large, finite volume by
passing to unimodular limits.
We develop a structure theory for unimodular measures on ,
characterizing them via invariance under a certain geodesic flow, and showing
that they correspond to transverse measures on a foliated `desingularization'
of . 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
-manifolds with finitely generated fundamental group.Comment: 81 page
Invariant random subgroups of semidirect products
We study invariant random subgroups (IRSs) of semidirect products . In particular, we characterize all IRSs of parabolic subgroups
of , and show that all ergodic IRSs of are either of the form for some IRS of , or are induced from IRSs of
, where is a
lattice.Comment: 16 page
Extending pseudo-Anosov maps to compression bodies
We show that a pseudo-Anosov map on a boundary component of an irreducible
3-manifold has a power that partially extends to the interior if and only if
its (un)stable lamination is a projective limit of meridians. The proof is
through 3-dimensional hyperbolic geometry, and involves an investigation of
algebraic limits of convex cocompact compression bodies.Comment: 29 page
A finiteness theorem for hyperbolic 3-manifolds
We prove that there are only finitely many closed hyperbolic 3-manifolds with
injectivity radius and first eigenvalue of the Laplacian bounded below whose
fundamental groups can be generated by a given number of elements. An
application to arithmetic manifolds is also given.Comment: 20 pages, to appear in Journal of the London Mathematical Societ
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