Ian Biringer

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.

Unimodularity of invariant random subgroups

An invariant random subgroup H≤G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H≤G there almost surely exists an invariant measure on G/H. Equivalently, the modular function of H is almost surely equal to the modular function of G, restricted to H. 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.

Automorphisms of the compression body graph

When

Invariant random subgroups of semidirect products

S

On the growth of L 2-invariants for sequences of lattices in Lie groups

is a closed, orientable surface with genus

On the growth of Betti numbers of locally symmetric spaces

g(S) \geq 2

Unimodular measures on the space of all Riemannian manifolds

, we show that the automorphism group of the compression body graph

Ranks of mapping tori via the curve complex

\mathcal{CB}(S)

On the Growth of

is the mapping class group. Here, vertices are compression bodies with exterior boundary

L^2

S