On Discreteness of Commensurators

Commensurations and Metric Properties of Houghton’s Groups

We describe the automorphism groups and the abstract commensurators of Houghton’s groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.

Commensurators of Cusped Hyperbolic Manifolds

This paper describes a general algorithm for finding the commensurator of a non-arithmetic hyperbolic manifold with cusps, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all non-arithmetic hyperbolic once-punctur...

Abstract Commensurators of Profinite Groups

In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group G is a group Comm(G) which depends only on the commensurability class of G. We study various properties of Comm(G); in particular, we find two natural ways to turn it into a topological group. We also use Comm(G) to study topological groups which conta...

Rigidity of Commensurators and Irreducible Lattices

Abstract Commensurators of Braid Groups

Let Bn be the braid group on n ≥ 4 strands. We show that the abstract commensurator of Bn is isomorphic to Mod(S)⋉ (Q ⋉ Q), where Mod(S) is the extended mapping class group of the sphere with n + 1 punctures.