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Topology/Geometry Seminar

On local residual finiteness of abstract commensurators of Fuchsian groups 

Khalid Bou-Rabee (CUNY)

Location:  Room 005
Date & time: Tuesday, 19 February 2019 at 3:30PM - 4:30PM

Abstract:  The abstract commensurator (aka “virtual automorphisms”) of a group encodes “hidden symmetries”, and is a natural generalization of the automorphism group. In this talk, I will give an introduction to these mysterious and classical groups and then discuss their residual finiteness. Recall that residual finiteness is a property enjoyed by linear groups (by A. I. Malcev), mapping class groups of closed oriented surfaces (by EK Grossman), and branch groups (by definition!). Moreover, by work of Armand Borel, Gregory Margulis, G. D. Mostow, and Gopal Prasad, the abstract commensurator of any irreducible lattice in any “nice enough” semisimple Lie group is locally residually finite (a property is termed “local” if it is satisfied by every finitely generated subgroup of the group). “Nice enough” is sufficiently broad that the only remaining unknown case is PSL_2(R). Are abstract commensurators of lattices in PSL_2(R) locally residually finite? Lattices here are commensurable with either a free group of rank 2 or the fundamental group of an oriented surface of genus 2.  I will present a complete answer to this question with a proof that is computer-assisted. Our answer and methods open up new questions and research directions, so graduate students are especially encouraged to attend.

This talk covers joint work with Daniel Studenmund.


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