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2016 Wolfe Lecture in Mathematics

John Pardon

Stanford University


Thursday April 14, 2016
4:00pm, Herman Brown 227


Title: Totally disconnected groups (not) acting on three-manifold

 

Abstract:

    Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery--Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery--Zippin) that it suffices to rule out the case of the additive group of $p$-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.

 

 

The Annual Wolfe Lectures are supported through a generous gift from the estate of Raquel A. Wolfe in loving memory of her husband, Dr. Alfred S. Wolfe, and his love for math which he instilled in both his children.

 

Updated April 7, 2016