Expander graphs arising from finite groups

Martin Kassabov

Cornell University

Informally, expander graphs are  graphs which can not be easily

disconnected.  In the case of bounded degree, this property is

equivalent to a spectral gap in the Laplacian matrix of the graph.

Margulis was the first to find an explicit construction of expander

graphs, relating expansion to Kazhdan’s property $T$. I will outline

this connection and construct several families of expander graphs.

Groups with property $T$ and infinitely many alternating quotients

Martin Kassabov

Cornell University

I will outline several methods for showing property $T$ — one of the

methods which originated in the work of Dymara and Januszkiewicz

uses the geometry of Hilbert spaces. I will use an extension of the

method to show that certain subgroups of

$\mathsf{Aut}(\mathbb{F}_p[x,y,z])$ have property $T$. As a

consequence for any prime $p$, we construct 3 permutations in

$\mathsf{Alt}(p^3-1)$ which not only generate the group but also

make the resulting Cayley graphs expanders.

This is joint work with Pierre-Emmanuel Caprace.