##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Brian Longo

#### UCSD

## Super-approximation for linear groups in positive characteristic.

##### Abstract:

Let $\Omega\subseteq {\rm GL}_n(F_p(t))$ be a finite symmetric set containing the identity. Let $\Gamma$ be the group generated by $\Omega$ and let $\mathbb{G}$ be the Zariski-closure of $\Gamma$. We discuss conditions on which the family of Cayley graphs $\{{\rm Cay}(\Gamma ({\rm mod} Q), \Omega)\}$ is a family of Cayley graphs as $Q$ ranges through a certain subset $\Sigma$ of $F_p[t]$. This problem is a positive characteristic variation of the work of Bourgain-Gamburd, Varju, Salehi Golsefidy-Varju, and others. We focus on the difficulties that arise in positive characteristic. Please note: There will be pre-talk for graduate students from 2:30 - 3:00. The regular talk will begin at 3:00.

### November 2, 2015

### 2:00 PM

### AP&M 7218

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