##### Department of Mathematics,

University of California San Diego

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### Math 211B - Group Actions Seminar

## Michael Zshornack

#### UC Santa Barbara

## Twist flows and the arithmetic of surface group representations

##### Abstract:

Margulis's work on lattices and a number of questions on the existence of surface subgroups motivate the need for understanding arithmetic properties of spaces of surface group representations. In recent work with Jacques Audibert, we outline one possible approach towards understanding such properties for the Hitchin component, a particularly nice space of representations. We utilize the underlying geometry of this space to reduce questions about its arithmetic to questions about the arithmetic of certain algebraic groups, which in turn, allows us to characterize the rational points on these components. In this talk, I'll give an overview of the geometric methods behind the proof of our result and indicate some natural questions about the nature of the resulting surface group actions that follow.

Host: Brandon Seward

### March 7, 2024

### 10:00 AM

APM 7321

Research Areas

Ergodic Theory and Dynamical Systems****************************