##### Department of Mathematics,

University of California San Diego

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### Math 211B - Group Actions Seminar

## Prof. Joshua Bowman

#### Pepperdine University

## Horocycle flow on $\mathcal{H}(2)$ and the gap distribution for slopes of saddle connections

##### Abstract:

Saddle connections on a translation surface generalize both diagonals in a polygon and primitive vectors in a 2-dimensional lattice. Their slopes thus contain geometric and algebraic information about the surface. Slopes of saddle connections can be studied using the action of a horocycle subgroup of $\mathrm{SL}_2(\mathbb{R})$ on the moduli space of all translation surfaces. In particular, gaps between slopes are directly related to the return time function of a PoincarĂ© section for the horocycle flow.

In this talk, we will describe a PoincarĂ© section for horocycle flow in the smallest nontrivial stratum $\mathcal{H}(2)$ and see how to compute the return time function. Then we will examine some consequences for gap distributions. This is joint work with Anthony Sanchez.

Host: Brandon Seward

### May 23, 2024

### 10:00 AM

APM 7321

Research Areas

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