## Virtual Seminar on

## Geometry and Topology

### https://visgat.cayley.kr

## How to join

**Zoom Meeting ID** 739-035-2844

**Password** Zero minus the Euler characteristic of a closed oriented surface of genus 4000

**Time **Wed 4 pm or Thursday 9 am (or so) in KST (Korea Standard Time)

## 2021

**April 21 (W) 4 pm Korea **

Yi Liu (BICMR)

*Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups*

In this talk, I will outline a proof for showing that the profinite completion of the fundamental group determines finite-volume hyperbolic 3-manifolds up to finitely many possibilities. As one of the main steps, I will explain why the Thurston norm on the first (real) cohomology is determined by the completion.

**May 13 (Th) 4 pm Korea**

Masato Mimura (Tohoku U)

*The Green--Tao theorem for number fields*

Joint work with colleagues in Tohoku University: Wataru Kai, Shin-ichiro Seki, Akihiro Munemasa and Kiyoto Yoshino. Ben Green and Terence Tao have proved that the set of rational primes contains arbitrarily long arithmetic progressions. We establish generalizations of this theorem in the setting of (arbitrary) number fields. No serious background on number theory is assumed for this talk. For the preprint, see https://arxiv.org/abs/2012.15669

**May 27 (Th) 10 am Korea (= May 26 9 pm EST)**

Tarik Aougab (Haverford College)

*Simple length rigidity for covers*

Suppose X and Y are finite covers of a fixed hyperbolic surface S. We first show that if for all closed curves gamma on S, gamma admits a simple lift to X if and only if it does to Y, then X and Y are equivalent covers. Using similar ideas, we address the question of when two covers of a fixed hyperbolic surface are isometric when their unmarked simple length spectra agree. We outline some sufficient criteria on the covers for this and generate families of examples. This represents joint work with Max Lahn, Marissa Loving, and Nick Miller.