## 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)

## 2022

**Mar**** ****3, 4**** (****Th, F****) ****10 -11**** am Korea**

Lei Chen (University of Maryland, College Park)

*TBA*

## 2021

**December**** ****2**** (****Th****) ****9**** - 1****0**** am Korea**

Minju Lee (IAS)

*Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends.*

This is joint work with Hee Oh. We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\mathrm{SO}(d,1)$ acting on the space $\Gamma\backslash\mathrm{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash\mathbb{H}^d$ is a convex cocompact manifold with

Fuchsian ends. For $d = 3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate subgroups causes serious issues, but in the end, all orbit closures of unipotent flows are relatively

homogeneous. Our results imply the following: for any $k\geq 1$,

(1) the closure of any $k$-horosphere in $M$ is a properly immersed submanifold;

(2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold;

(3) an infinite sequence of maximal properly immersed geodesic $(k+1)$-planes intersecting $\mathrm{core} M$ becomes dense in $M$.

**November**** ****3**** (W) ****10**** - ****11**** ****a****m Korea**

Hongbin Sun (Rutgers University)

*All finitely generated 3-manifold groups are Grothendieck rigid*

A finitely generated residually finite group G is said to be Grothendieck rigid if for any finitely generated proper subgroup H<G, the inclusion induced homomorphism \hat{H}\to \hat{G} on their profinite completions is not an isomorphism. There do exist finitely presented groups that are not Grothendieck rigid. We will prove that, if we restrict to the family of finitely generated 3-manifold groups, then all these groups are Grothendieck rigid. The proof relies on a precise description on non-separable subgroups of 3-manifold groups.

**October 13**** (****W****) 4 - 5 pm Korea**

Javier de la Nuez GonzĂˇlez (KIAS)

**Title**: Formal solutions and their role in the study of the first order theory of free groups

**Abstract**: I will discuss the existence of formal solutions for positive and related sentences valid in free groups. I will also hint at their role in Sela's solution to the Tarski problem, together with that of other tools.