## Virtual Seminar on

## Geometry and Topology

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

## How to join

**Zoom Link** http://cayley.kr/vz

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

## 2022

**August 12 (F), 10 am Korea**** (hybrid)**

**Note the date/location change!**

**In-Person : KIAS ****8101**

**Online : ****http://cayley.kr/vz**

Mark Pengitore (UVa)

*Residual finiteness of the mapping class group with respect to solvable covers*

*In this talk, we will quantify residual finiteness of the mapping class group of finite type with respect to congruence quotients coming from characteristic finite index covers of the underlying surface with solvable deck group. We refer to these quoteints as congruence quotients of solvable type. In particular, we construct an infinite sequence of mapping classes where the minimal congruence quotient of solvable type that detects one of these mapping classes has size which is super polynomial in word length.*

**August 17, 18, 19 (W/Th/F), 10 am Korea**** (hybrid)**

**In-Person : KIAS 1503**

**Online : http://cayley.kr/vz**

Damian Osajda (Universitoy of Wrocław, IMPAN)

*Locally elliptic actions and nonpostitive curvature*

The talks are based on joint works with Karol Duda, Thomas Haettel, Sergey Norin, and Piotr Przytycki.

There are numerous questions concerning actions of torsion groups on nonpositivelycurved spaces. One example is a well-known conjecture stating that subgroups of CAT(0) groups, that is, of groups acting geometrically on CAT(0)spaces, do notc ontain infinite torsion (every element has finite order) subgroups.Proving this conjecture might be seen as a first step towards establishing the Tits Alternative for CAT(0) groups. Notethat for torsiongroups every element acts elliptically on the CAT(0) space, that is, fixes a point. This leads to a concept of locally elliptic actions, that is, actions in which the fixed point set of each element is nonempty.

Generalizing the above conjecture and a number of other related open questions we state the following Meta-Conjecture: Every locally elliptic actionof a finitely generated group on a finite dimensional nonpositively curved complex is elliptic, that is, has a global fixed point. Here, `nonpositively curved complex' could mean: CAT(0), Gromov hyperbolic, small cancellation, systolic, or Helly polyhedral complex, and many others.`Finite dimensional' could refer to finite dimensionality as a polyhedral complex.The assumptions in the Meta-Conjecture are important. Every not finitely generated group acts on a tree without a fixed point. Some infinite torsion groups, even some infinite Burnside groups (there is a common bound on orders of elements) act without fixed points on infinite dimensional CAT(0) spaces. Also the combinatorial setting, that is, considering complexes instead of arbitrary metric spaces is essential.

In the talks I will explain in details motivations for the Meta-Conjecture, some of its consequences, and relations to well-known open problems. I will present the actualconjectures being specifications of the Meta-Conjecture and explain state of the art,focusing on recent results of my collaborators and myself, and providing (ideas of) proofs.

**August 24 (W), 10 am Korea (hybrid)**

**In-Person : KIAS 1503**

**Online : http://cayley.kr/vz**

Seung-Yeon Ryoo (Princeton)

*Vertical versus horizontal inequalities on nilpotent Lie groups and groups of polynomial growth*

It is known that simply connected nonabelian nilpotent Lie groups and not virtually abelian groups of polynomial growth fail to embed bilipschitzly into uniformly convex Banach spaces, because these groups have Carnot groups as asymptotic cone and because Carnot groups fail to embed bilipschitzly into uniformly convex Banach spaces by the Pansu--Semmes nonembeddability argument. We quantify this fact by providing a lower bound on the distortion of balls in the aforementioned groups into uniformly convex spaces. In particular, we show that the $L^p$-distortion, $(1<p<\infty)$, of a ball of radius $n\ge 2$ in the aforementioned groups is exactly $(\log n)^{1/\max\{p,2\}}$ up to constants. We achieve this by establishing ``vertical-versus-horizontal Poincaré inequalities'' which are specifically tailored to measuring the bilipschitz distortion of these balls and which demonstrate a collapse of embeddings along central directions. We provide some conjectural bounds on the L^1-distortion of these balls.