## Virtual Seminar on

## Geometry and Topology

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

## Organizer

Hyungryul Baik (KAIST), Sang-hyun Kim (KIAS)

## How to join

Zoom http://cayley.kr/vz

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

## 2023

May 30, June 1, 10 - 11:15 am

Online http://shkim.org/vz

In person KIAS (1423)

Masato Mimura (Tohoku University)

Talk 1: An introduction to the theory of invariant quasimorphisms

Fix a group. A real valued function on it is called a quasimorphism if this satisfies the identity of being a group homomorphism up to uniformly bounded error. There are the following two types of quasimorphisms that might be considered as 'non-interesting' ones: genuine group homomorphisms and uniformly bounded maps. It is well known that the quotient vector space of that of all quasimorphisms over that of sums of functions of these two types is naturally isomorphic to the kernel of the comparison map from second bounded cohomology to second ordinary cohomology of the group. In many cases, this vector space is either the zero space or infinite dimensional.

In this talk, we take a pair (G,N) of a group G and its normal subgroup N. In this setting, we can define a notion of G-invariant quasimorphisms on N. We will present an introduction to this relatively new object. In particular, we can define a certain vector space related to invariant quasimorphisms; this can be finite-dimensional under a mild condition. For example, if G is the surface group of genus at least two and N is the commutator subgroup of G, then this vector space is one-dimensional. We will provide examples and motivations of quasimorphisms and invariant quasimorphisms in this first talk.

Talk 2: Applications of invariant quasimorphisms and stable mixed commutator length

In this second talk, we will present some applications of the theory of invariant quasimorphisms. Some examples are related to symplectic geometry and group actions on the circle. Also, for a group pair (G,N) in our setting, we can define the mixed commutator length on the mixed commutator subgroup [G,N], which is the word length with respect to the set of simple (G,N)-commutators. The stabilization of the mixed commutator length is called the mixed scl; this equals the scl when N=G. We study large scale behavior of the mixed scl; more precisely, we study it in the aspect of coarse groups, the concept recently developed by Leitner and Vigolo.

June 13, 2023

Online http://shkim.org/vz

In person KIAS (TBD)

Jiyoung Han (KIAS)

Homogeneous spaces of orthogonal groups of quadratic forms over real and p-adic fields

In the real case, the special orthogonal groups of indefinite quadratic forms of low dimensions are orientation-preserving isometry groups of real hyperbolic spaces or the product of two copies of the isometry group of the hyperbolic plane. Hence these spaces can be viewed as quotient spaces of special orthogonal groups of indefinite quadratic forms. In this talk, I will introduce quotient spaces of special orthogonal groups of indefinite quadratic forms over the p-adic field, which could be analogs of hyperbolic spaces from this point of view. And we will see how one can use this geometry of “hyperbolic spaces” in the generic quantitative Oppenheim conjecture in low dimensions.

June 20, 22 (tentative), 2023

Christian Rosendal (UMCP)

Online http://shkim.org/vz

In person KIAS (TBD)

KIAS--Springer Lectures Geometric group theory beyond locally compact groups

These lectures will provide an introduction to the geometrisation of topological groups, in particular, the large scale geometric aspects of topological transformation groups, such as homeomorphism groups of compact manifolds, mapping class groups of infinite-type surfaces, and automorphism groups of countable structures. We will show how general considerations lead to a canonical large scale geometric structure on every topological group and provide criteria for its metrisability and for when the structure can be further improved to a quasi-metric structure. We apply the framework to a few significant examples including homeomorphism groups and automorphism groups of graphs. Finally, we address the interplay between model theory of countable structures and the geometry of their automorphism groups.

June 27, 2023

Online http://shkim.org/vz

In person KIAS (TBD)

Minkyu Kim (KIAS)

TBA

TBA

July 4, 2023

Online http://shkim.org/vz

In person KIAS (TBD)

Seung Uk Jang (Rennes)

TBA

TBA

Auguest 29, 31 (tentative)

Online http://shkim.org/vz

In person KIAS (TBD)

Alan Reid (Rice)

TBA

TBA

September 19, 21 (tentative)

Online http://shkim.org/vz

In person KIAS (TBD)

Andrés Navas (Universidad de Santiago de Chile)

TBA

TBA

September 26 (tentative)

Online http://shkim.org/vz

In person KIAS (TBD)

Cristóbal Rivas (Universidad de Chile)

TBA

TBA

TBA (2023): Ser Peow Tan, Ken'ichi Ohshika, Jason Behrstock