# Geometry, Topology and Dynamics Seminar: Entropy, metrics and quasi-morphisms, by Michael Brandenbursky

September 16, 2019

3:00 PM - 3:50 PM

Michael Brandenbursky (Ben Gurion University): Entropy, metrics and quasi-morphisms

One of the mainstream and modern tools in the study of non abelian groups are quasi-morphisms.

These are functions from a group to the reals which satisfy homomorphism condition up to a bounded error.

Nowadays they are used in many fields of mathematics.

For instance, they are related to bounded cohomology, stable commutator length,

metrics on diffeomorphism groups, displacement of sets in symplectic topology, dynamics, knot theory,

orderability, and the study of mapping class groups and of concordance group of knots.

Let $S$ be a compact oriented surface. In this talk I will discuss several invariant metrics and

quasi-morphisms on the identity component ${\rm Diff}_0(S, {\rm area})$ of the group

of area preserving diffeomorphisms

of $S$. In particular, I will show that some quasi-morphisms on ${\rm Diff}_0(S, {\rm area})$

are related to the topological entropy.

More precisely, I will discuss a construction of infinitely many linearly independent

quasi-morphisms on ${\rm Diff}_0(S, {\rm area})$ whose absolute values bound from below the topological entropy.

If time permits, I will define a bi-invariant metric on this group, called the entropy metric,

and show that it is unbounded. Based on a joint work with M. Marcinkowski.

Please click here to make changes to, or delete, this seminar announcement.

## Date posted

Sep 20, 2019

## Date updated

Sep 20, 2019