250090 VO Geometric and asymptotic group theory (2014W)

The purpose of this course is to give an introduction to coarse embeddings of infinite graphs and groups.

A coarse embedding is a far-reaching generalization of an isometric embedding. The concept was introduced by Gromov in 1993. It plays a crucial role in the study of large-scale geometry of infinite groups and the Novikov higher signature conjecture. Coarse amenability, also known as Guoliang Yu's property A, is a weak amenability-type condition that is satisfied by many known metric spaces. It implies the existence of a coarse embedding into a Hilbert space.

Coarse embeddings and related constructions find applications in modern geometric group theory, algebraic topology, and theoretical computer science.

In this introductory course, we discuss the interplay between infinite expander graphs, coarse amenability, and coarse embeddings. We present several 'monster' constructions in the setting of metric spaces of bounded geometry and finitely generated groups.

The course is open to students of all degrees (Bachelor, Master or PhD). The knowledge of the following fundamental concepts is required: graph, group, free group, presentation of a group by generators and relators, fundamental group.