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Geometric and asymptotic group theory

Coarse embeddings of infinite graphs and groups

Lecture by Goulnara Arzhantseva
Problem session by Martin Finn-Sell
Dienstag, 10:00-12:00, Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock

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.

Exam information: The date for the first exam will be the 30th January 2015 and will take place at 9.00am in Seminarroom 11 (SR11). The exam itself will be a oral exam in which you will prepare a question from the list of questions chosen at random and present your solution. You may bring one side of A4 (that is one page of A4 written on only one side) of notes concerning the course with you to the exam.

To repeat: The first exam date is 30/01/2015, 9.00am in Seminarroom 11 (SR11)

List of exam questions

Lists of problems:

Blatt 1Blatt 2Blatt 3

Additional notes:

Notes on coarse embeddings

"Coarse non-amenability and coarse embeddings" Goulnara Arzhantseva, Erik Guentner and Jan Spakula, 2012, Geometry and Functional Analyis (GAFA) - This paper concerns coarse embeddability of homology 2-covers arising from finite quotients of free groups

If the above link is inaccessible (or asks you to pay), a copy is available at: Paper copy from Goulnara Arzhantseva's website