# C3.2 Geometric Group Theory - Material for the year 2019-2020

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Some familiarity with Cayley graphs, fundamental group and covering spaces (as for example in the course B3.5 Topology & Groups) would be a helpful though not essential prerequisite.

16 lectures.

### Assessment type:

- Written Examination

The aim of this course is to introduce the fundamental methods and problems of geometric group theory and discuss their relationship to topology and geometry.

The first part of the course begins with an introduction to presentations and the list of problems of M. Dehn. It continues with the theory of group actions on trees and the structural study of fundamental groups of graphs of groups.

The second part of the course focuses on modern geometric techniques and it provides an introduction to the theory of Gromov hyperbolic groups.

Free groups. Group presentations. Dehn's problems. Residually finite groups.

Group actions on trees. Amalgams, HNN-extensions, graphs of groups, subgroup theorems for groups acting on trees.

Quasi-isometries. Hyperbolic groups. Solution of the word and conjugacy problem for hyperbolic groups.

If time allows: Small Cancellation Groups, Stallings Theorem, Boundaries.

- J.P. Serre,
*Trees*(Springer Verlag 1978). - M. Bridson, A. Haefliger,
*Metric Spaces of Non-positive Curvature, Part III*(Springer, 1999), Chapters I.8, III.H.1, III. Γ 5. - H. Short
*et al.*, 'Notes on word hyperbolic groups', Group Theory from a Geometrical Viewpoint, Proc. ICTP Trieste (eds E. Ghys, A. Haefliger, A. Verjovsky, World Scientific 1990), available online at: http://www.cmi.univ-mrs.fr/~hamish/ - C.F. Miller,
*Combinatorial Group Theory*, notes: http://www.ms.unimelb.edu.au/~cfm/notes/cgt-notes.pdf

- G. Baumslag,
*Topics in Combinatorial Group Theory*(Birkhauser, 1993). - O. Bogopolski,
*Introduction to Group Theory*(EMS Textbooks in Mathematics, 2008). - R. Lyndon, P. Schupp,
*Combinatorial Group Theory*(Springer, 2001). - W. Magnus, A. Karass, D. Solitar,
*Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations*(Dover Publications, 2004). - P. de la Harpe,
*Topics in Geometric Group Theory*, (University of Chicago Press, 2000).