Content: This module is an introduction to the field of geometric group theory. The basic premise of this field is that topological and geometric methods can be applied to the study of finitely generated groups by studying actions of these groups on various spaces. For example restrictions on the topology and curvature of the space can have strong algebraic consequences for groups that act "properly." Although this basic idea can be traced back a century or more, the subject exploded in the 1980s with work of Thurston and Gromov, and has become a major area of current research. Prominent roles in geometric group theory are played by low dimensional topology and hyperbolic geometry, but it has points of contact with and borrows techniques from a wide range of mathematical subjects.

Learning outcomes: Familiarity with classes of groups commonly studied in geometric group theory, the spaces they act on and ability to use features of these spaces to extract information about the groups.

Books:

Löh, Geometric Group Theory, An Introduction : Universitext, Springer (2017)
P. de la Harpe, Topics in Geometric Group Theory : Chicago lectures in mathematics, University of Chicago Press (2000)
M. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature : Grundlehren der Math. Wiss. No. 319, Springer (1999)
C. Druţu, M. Kapovich, Geometric Group Theory : Colloquium publications, Vol. 63, American Mathematics Society (2018)
A. Casson and S. Bleiler, Automorphisms of Surfaces after Thurston. Cambridge University Press (1988)