# MA4H4 Geometric Group Theory

**Lecturer:** Brian Bowditch

**Term(s):** Term 2

**Status for Mathematics students:** List C

**Commitment:** 30 lectures

**Assessment:** 3 hour written examination (100%)

**Prerequisites:** MA222 Metric Spaces (strongly recommended), MA243 Geometry (recommended) MA3F1 Introduction to topology (recommended). Some connections with MA448 Hyperbolic Geometry.

**Leads To: **

**Content**: This will be an introduction to the basic ideas of geometric group theory. The main aim of subject is to apply geometric constructions to understand finitely generated groups. Although many of the ideas can be traced back a century or more, the modern subject has its origins in the 1980s and has rapidly grown into a major field in its own right. It draws on ideas from many subjects, though two particular sources of inspiration are low dimensional topology and hyperbolic geometry. A significant insight is that ``most'' finitely presented groups are ``hyperbolic'' in a broad sense. This has many profound applications. Some familiarity with group presentations will be useful. Beyond that, geometric or topological background is probably more relevant than algebraic background.

Learning outcomes: An understanding of the main notions of quasi-isometry, quasi-isometry invariants, and hyperbolic groups. To be able to apply these in particular examples.

**Books**:

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).

B. H. Bowditch, *A course on geometric group theory* : MSJ Memoirs, Vol 16, Mathematical Society of Japan (2006).