# Combinatorial and geometric group theory

Supervisor: Adam Piggott

The symmetries of an object have an algebraic structure. Group theory is the study of such structures. In combinatorial group theory, groups are specified via group presentations. This means that we specify an alphabet of symbols, often only a few symbols, and some algebra rules which hold in the group. Everything else about the group must be deduced from the rules we specify. In geometric group theory, we exploit deep connections between groups and geometric structures. There is a sense in which a group itself is a geometric object, and every geometric object comes equipped with a group of symmetries (the isometries). We can use geometry to learn about groups, and we can learn about geometric structures using group theory.

I will be happy to talk to any honours, masters or Ph.D. student interested in combinatorial and/or geometric group theory to see if we can find a topic which suits their interests. An interested student may wish to peruse the book Office Hours with a Geometric Group Theorist, edited by Matt Clay & Dan Margalit to get a feel for some of the topics in geometric group theory. For combinatorial group theory I suggest browsing through Combinatorial Group Theoryby Lyndon and Shupp.