# Geometric Group Theory (GGT)

### Computational Aspects of Geometric Group Theory

Directed by Kim Ruane and Genevieve Walsh

*Recommended Book to get yourself ready: *“Groups, Graphs and Trees”

by John Meier. The link is https://www.cambridge.org/core/books/groups-graphs-and-trees/4D0D919000500E7ED9964962EEFCB871

*Description:* We’ll use computational group theory and hyperbolic geometry tools (GAP, MAGMA, and SnapPY) to understand interesting subgroups of infinite groups. We are focusing on certain well-known classes of groups, right-angled Coxeter and right-angled Artin groups. The figure above is a fundamental domain for a right-angled Coxeter group. We’ll start off by studying group presentations, Cayley graphs, and geometric examples. We’ll finish up by analyzing some specific kernels of maps to *Z*, and natural extensions of this.

*Topics
may include:*

- What algorithms are and how they can be used effectively in group theory;
- Learn how to use the computer program GAP and understand how the algorithms for this program work;
- Investigate how subgroups and various types of growth of subgroups can be detected;
- Normal forms and effective algorithms.

The students will become familiar with GAP and also with lim, a program that maps the orbits under Fuchsian and Kleinian groups.

*Level of Students and prerequisites: *A beginning course in abstract algebra, including basic group theory. Exceptions may be considered.

*Who
we are: Kim Ruane and Genevieve
Walsh: *We are
experienced researchers in geometric group
theory and topology who have “supervised” many students at various stages.
We love to see what students bring to the table.