## University of Sydney Algebra Seminar

# Scott Murray (University of Sydney)

## Friday 4th December, 12.05-12.55pm, Carslaw 375

### Computing fundamental domains for congruence subgroups of SL2

This is joint work with Lisa Carbone and Leigh Cobbs of Rutgers University.

The Bass-Serre theory of groups acting on trees is vital to the structure
theory of certain infinite groups. In this talk, I consider the action of
SL_{2}(F_{q}((t))) and PGL_{2}(F_{q}((t))) on their Bruhat-Tits graph (or
building). By quotienting out the action of congruence subgroups, we get
a class of graphs called fundamental domains. In addition to their
importance in group theory, this construction is believed to give families
of expander graphs.

Before our work, very few of these graphs had been explicitly constructed. We used Magma to construct them. These constructions allowed us to make new conjectures on the structure of these graphs, some of which we have been able to prove. We also found and corrected some errors in previously published results.