Fundamental domains for congruence subgroups of SL2 in positive characteristic

Lisa Carbone; Leigh Cobbs; Scott H. Murray

paper

Fundamental domains for congruence subgroups of SL2 in positive characteristic

arXiv:0909.0062

Abstract

In this work, we construct fundamental domains for congruence subgroups of $S L_{2} (F_{q} [t])$ and $P G L_{2} (F_{q} [t])$ . Our method uses Gekeler's description of the fundamental domains on the Bruhat- Tits tree $X = X_{q + 1}$ in terms of cosets of subgroups. We compute the fundamental domains for a number of congruence subgroups explicitly as graphs of groups using the computer algebra system Magma.