Fundamental domains for congruence subgroups of SL2 in positive characteristic

In this work, we construct fundamental domains for congruence subgroups of $S{L}_2(F_q[t])$ and $PG{L}_2(F_q[t])$. 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.