Fundamental domains for congruence subgroups of SL2 in positive characteristic

TL;DR

This paper constructs explicit fundamental domains for congruence subgroups of SL_2 and PGL_2 over polynomial rings in positive characteristic, using Bruhat-Tits trees and computer algebra for detailed computations.

Contribution

It introduces a method to explicitly compute fundamental domains for these subgroups in positive characteristic using coset descriptions and computational tools.

Findings

Explicit fundamental domains for several congruence subgroups

Graph of groups representations of the domains

Use of Magma for computational verification

Abstract

In this work, we construct fundamental domains for congruence subgroups of $SL_2(F_q[t])$ and $PGL_2(F_q[t])$. 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.