On computing quaternion quotient graphs for function fields

TL;DR

This paper presents an algorithm to compute fundamental domains and group presentations for quaternion unit groups acting on Bruhat-Tits trees over function fields, extending classical Fuchsian group actions.

Contribution

It introduces a novel algorithm for fundamental domain computation and explicit group presentation for quaternion unit groups over function fields.

Findings

Algorithm computes fundamental domains efficiently.

Provides explicit generators and relations for quaternion unit groups.

Establishes an upper bound on the algorithm's running time.

Abstract

Let $\Lambda$ be a maximal $\mathbb{F}_q[T]$-order in a division quaternion algebra over $\mathbb{F}_q(T)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units $\Lambda^*$ on the Bruhat-Tits tree $\mathcal{T}$ associated to $PGL_2(\mathbb{F}_q((1/T)))$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group $\Lambda^*$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that $\Lambda^*\backslash\mathcal{T}$ is {\em almost} Ramanujan.