Discrete two-generator subgroups of ${\rm PSL_2}$ over non-archimedean local fields

TL;DR

This paper establishes criteria and algorithms for determining the discreteness and density of two-generator subgroups in ${ m PSL_2}$ over non-archimedean local fields, with applications to isometry groups of $ ext{Λ}$-trees.

Contribution

It provides necessary and sufficient conditions and practical algorithms for analyzing two-generator subgroups' discreteness and density in ${ m PSL_2}$ over non-archimedean fields, extending to actions on $ ext{Λ}$-trees.

Findings

Criteria for discreteness of subgroups in ${ m PSL_2}(K)$

Algorithms to decide subgroup discreteness and density

Structure theorem for two-generator groups acting on $ ext{Λ}$-trees

Abstract

Let $K$ be a non-archimedean local field with residue field of characteristic $p$. We give necessary and sufficient conditions for a two-generator subgroup $G$ of ${\rm PSL_2}(K)$ to be discrete, where either $K=\mathbb{Q}_p$ or $G$ contains no elements of order $p$. We give a practical algorithm to decide whether such a subgroup $G$ is discrete. We also give practical algorithms to decide whether a two-generator subgroup of either ${\rm SL_2}(\mathbb{R})$ or ${\rm SL_2}(K)$ (where $K$ is a finite extension of $\mathbb{Q}_p$) is dense. A crucial ingredient for this work is a structure theorem for two-generator groups acting by isometries on a $\Lambda$-tree.