Trace Minimization and Roots in ${\rm PSL}(2,\mathbb{R})$

TL;DR

This paper investigates conditions under which subgroups generated by roots of elements in a discrete free group in ${ m PSL}(2, eal)$ are also discrete and free, providing explicit formulas and an algorithm for trace minimization.

Contribution

It introduces new formulas for powers and roots in ${ m PSL}(2, eal)$, and develops an explicit Trace Minimization Algorithm for analyzing subgroup discreteness.

Findings

Explicit formula for $ au \\le -2$ case

Trace Minimization Algorithm for $ au > 2$

Formulas for powers, roots, and traces in ${\rm PSL}(2,\real)$

Abstract

Suppose that $A,B \in {\rm PSL}(2,\mathbb{R})$ generate a discrete and free group of rank 2, and let $m,n\ge 1$. We consider subgroups $\langle R,S\rangle$ of ${\rm PSL}(2,\mathbb{R})$ generated by roots of $A$ and $B$, i.e., by elements such that $R^m=A$ and $S^n=B$. Depending on whether the commutator trace $\tau={\rm tr}([A,B])$ is larger or smaller than 2, we describe necessary and sufficient conditions for $\langle R,S\rangle$ to be discrete and free of rank 2. For $\tau\le -2$, this can be checked with an explicit formula. For $\tau > 2$, one has to use the Trace Minimization Algorithm. Besides an explicit formulation of this algorithm, we prove new formulas for the powers and roots of elements of ${\rm PSL}(2,\mathbb{R})$, their traces and their commutator traces. The case of positive rational exponents $m,n$ is treated, as well.