Trace Minimization and Roots in ${\rm PSL}(2,\mathbb{R})$
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| Format: | Preprint |
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2025
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| author | Kreuzer, Martin Moldenhauer, Anja Rosenberger, Gerhard |
| author_facet | Kreuzer, Martin Moldenhauer, Anja Rosenberger, Gerhard |
| contents | 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 $τ={\rm tr}([A,B])$ 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 $τ\le -2$, this can be checked with an explicit formula. For $τ> 2$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trace Minimization and Roots in ${\rm PSL}(2,\mathbb{R})$ Kreuzer, Martin Moldenhauer, Anja Rosenberger, Gerhard Group Theory 20G20 (Primary) 20-08, 20E05, 20G07 (Secondary) 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 $τ={\rm tr}([A,B])$ 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 $τ\le -2$, this can be checked with an explicit formula. For $τ> 2$, 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. |
| title | Trace Minimization and Roots in ${\rm PSL}(2,\mathbb{R})$ |
| topic | Group Theory 20G20 (Primary) 20-08, 20E05, 20G07 (Secondary) |
| url | https://arxiv.org/abs/2508.06185 |