Non-freeness of Groups Generated by Two Parabolic Elements with Small Rational Parameters
| Autor: | Thomas Koberda, Sang-hyun Kim |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Discrete mathematics
Code (set theory) General Mathematics 010102 general mathematics Geometric Topology (math.GT) Group Theory (math.GR) 16. Peace & justice Mathematical proof 01 natural sciences Upper and lower bounds Mathematics - Geometric Topology 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Mathematics - Group Theory Mathematics |
| Zdroj: | Michigan Mathematical Journal. 71 |
| ISSN: | 0026-2285 |
| DOI: | 10.1307/mmj/20205868 |
| Popis: | Let $q\in\mathbb{C}$, let \[a=\begin{pmatrix} 1&0\\1&1\end{pmatrix},\quad b_q=\begin{pmatrix} 1&q\\0&1\end{pmatrix},\] and let $G_q27$, we prove that the lower density of denominators $r\in \mathbb{N}$ for which $G_{s/r}$ is non-free has a lower bound \[ 1- \left(1-\frac{11}{s}\right) \prod_{n=1}^\infty \left(1-\frac{4}{s^{2^n-1}}\right). \] Finally, we show that for a fixed $s$, there are arbitrarily long sequences of consecutive denominators $r$ such that $G_{s/r}$ is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation. Comment: 26 pages. To appear in the Michigan Mathematical Journal |
| Databáze: | OpenAIRE |
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