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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.10422 |
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| _version_ | 1866912708705648640 |
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| author | Buyalos, Christopher Thadani, Jayden Wang, Xinbei Zykoski, Bradley Zshornack, Michael |
| author_facet | Buyalos, Christopher Thadani, Jayden Wang, Xinbei Zykoski, Bradley Zshornack, Michael |
| contents | For any $q\in\mathbb{R}$, let $A:=\left(\begin{smallmatrix}1 & 1\\0 & 1\end{smallmatrix}\right), B_q:=\left(\begin{smallmatrix}1 & 0\\q & 1\end{smallmatrix}\right)$ and let $G_q:=\langle A,B_q\rangle\leqslant\operatorname{SL}(2,\mathbb{R})$. Kim and Koberda conjecture that for every $q\in\mathbb{Q}\cap(-4,4)$, the group $G_q$ is not freely generated by these two matrices. We generalize work of Smilga and construct families of $q$ satisfying the conjecture that accumulate at infinitely many different points in $(-4,4)$. We give different constructions of such families, the first coming from applying tools in Diophantine geometry to certain polynomials arising in Smilga's work, the second from sums of geometric series and the last from ratios of Pell and Half-Companion Pell Numbers accumulating at $1+\sqrt{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A family of accumulation points of non-free rational numbers Buyalos, Christopher Thadani, Jayden Wang, Xinbei Zykoski, Bradley Zshornack, Michael Group Theory 20E05, 11D09 For any $q\in\mathbb{R}$, let $A:=\left(\begin{smallmatrix}1 & 1\\0 & 1\end{smallmatrix}\right), B_q:=\left(\begin{smallmatrix}1 & 0\\q & 1\end{smallmatrix}\right)$ and let $G_q:=\langle A,B_q\rangle\leqslant\operatorname{SL}(2,\mathbb{R})$. Kim and Koberda conjecture that for every $q\in\mathbb{Q}\cap(-4,4)$, the group $G_q$ is not freely generated by these two matrices. We generalize work of Smilga and construct families of $q$ satisfying the conjecture that accumulate at infinitely many different points in $(-4,4)$. We give different constructions of such families, the first coming from applying tools in Diophantine geometry to certain polynomials arising in Smilga's work, the second from sums of geometric series and the last from ratios of Pell and Half-Companion Pell Numbers accumulating at $1+\sqrt{2}$. |
| title | A family of accumulation points of non-free rational numbers |
| topic | Group Theory 20E05, 11D09 |
| url | https://arxiv.org/abs/2511.10422 |