A family of accumulation points of non-free rational numbers
Christopher Buyalos, Jayden Thadani, Xinbei Wang, Bradley Zykoski, Michael Zshornack
TL;DR
The paper advances the understanding of when G_q, generated by opposite parabolic matrices in SL(2, R), is not free for rational q in (-4,4). It extends Smilga’s half-relations framework and uses Diophantine geometry to produce infinite families of rational non-free q accumulating at multiple points, including 1-step relation numbers, via length-5 half-relations and conic discriminants. It also supplies alternative constructions from partial geometric sums and from Pell-type sequences that yield accumulation near 1+\sqrt{2}, and discusses connections to Greenberg–Shalom and arithmeticity, while posing questions about density and higher-degree algebraic cases. These results collectively broaden the known accumulation points of non-free rational numbers and highlight rich interactions between group theory, Diophantine geometry, and number theory.
Abstract
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 $1+\sqrt{2}$.