A family of accumulation points of non-free rational numbers

TL;DR

This paper constructs multiple families of rational numbers within (-4,4) that accumulate at infinitely many points, supporting a conjecture about the non-freeness of certain matrix groups in SL(2,R).

Contribution

It generalizes previous work by providing new constructions of accumulation points for families of rational parameters related to matrix groups.

Findings

Families of rational numbers accumulating at multiple points in (-4,4) are constructed.

Different methods, including Diophantine geometry and number sequences, are used for these constructions.

Supports the conjecture that certain groups are not freely generated for rational parameters in (-4,4).

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 $1+\sqrt{2}$.