New sequences of non-free rational points

TL;DR

This paper presents new infinite families of rational values of 4, including some squares of rationals, where the generated matrix groups are not free, expanding understanding of these algebraic structures.

Contribution

It introduces novel sequences of rational 4 values that lead to non-free matrix groups, providing new examples and insights into their algebraic properties.

Findings

Identifies new rational 4 sequences with non-free group structures

Shows some 4 values are squares of rationals

Expands known examples of non-free matrix groups

Abstract

We exhibit some new infinite families of rational values of $\tau$, some of them squares of rationals, for which the group or even the semigroup generated by the matrices $\left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 0 \\ \tau & 1 \end{smallmatrix} \right)$ is not free.