## Freeness of a matrix semigroup

Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the matrices of determinant $\pm1$ which are proportional to $$A_1:=\left(\begin{array}{cc}0.85&0.04\\-0.04&0.85\end{array}\right),\quad A_2:=\left(\begin{array}{cc}0.20&-0.26\\0.23&0.22\end{array}\right),\quad A_3:=\left(\begin{array}{cc}-0.15&0.28\\0.26&0.24\end{array}\right).$$ In fact, our question is slightly more specific: we would like to know that … Read more