Some background: my coauthors and I are working on a problem which deals with the exponential growth rates of certain infinite products of matrices. One of the sub-problems which arises in this project is to prove that a particular function from the unit interval to the reals, which describes the growth rates of a family of related infinite matrix products, does not have any line segments in its graph. Now, the existence of a line segment would imply the existence of a pair of matrices some of whose characteristics coincide in a precise way. We have found an analytic proof that the graph cannot contain line segments, but it is long and displeasingly ad-hoc, and we would rather like to replace it with a shorter argument! Hence, we have found ourselves wondering whether there is an algebraic obstacle which would prevent these coincidences from occurring.

The question which arises from this research, then, is:

Do there exist matrices $A,B \in SL(2,\mathbb{Z})$ which satisfy the following constraints: the matrices $A$ and $B$ have only positive entries, do not commute, and satisfy $$\rho(AB)=\rho(A)\rho(B),$$ $$\mathbb{Q}\left(\rho(A)\right)=\mathbb{Q}\left(\rho(B)\right) \neq \mathbb{Q},$$ where $\rho$ denotes spectral radius?

The non-existence of such a pair of matrices would imply the non-existence of the line segments mentioned above.

One of my coauthors suspects that the spectral radius constraints on $A$ and $B$ cannot be met unless $A=C^k$, $B=C^\ell$ for some $C \in SL(2,\mathbb{Z})$ and $k,\ell \in \mathbb{N}$, which would contradict the hypothesis that $A$ and $B$ do not commute. Any solutions which end up being used in our paper will of course be gratefully acknowledged!