Decidability of the Membership Problem for $2times 2$ integer matrices

The main result of this paper is the decidability of the membership problem for $2times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2times 2$ integer matrices $M_1,dots,M_n$ and $M$ decides whether $M$ belongs to the semigroup generated by ${M_1,dots,M_n}$. Our algorithm relies on a translation of the numerical problem on matrices into combinatorial problems on words. It also makes use of some algebraical properties of well-known subgroups of $mathrm{GL}(2,mathbb{Z})$ and various new techniques and constructions that help to limit an infinite number of possibilities by reducing them to the membership problem for regular languages.