The main result of this paper is the decidability of the membership problem
for 2×2 nonsingular integer matrices. Namely, we will construct the
first algorithm that for any nonsingular 2×2 integer matrices
M1,…,Mn and M decides whether M belongs to the semigroup generated
by {M1,…,Mn}.
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 GL(2,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