In this article we survey recent progress in the algorithmic theory of matrix
semigroups. The main objective in this area of study is to construct algorithms
that decide various properties of finitely generated subsemigroups of an
infinite group G, often represented as a matrix group. Such problems might
not be decidable in general. In fact, they gave rise to some of the earliest
undecidability results in algorithmic theory. However, the situation changes
when the group G satisfies additional constraints. In this survey, we give an
overview of the decidability and the complexity of several algorithmic problems
in the cases where G is a low-dimensional matrix group, or a group with
additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New