The membership problem for subsemigroups of GL 2 ( Z ) is NP-complete

We show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2 × 2 matrices from the General Linear Group GL 2 ( Z ) is solvable in NP . We extend this to prove that the membership problem is decidable in NP for GL 2 ( Z ) and for any arbitrary regular expression over matrices from the Special Linear group SL 2 ( Z ). We show that determining if a given finite set of matrices from SL 2 ( Z ) or the modular group PSL 2 ( Z ) generates a group or a free semigroup are decidable in NP . Previous algorithms, shown in 2005 by Choffrut and Karhumäki, were in EXPSPACE . Our algorithm is based on new techniques allowing us to operate on compressed word representations of matrices without explicit expansions. When combined with known NP -hard lower bounds, this proves that the membership problem over GL 2 ( Z ) is NP -complete, and the group problem and the non-freeness problem in SL 2 ( Z ) are NP -complete. 1