On the Identity Problem for the Special Linear Group and the Heisenberg Group

We study the identity problem for matrices, i.e., whether the identity matrix is in a semigroup generated by a given set of generators. In particular we consider the identity problem for the Special Linear Group following recent NP-completeness result for SL(2,Z) and the undecidability for SL(4,Z) generated by 48 matrices. In the seminal paper of Paterson in 1970, an injective morphism from pairs of words into 3 × 3 integral matrices was used to prove undecidability of mortality and later led to many undecidability results of matrix problems in dimension three. In our paper we show that there is no embedding from pairs of words into 3×3 integral matrices with determinant one, i.e., into SL(3,Z), which is a strong evidence that computational problems in SL(3,Z) are decidable as all known undecidability techniques for low-dimensional matrices are based on encoding of Turing machine computations via the PCP which cannot be applied in SL(3,Z) following our result. In case of the PCP encoding, the matrix products extended by the right multiplication correspond to the Turing machine simulation and the only known proof alternatives rely on recursively enumerable sets and the Hilbert’s Tenth Problem, but provide undecidability for matrix equations of very high dimensions. As the decidability status of the identity problem in dimension three is still a long standing open problem, we look for an important subgroup of SL(3,Z), the Heisenberg group H(3,Z), for which the identity problem could be decidable following our result on non-existence of embedding. We show that the identity problem for a matrix semigroup generated by matrices from H(3,Z) and even H(3,Q) is decidable in polynomial time. Furthermore, we extend the decidability result for H(n,Q) in any dimension n. Moreover we are tightening the gap between decidability and undecidability results by improving the undecidability result for the identity problem substantially reducing the bound on the size of the generator set from 48 to 8 in SL(4,Z) by developing a novel reduction technique.