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

On the Identity Problem for Unitriangular Matrices of Dimension Four

“…In fact, starting with k = card(G) matrices, we need to solve at most O(k) linear equations, O(k) linear programming instances and one Identity Problem in H 3 or H 5 before either card(G) decreases or a conclusion on decidability is reached. All these problems have input of linear size in k, and are known to have polynomial complexity except for the Identity Problem in H 5 (whose complexity is unknown, see [7]).…”

“…Nevertheless, there have been some positive decidability results. The Identity Problem has been shown to be decidable for the group of 3 × 3 unitriangular integer matrices UT(3, Z) and the Heisenberg groups H 2n+1 , in [7]. Shortly after, the decidability result was extended to the Membership Problem [5].…”

The Identity Problem in the special affine group of $\mathbb{Z}^2$

“…It remains an intricate open problem whether any of these four algorithmic problems is decidable in SL(3, Z). Nevertheless, Ko, Niskanen and Potapov [23] recently showed that SL(3, Z) cannot embed pairs of words over an alphabet of size two, suggesting that all four problems in SL(3, Z) might be decidable. The following table I summarizes the state of art as well as our result, the group SA(2, Z) will be introduced in the next subsection.…”