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

## TL;DR

This paper investigates the computational complexity of the matrix identity problem in special linear groups and the Heisenberg group, providing new decidability results for certain subgroups and improved undecidability bounds for larger groups.

## Contribution

It proves no embedding of pairs of words into SL(3,Z), shows polynomial-time decidability in the Heisenberg group, and significantly reduces the generator set size for undecidability in SL(4,Z).

## Key findings

- No embedding of pairs of words into SL(3,Z)
- Decidability of the identity problem in polynomial time for H(3,Z)
- Reduced the undecidability bound in SL(4,Z) from 48 to 8 generators

## Abstract

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 ${\rm SL}(2,\mathbb{Z})$ and the undecidability for ${\rm SL}(4,\mathbb{Z})$ generated by $48$ matrices. First we show that there is no embedding from pairs of words into $3\times3$ integer matrices with determinant one, i.e., into ${\rm SL}(3,\mathbb{Z})$ extending previously known result that there is no embedding into $\mathbb{C}^{2\times 2}$. Apart from theoretical importance of the result it can be seen as a strong evidence that the computational problems in \slthreez are decidable. The result excludes the most natural possibility of encoding the Post correspondence problem into ${\rm SL}(3,\mathbb{Z})$, where the matrix products extended by the right multiplication correspond to the Turing machine simulation. Then we show that the identity problem is decidable in polynomial time for an important subgroup of ${\rm SL}(3,\mathbb{Z})$, the Heisenberg group ${\rm H}(3,\mathbb{Z})$. Furthermore, we extend the decidability result for ${\rm H}(n,\mathbb{Q})$ in any dimension $n$. Finally we are tightening the gap on decidability question for this long standing open problem by improving the undecidability result for the identity problem in ${\rm SL}(4,\mathbb{Z})$ substantially reducing the bound on the size of the generator set from $48$ to $8$ by developing a novel reduction technique.