On Reachability Problems for Low-Dimensional Matrix Semigroups

On the Identity Problem for Unitriangular Matrices of Dimension Four

“…Another natural follow-up question is the Membership Problem for UT(4, Z). An interesting idea would be to adapt the Register Automata method introduced in [5] for passing from the Identity Problem to the Membership Problem.…”

“…Equation (5) shows that B(σ, t) ∈ U 1 for all σ ∈ S 4 , t ∈ Z ≥0 . The main idea of this case is that D σ , E σ , F σ do not vanish and the asymptotic behaviour of τ (B(σ, t)) approaches the vector (t 2 D σ , t 2 E σ , t 3 F σ ).…”

“…The idea of this proof is similar to the solution to the Identity Problem in H 3 (see [5]). Consider any product…”

“…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]. It remained an open problem whether the Identity Problem in UT(n, Z) is decidable for n ≥ 4, as well as finding the smallest n for which the Membership Problem for UT(n, Z) becomes undecidable.…”

“…This further tightens the gap between decidability and undecidability and can be regarded as a first step towards the Membership Problem for UT (4, Z). The foundation of our method is the arguments developed in [5] for the Membership Problem of UT (3, Z). However, in order to pass from dimension three to four, we need to introduce additional methods from convex geometry, linear programming and algebraic geometry.…”