A characterization of wreath products where knapsack is decidable

Abstract

The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group G and takes as input group elements g1, . . . , gn, g ∈ G and asks whether there are x1, . . . , xn ≥ 0 withWe study the knapsack problem for wreath products G ≀ H of groups G and H.Our main result is a characterization of those wreath products G ≀ H for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors G and H. To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem.Moreover, we apply our main result to H3(Z), the discrete Heisenberg group, and to Baumslag-Solitar groups BS(1, q) for q ≥ 1. First, we show that the knapsack problem is undecidable for G ≀ H3(Z) for any G ̸ = 1. This implies that for G ̸ = 1 and for infinite and virtually nilpotent groups H, the knapsack problem for G ≀ H is decidable if and only if H is virtually abelian and solvability of systems of exponent equations is decidable for G. Second, we show that the knapsack problem is decidable for G ≀ BS(1, q) if and only if solvability of systems of exponent equations is decidable for G. ACM Subject ClassificationTheory of computation → Problems, reductions and completeness; Theory of computation → Theory and algorithms for application domains Keywords and phrases knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, Baumslag-Solitar groups Digital Object Identifier 10.4230/LIPIcs...

Exponent equations in HNN-extensions

“…As mentioned in the introduction, the class of knapsack semilinear groups is very rich. Examples of a groups, where knapsack is decidable but solvability of systems of exponent equations is undecidable are the Heisenberg group H 3 (Z) (the group of all upper triangular (3 × 3)-matrices over the integers, where all diagonal entries are 1) [22] and the Baumslag-Solitar group BS(1, 2) [3,25]. These groups are not knapsack semilinear in a strong sense: there are knapsack expressions e such that sol H3(Z) (e) (resp.…”

Exponent equations in HNN-extensions

“…There are groups with a decidable knapsack problem but an undecidable solvability problem for systems of exponent equations. Examples are the discrete Heisenberg group [25] and the Baumslag-Solitar group BS(1, 2) [5,Theorem E.1]. Let us also remark that the variants of these problems, where the variables x i range over Z are not harder, since one can replace a power g x i i with x i ranging over Z by g x i i (g −1 i ) y i with x i , y i ranging over N.…”