Bergsträßer, Pascal ;
Ganardi, Moses ;
Zetzsche, Georg
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 g_1,…,g_n,g ∈ G and asks whether there are x_1,…,x_n ≥ 0 with g_1^{x_1}⋯ g_n^{x_n} = g. We 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 BaumslagSolitar 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.
BibTeX  Entry
@InProceedings{bergstraer_et_al:LIPIcs.STACS.2021.11,
author = {Bergstr\"{a}{\ss}er, Pascal and Ganardi, Moses and Zetzsche, Georg},
title = {{A Characterization of Wreath Products Where Knapsack Is Decidable}},
booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
pages = {11:111:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771801},
ISSN = {18688969},
year = {2021},
volume = {187},
editor = {Bl\"{a}ser, Markus and Monmege, Benjamin},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13656},
URN = {urn:nbn:de:0030drops136566},
doi = {10.4230/LIPIcs.STACS.2021.11},
annote = {Keywords: knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, BaumslagSolitar groups}
}
10.03.2021
Keywords: 

knapsack, wreath products, decision problems in group theory, decidability, discrete Heisenberg group, BaumslagSolitar groups 
Seminar: 

38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Issue date: 

2021 
Date of publication: 

10.03.2021 