Topic
Polycyclic group
About: Polycyclic group is a research topic. Over the lifetime, 155 publications have been published within this topic receiving 2065 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, the authors give a new proof that compact infra-solvmanifolds with isomorphic fundamental groups are smoothly diffeomorphic and prove rigidity results for manifolds which are constructed using affine actions of virtually polycyclic groups on solvable Lie groups.
73 citations
Posted Content•
TL;DR: It is shown that the knapsack problem is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2) and that for every co-context-free group, the knapping problem is decidable.
Abstract: It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the discrete Heisenberg group itself, the knapsack problem is decidable. Hence, decidability of the knapsack problem is not preserved under direct products. It is also shown that for every co-context-free group, the knapsack problem is decidable. For the subset sum problem (also introduced by Myasnikov, Nikolaev, and Ushakov) we show that it belongs to the class NL (nondeterministic logspace) for every finitely generated virtually nilpotent group and that there exists a polycyclic group with an NP-complete subset sum problem.
61 citations
TL;DR: In this article, it was shown that every locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group by proving the Hilbert basis theorem.
43 citations
1 Mar 1967
TL;DR: Auslander as mentioned in this paper showed that every polycyclic group has a faithful representation in GL(n, Z) for some n, thus solving a problem of P. Hall [2] that involves considerable knowledge of the theory of Lie groups.
Abstract: L. Auslander [1] has recently shown that every polycyclic group' has a faithful representation in GL(n, Z) for some n, thus solving a problem of P. Hall [2]. His proof involves considerable knowledge of the theory of Lie groups. Since the result obtained is purely algebraic, it is of interest to find a purely algebraic proof of it. It struck me that the proof of Ado's theorem [5] could be adapted to this purpose and I will show here that this is indeed the case. I would like to thank J. Thompson and J. Alperin for calling this problem to my attention. Recall that a matrix (aij) is called uni-triangular if aij = 0 for j
42 citations
36 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 1 |
| 2020 | 3 |
| 2019 | 2 |
| 2018 | 6 |
| 2017 | 4 |