The Isomorphism Problem for a Family of One-Relator Groups
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The Isomorphism Problem for a Family of One-Relator Groups Vu The Khoi1 Received: 10 March 2019 / Revised: 12 September 2019 / Accepted: 3 October 2019 / Published online: 15 September 2020 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract In this paper, we use the Alexander ideals of groups to solve the isomorphism problem for a family of groups G(m, n, p) which coincide with the Baumslag-Solitar groups when p = 1. Keywords Group isomorphism problem · Alexander ideal Mathematics Subject Classification (2010) Primary 20Exx · Secondary 57M05
1 Introduction and Main Results The isomorphism problem is a fundamental problem in group theory in which we have to decide whether two finitely presented groups are isomorphic. It is well-known that, in the most general form, the isomorphism problem was unsolvable. Therefore, people often study the problem for a special class of groups. The Baumslag-Solitar groups B(m, n), where m = 0, n = 0, were first defined by Baumslag and Solitar [3] as follows B(m, n) = a, b a −1 bm a = bn . The Baumslag-Solitar groups play a special role since they serve as a rich source of examples and counterexamples for many questions in group theory. The isomorphism problem for the Baumslag-Solitar groups was solved in [4, 8, 11]. A bigger family of groups which contains all the Baumslag-Solitar groups is defined as G(m, n, p) = a, b a −p bm a p = bn , where m = 0, n = 0, p = 0. The family of groups G(m, n, p) also attracts many attentions. Group-theoretical properties such as residual finiteness and conjugacy separability of the family G(m, n, p) have been studied in [1, 9, 10].
Vu The Khoi
[email protected] 1
Institute of Mathematics, VAST, 18 Hoang Quoc Viet, 10307, Hanoi, Vietnam
V.T. Khoi
898
The purpose of this paper is to solve the isomorphism problem for the family G(m, n, p). Our method follows the line of [8]. By using the Alexander ideals of groups in the family we can tell if two groups are not isomorphic. In the case of the Baumslag-Solitar groups, by simply counting the number of connected components of the zero locus of their ideals, we can distinguish the groups. For the family G(m, n, p), the algebra is more complicated due to the fact that it has three parameters. As a result, the arguments as in the case of the Baumslag-Solitar groups do not work. Instead, we have to use direct algebraic arguments to show that there is no monomial automorphism mapping the Alexander ideal of one group to that of the other. Our main result is the following theorem. Theorem 1.1 The groups G(m, n, p) and G(m , n , p ) are isomorphic if and only if one of the following conditions satisfies: (i) For a suitable 1 = ±1, 2 = ±1 we have p = 1 p , m = 2 m and n = 2 n . (ii) For a suitable 1 = ±1, 2 = ±1 we have p = 1 p , m = 2 n and n = 2 m . (iii) For a suitable 1 = ±1, 2 = ±1 we have m = n = 1 p and m = n = 2 p. The paper consists of three sections. In Section 2, we
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