Binate towers of groups
type: Item
instance of: scholarly article
title: Binate towers of groups
zbMATH Open document ID: 0815.20015
DOI: 10.1007/BF01198664
author: A. J. Berrick
author: K. Varadarajan
published in: Archiv der Mathematik
publication date: 1995-04-18T00:00:00Z
review text: For any group \(H\) it is shown that the category \(B\text{Tow}_ H\) of binate towers with base \(H\) admits an initial object \(U(H) = \{u(H)_ i,\theta_ i,t_ i\}\). Since \(U(H)\) is obtained from \(H\) by repeated HNN extensions it is referred to as the HNN binate tower with base \(H\). Given a tower \(\{G_ i\}\) with base \(G_ 0 = H\) and surjective homomorphisms \(\{\pi_ i\} : \{U(H)_ i\} \to \{G_ i\}\) of towers, necessary and sufficient conditions are determined for getting a binate tower \(\{G_ i,\alpha_ i,u_ i\}\) rendering \(\{\pi_ i\} : \{U(H)_ i,\theta_ i,t_ i\} \to \{G_ i,\alpha_ i,u_ i\}\) a morphism in \(B \text{Tow}_ H\). For any \(\{H_ i,P_ i,v_ i\}\) in \(B\text{Tow}_ H\) the limit group \({\underset \longrightarrow \lim} H_ i\) is acyclic. The binate tower \(U(1)\) has a number of remarkable properties. The limit group \(U {\underset \longrightarrow \lim} U(1)\) is very badly non- Hopfian and very badly non co-Hopfian. While the group \({\underset \longrightarrow \lim} U(1)\) is acyclic, the groups \(U(1)_ i\) have a total integral homology free abelian of \(\text{rank} > 2^{2i - 2} + 1\) for \(i \geq 3\). Also, some connections between the problem of Kervaire on normal generators of free products with the infinite cyclic group and study of binate towers are indicated. In particular, it is noted that affirming the Kervaire conjecture for all binate groups would suffice to affirm it for all groups. For certain binate groups this is proved.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20J15
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 743511
zbMATH Keywords: category of binate towers
zbMATH Keywords: acyclic groups
zbMATH Keywords: co-Hopfian groups
zbMATH Keywords: initial object
zbMATH Keywords: HNN extensions
zbMATH Keywords: surjective homomorphisms
zbMATH Keywords: integral homology
zbMATH Keywords: normal generators
zbMATH Keywords: free products
zbMATH Keywords: Kervaire conjecture
zbMATH Keywords: binate groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 743511