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Binate towers of groups. (English) Zbl 0815.20015
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.

MSC:
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F38 Other groups related to topology or analysis
20J05 Homological methods in group theory
20J15 Category of groups
20E36 Automorphisms of infinite groups
20F05 Generators, relations, and presentations of groups
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