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Exponential groups. II: Extensions of centralizers and tensor completion of CSA-groups. (English) Zbl 0866.20014
[For part I cf. Sib. Mat. Zh. 35, No. 5, 1106-1118 (1994; Zbl 0851.20050).]
The theory of exponential groups begins with results of A. Mal’cev, P. Hall, G. Baumslag and R. Lyndon. Let $$A$$ be an arbitrary associative ring with identity. A group $$G$$ is called an $$A$$-group if there exists an action of $$A$$ on $$G$$ ($$g^\alpha$$ is the result of the action of $$a\in A$$ on $$g\in G$$) satisfying the following axioms:
\begin{alignedat}{2}2&1.\;g^1=g,\;g^0=1,\;1^\alpha =1;&\quad&2.\;g^{\alpha+\beta}=g^\alpha\cdot g^\beta,\;g^{\alpha\beta}=(g^\alpha)^\beta;\\ &3.\;(h^{-1}gh)^\alpha=h^{-1}g^\alpha h;&\quad&4.\;[g,h]=1\Rightarrow (gh)^\alpha=g^\alpha h^\alpha.\end{alignedat}
For example, unipotent groups over a field $$k$$ of characteristic zero are $$k$$-groups, pro-$$p$$-groups are $$\mathbb{Z}_p$$-groups over the ring of $$p$$-adic integers, etc. The aim of the article is to construct the general theory of tensor $$A$$-completions of groups with an emphasis on $$A$$-free groups. For arbitrary groups it is difficult to give a concrete description of their $$A$$-completion. The authors introduce a class of CSA-groups (it contains abelian, free, hyperbolic torsion-free groups and groups acting freely on $$\Lambda$$-trees), for which a good and concrete description of tensor completion exists. As a corollary they study basic properties of $$A$$-free groups such as canonical and reduced forms of elements, commuting and conjugate elements. Some interesting open problems on this area are formulated.

##### MSC:
 20E08 Groups acting on trees 20E22 Extensions, wreath products, and other compositions of groups 20F06 Cancellation theory of groups; application of van Kampen diagrams 20J15 Category of groups 20F65 Geometric group theory 16W20 Automorphisms and endomorphisms
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