×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI