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.
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.
Reviewer: A.G.Myasnikov (New York)
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 |
