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Pseudo-mitotic groups. (English) Zbl 0569.20039
The concept of a mitotic group was used by G. Baumslag, E. Dyer and A. Heller [J. Pure Appl. Algebra 16, 1-47 (1980; Zbl 0419.20026)] to prove results on embeddings of groups into acyclic groups. In the paper under review this concept is generalized to that of what is called a pseudo-mitotic group, and it is poved that pseudo-mitotic groups are acyclic. Since the group \(G_ n\) of homeomorphisms of \({\mathbb{R}}^ n\) with compact support is pseudo-mitotic but not known to be mitotic in this way a uniform proof for the acyclicity of all \(G_ n\)- which is actually due already to Mather - and of all mitotic groups is obtained. It is also shown that it is not possible to get functorial embeddings of groups into algebraically closed groups.
Reviewer: J.Huebschmann

MSC:
20J05 Homological methods in group theory
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F38 Other groups related to topology or analysis
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E07 Subgroup theorems; subgroup growth
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References:
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