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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

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
Full Text: DOI
[1] Baumslag, G.; Dyer, E.; Heller, A., The topology of discrete groups, J. pure appl. algebra, 16, 1-47, (1980) · Zbl 0419.20026
[2] Kan, D.M.; Thurston, W.P., Every connected space has the homology of a K(π, 1), Topology, 15, 253-258, (1976) · Zbl 0355.55004
[3] Lyndon, R.C.; Schupp, P.E., Combinatorial group theory, () · Zbl 0368.20023
[4] Mather, J., The vanishing of the homology of certain groups of homeomorphisms, Topology, 10, 297-298, (1971) · Zbl 0207.21903
[5] Neumann, B.H., A note on algebraically closed groups, J. lond. math. soc., 27, 227-242, (1952) · Zbl 0046.24802
[6] Neumann, B.H., The isomorphism problem for algebraically closed groups, (), 553-562
[7] Scott, W.R., Algebraically closed groups, (), 118-121 · Zbl 0043.02302
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