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Localization, completions and metabelian groups | | Gilbert Baumslag
; Roman Mikhailov
; Kent Orr
; | | Date: |
23 Jan 2013 | | Abstract: | For a pair of finitely generated residually nilpotent groups $G,H$, the group
$H$ is called para-$G$ if there exists a homomorphism $G o H$ which induces
isomorphisms of all lower central quotients. Groups $G$ and $H$ are called
para-equivalent if $H$ is para-$G$ and $G$ is para-$H$. In this paper we
consider the para-equivalence relation for the class of metabelian groups. For
a metabelian group $G$, we show that all para-$G$ groups naturally embed in a
type of completion of the group $G$, a smaller and simpler analog of the
pro-nilpotent completion of $G$, which is called the Telescope of $G$. This
places strong restrictions on para-equivalent groups. In particular, for
finitely generated metabelian groups, para-equivalence preserves the property
of being finitely presented. Numerous examples illustrate our approach. We
construct pairs of non-isomorphic para-equivalent polycyclic groups, as well as
groups determined by their lower central quotients. | | Source: | arXiv, 1301.5533 | | Services: | Forum | Review | PDF | Favorites | |
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