Profinite and pro-$p$ completions of Poincaré duality groups of dimension 3
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- by Dessislava H. Kochloukova and Pavel A. Zalesskii PDF
- Trans. Amer. Math. Soc. 360 (2008), 1927-1949 Request permission
Abstract:
We establish some sufficient conditions for the profinite and pro-$p$ completions of an abstract group $G$ of type $FP_m$ (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type $FP_m$ over the field $\mathbb {F}_p$ for a fixed natural prime $p$ (resp. of finite cohomological $p$-dimension, of finite Euler $p$-characteristic). We apply our methods for orientable Poincaré duality groups $G$ of dimension 3 and show that the pro-$p$ completion $\widehat {G}_p$ of $G$ is a pro-$p$ Poincaré duality group of dimension 3 if and only if every subgroup of finite index in $\widehat {G}_p$ has deficiency 0 and $\widehat {G}_p$ is infinite. Furthermore if $\widehat {G}_p$ is infinite but not a Poincaré duality pro-$p$ group, then either there is a subgroup of finite index in $\widehat {G}_p$ of arbitrary large deficiency or $\widehat {G}_p$ is virtually $\mathbb {Z}_p$. Finally we show that if every normal subgroup of finite index in $G$ has finite abelianization and the profinite completion $\widehat {G}$ of $G$ has an infinite Sylow $p$-subgroup, then $\widehat {G}$ is a profinite Poincaré duality group of dimension 3 at the prime $p$.References
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Additional Information
- Dessislava H. Kochloukova
- Affiliation: IMECC-UNICAMP, Cx. P. 6065, 13083-970 Campinas, SP, Brazil
- Email: desi@ime.unicamp.br
- Pavel A. Zalesskii
- Affiliation: Department of Mathematics, University of Brasília, 70910-900 Brasília DF, Brazil
- MR Author ID: 245312
- Email: pz@mat.unb.br
- Received by editor(s): December 1, 2005
- Published electronically: October 22, 2007
- Additional Notes: Both authors were partially supported by “bolsa de produtividade de pesquisa” from CNPq, Brazil
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 1927-1949
- MSC (2000): Primary 20E18
- DOI: https://doi.org/10.1090/S0002-9947-07-04519-9
- MathSciNet review: 2366969