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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Profinite and pro-$ p$ completions of Poincaré duality groups of dimension 3


Authors: Dessislava H. Kochloukova and Pavel A. Zalesskii
Journal: Trans. Amer. Math. Soc. 360 (2008), 1927-1949
MSC (2000): Primary 20E18
DOI: https://doi.org/10.1090/S0002-9947-07-04519-9
Published electronically: October 22, 2007
MathSciNet review: 2366969
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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$.


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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
Email: pz@mat.unb.br

DOI: https://doi.org/10.1090/S0002-9947-07-04519-9
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
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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