Profinite and pro- completions of Poincaré duality groups of dimension 3 |
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Authors: | Dessislava H Kochloukova Pavel A Zalesskii |
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Institution: | IMECC-UNICAMP,~Cx. P.~6065, 13083-970 Campinas,~SP,~Brazil ; Department of Mathematics, University of Brasília, 70910-900 Brasília DF, Brazil |
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Abstract: | We establish some sufficient conditions for the profinite and pro- completions of an abstract group of type (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type over the field for a fixed natural prime (resp. of finite cohomological -dimension, of finite Euler -characteristic). We apply our methods for orientable Poincaré duality groups of dimension 3 and show that the pro- completion of is a pro- Poincaré duality group of dimension 3 if and only if every subgroup of finite index in has deficiency 0 and is infinite. Furthermore if is infinite but not a Poincaré duality pro- group, then either there is a subgroup of finite index in of arbitrary large deficiency or is virtually . Finally we show that if every normal subgroup of finite index in has finite abelianization and the profinite completion of has an infinite Sylow -subgroup, then is a profinite Poincaré duality group of dimension 3 at the prime . |
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Keywords: | |
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