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

We establish some sufficient conditions for the profinite and prop completions of an abstract group G of type FPm (resp of finite cohomological dimension, of finite Euler characteristics) to be of type FPm over the field Fp for a fixed natural prime p (resp. of finite cohomological p-dimension, of finite Euler p-characteristics). We apply our methods for orientable Poincaré duality groups G of dimension 3 and show that the pro-p completion Ĝp of G is a pro-p Poincaré duality group of dimension 3 if and only if every subgroup of finite index in Ĝp has deficiency 0 and Ĝp is infinite. Furthermore if Ĝp is infinite but not a Poincaré duality pro-p group then either there is a subgroup of finite index in Ĝp of arbitrary large deficiency or Ĝp is virtually Zp. Finally we show that if every normal subgroup of finite index in G has finite abelianization and the profinite completion Ĝ of G has an infinite Sylow p-subgroup then Ĝ is a profinite Poincaré duality group of dimension 3 at the prime p. Mathematics Subject Classification : 20E18 ∗Both authors are partially supported by ”bolsa de produtividade de pesquisa” from CNPq, Brazil