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

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 $p$.