Dessislava H. Kochloukova

PROFINITE AND PRO-p COMPLETIONS OF POINCARÉ DUALITY GROUPS OF DIMENSION 3

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 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 Poincare duality groups G of dimension 3 and show that the pro-p completion Ĝp of G is a pro-p Poincare duality group of dimension 3 if and only if every subgroup of finite index in Ĝ p has deficiency 0 and Ĝ p is infinite. Furthermore if Ĝ p is infinite but not a Poincare duality pro-p group, then either there is a subgroup of finite index in Ĝ p of arbitrary large deficiency or Ĝ p 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 Poincare duality group of dimension 3 at the prime p.

COHOMOLOGICAL FINITENESS PROPERTIES OF THE BRIN-THOMPSON-HIGMAN GROUPS 2V AND 3V

We show that Brins generalisations 2V and 3V of the Thompson-Higman group V are of type FP1. Our methods also give a new proof that both groups are nitely presented.

On subdirect products of type FP m of limit groups

Abstract We show that limit groups are free-by-(torsion-free nilpotent) and have non-positive Euler characteristic. We prove that for any non-abelian limit group the Bieri–Neumann–Strebel–Renz Σ-invariants are the empty set. Let s ⩾ 3 be a natural number and G be a subdirect product of non-abelian limit groups intersecting each factor non-trivially. We show that the homology groups of any subgroup of finite index in G, in dimension i ⩽ s and with coefficients in ℚ, are finite-dimensional if and only if the projection of G to the direct product of any s of the limit groups has finite index. The case s = 2 is a deep result of M. Bridson, J. Howie, C. F. Miller III and H. Short.

Volume gradients and homology in towers of residually-free groups

We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups

Profinite completions of orientable Poincaré duality groups of dimension four and Euler characteristic zero

G_n<G

HOMOLOGICAL FINITENESS PROPERTIES OF WREATH PRODUCTS

Gn<G of increasing finite index in a fixed finitely generated group G, assuming

Centralisers of finite subgroups in soluble groups of type FPn

\bigcap _n G_n =1