Planar groups and the Seifert conjecture

Bowditch, Brian H. (2004) Planar groups and the Seifert conjecture. Journal für die reine und angewandte Mathematik, 576, 11-62.

Record type: Article

Abstract

We describe a number of characterisations of virtual surface groups which are based on the following result. Let G be a group and F be a field. We show that if G is FP2 over F and if H2ðG; FÞ, thought of as a k-vector space, contains a 1-dimensional Ginvariant subspace, then G is a virtual surface group (i.e. contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane). In particular, this applies to rational Poincare´ duality groups.
We also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group. We recover the result of Mess which characterises such groups as groups which are quasiisometric to complete riemannian planes. We also give a cohomological version of the Seifert conjecture, from which the topological Seifert conjecture (proven by Tukia, Mess, Gabai, Casson and Jungreis) can be recovered via work of Zieschang and Scott.

This record has no associated files available for download.