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Planar groups and the Seifert conjecture

Planar groups and the Seifert conjecture
Planar groups and the Seifert conjecture
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.
0075-4102
11-62
Bowditch, Brian H.
559a0b03-4ffd-49b0-aafe-4764e7de5143
Bowditch, Brian H.
559a0b03-4ffd-49b0-aafe-4764e7de5143

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.

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Published date: 2004

Identifiers

Local EPrints ID: 29775
URI: http://eprints.soton.ac.uk/id/eprint/29775
ISSN: 0075-4102
PURE UUID: 48fcb35f-c1ce-4398-8958-ac8a6ce0c9e0

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Date deposited: 12 May 2006
Last modified: 15 Jul 2019 19:08

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