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.

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