2004/01/03 by Brian H. Bowditch · 1 citation
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research #Conjecture #Planar #Combinatorics #Mathematics #Computer science #Computer graphics (images)
paper · doi:10.1515/crll.2004.084
openalex publication_date 2004/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/04/05
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. <br/>We also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental\ngroup 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.