## Planar groups and the Seifert conjecture

### Brian H. Bowditch

We describe a number of characterisations of virtual surface groups which
are based on the following result.
Let $ G $ be a group and $ k $ be a field.
We show that if $ G $ is $ FP_2 $ over $ k $ and if
$ H^2(G;k) $, thought of as a $ k $-vector space, contains a 1-dimensional
$ G $-invariant 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 reimannian planes.
We also give a cohomological version of the Seifert conjecture, from which
the topological Seifert conjecture (due to Tukia, Mess, Gabai, Casson and
Jungreis) can be recovered via work of Zieschang and Scott.

2000 Subject Classification: 20F65, 57M05, 20J06.

Preprint, Southampton 1999.

J. reine angew. Math. **576** (2004) 11-62.

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