Circular groups, planar groups, and the Euler class

We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the Euler class of C^infty diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus >1 admits a C^infty action with arbitrary Euler class. On the other hand, we show that Z oplus Z actions satisfy a homological rigidity property: every orientation-preserving C^1 action of Z oplus Z on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R^2 in every degree of smoothness.Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper15.abs.htm

Coxeter groups and random groups

For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgroups commensurable with some member of the family, with overwhelming probability.Comment: 18 pages, 14 figures; version 2 incorporates referee's correction

Quasimorphisms and laws

Stable commutator length vanishes in any group that obeys a law

Stable commutator length in subgroups of PL(I)

Let G be a subgroup of PL(I). Then the stable commutator length of every element of [G,G] is zero.Comment: 6 pages; version 2 incorporates referee's comment

Real places and torus bundles

If M is a hyperbolic once-punctured torus bundle over the circle, then the trace field of M has no real places.Comment: 15 pages; v4 incorporates referee's comment

Bridgeman's orthospectrum identity

We give a short derivation of an identity of Bridgeman concerning orthospectra of hyperbolic surfaces.Comment: 5 pages, 3 figures; v3 minor errors correcte