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

Scl, sails and surgery

We establish a close connection between stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abelian groups, and that it can be computed via integer programming. Furthermore, we show that the scl spectrum of nonabelian free groups contains elements congruent to every rational number modulo $\mathbb{Z}$, and contains well-ordered sequences of values with ordinal type $\omega^\omega$. Finally, we study families of elements $w(p)$ in free groups obtained by surgery on a fixed element $w$ in a free product of Abelian groups of higher rank, and show that \scl(w(p)) \to \scl(w) as $p \to \infty$.Comment: 23 pages, 4 figures; version 3 corrects minor typo