Dave Witte Morris

Amenable groups that act on the line

Let Gamma be a finitely generated, amenable group. Using an idea of E Ghys, we prove that if Gamma has a nontrivial, orientation-preserving action on the real line, then Gamma has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Gamma has a faithful action on the circle, then some finite-index subgroup of Gamma has the property that all of its nontrivial, finitely generated subgroups have infinite, cyclic quotients. It also means that every left-orderable, amenable group is locally indicable. This answers a question of P Linnell.

Unipotent flows on the space of branched covers of Veech surfaces

There is a natural action of SL

Hamiltonian cycles in Cayley graphs whose order has few prime factors

(2,\mathbb{R})

Cartan-decomposition subgroups of (2,)

on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup

Cayley graphs of order 16p are hamiltonian

U = \big\{\big(\begin{smallmatrix}1 & * \\ 0 & 1\end{smallmatrix}\big)\big\}

Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian

. We classify the U -invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL

Tessellations of homogeneous spaces of classical groups of real rank two

(2,\mathbb{R})

Cayley Graphs of Order 27p Are Hamiltonian

-orbit of the set of branched covers of a fixed Veech surface.) For the U -action on these submanifolds, this is an analogue of Ratners theorem on unipotent flows. The result yields an asymptotic estimate of the number of periodic trajectories for billiards in a certain family of non-Veech rational triangles, namely, the isosceles triangles in which exactly one angle is

On colour-preserving automorphisms of Cayley graphs

2 \pi/n

Odd-order Cayley graphs with commutator subgroup of order pq are hamiltonian

, with