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

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

Cosmology and Hierarchy in Stabilized Randall-Sundrum Models

We consider the cosmology and hierarchy of scales in models with branes immersed in a five-dimensional curved spacetime subject to radion stabilization. The universe naturally find itself in the radiation-dominated epoch when the inter-brane spacing is static and stable, independent of the form of the stabilizing potential. We recover the standard Friedmann equations without assuming a specific form for the bulk energy-momentum tensor. We address the hierarchy problem in the context of a quartic and exponential stabilizing potential, and find that in either case the presence of a negative tension brane is required and that the string scale can be as low as the electroweak scale. In the situation of self-tuning branes (corresponding to an exponential potential) where the bulk cosmological constant is set to zero, the brane tensions have hierarchical values.Comment: 3 pages, 1 figure, 1 table. Talk given at DPF 2000, Columbus, OH, August 12, 200

Non-termination Analysis of Logic Programs with Integer arithmetics

In the past years, analyzers have been introduced to detect classes of non-terminating queries for definite logic programs. Although these non-termination analyzers have shown to be rather precise, their applicability on real-life Prolog programs is limited because most Prolog programs use non-logical features. As a first step towards the analysis of Prolog programs, this paper presents a non-termination condition for Logic Programs containing integer arithmetics. The analyzer is based on our non-termination analyzer presented at ICLP 2009. The analysis starts from a class of queries and infers a subclass of non-terminating ones. In a first phase, we ignore the outcome (success or failure) of the arithmetic operations, assuming success of all arithmetic calls. In a second phase, we characterize successful arithmetic calls as a constraint problem, the solution of which determines the non-terminating queries.Comment: 15 pages, 2 figures, journal TPLP (special issue on the international conference of logic programming

Certifying incompressibility of non-injective surfaces with scl

Cooper-Manning and Louder gave examples of maps of surface groups to PSL(2,C) which are not injective, but are incompressible (i.e. no simple loop is in the kernel). We construct more examples with very simple certificates for their incompressibility arising from the theory of stable commutator length.Comment: 5 pages; version 2 incorporates referee's suggestion