Circle homeomorphisms and shears

We give parameterizations of homeomorphisms, quasisymmetric maps and symmetric maps of the unit circle in terms of shear coordinates for the Farey tesselation.Comment: 26 pages, 3 figure

A variation of McShane's identity for 2-bridge links

We give a variation of McShane's identity, which describes the cusp shape of a hyperbolic 2-bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants of $SL(2,\mathbb{C})$-characters of the once-punctured torus corresponding to the holonomy representations of the complete hyperbolic structures of 2-bridge link complements.Comment: 40 pages, 18 figure

Parent training for preschool ADHD: a randomized controlled trial of specialized and generic programs

BackgroundThe New Forest Parenting Package' (NFPP), an 8-week home-based intervention for parents of preschoolers with attention-deficit/hyperactivity disorder (ADHD), fosters constructive parenting to target ADHD-related dysfunctions in attention and impulse control. Although NFPP has improved parent and laboratory measures of ADHD in community samples of children with ADHD-like problems, its efficacy in a clinical sample, and relative to an active treatment comparator, is unknown. The aims are to evaluate the short- and long-term efficacy and generalization effects of NFPP compared to an established clinic-based parenting intervention for treating noncompliant behavior [Helping the Noncompliant Child' (HNC)] in young children with ADHD. MethodsA randomized controlled trial with three parallel arms was the design for this study. A total of 164 3-4-year-olds, 73.8% male, meeting DSM-IV ADHD diagnostic criteria were randomized to NFPP (N=67), HNC (N=63), or wait-list control (WL, N=34). All participants were assessed at post-treatment. NFPP and HNC participants were assessed at follow-up in the next school year. Primary outcomes were ADHD ratings by teachers blind to and uninvolved in treatment, and by parents. Secondary ADHD outcomes included clinician assessments, and laboratory measures of on-task behavior and delay of gratification. Other outcomes included parent and teacher ratings of oppositional behavior, and parenting measures. (Trial name: Home-Based Parent Training in ADHD Preschoolers; Registry: ClinicalTrials.gov Identifier: NCT01320098; URL: ). ResultsIn both treatment groups, children's ADHD and ODD behaviors, as well as aspects of parenting, were rated improved by parents at the end of treatment compared to controls. Most of these gains in the children's behavior and in some parenting practices were sustained at follow-up. However, these parent-reported improvements were not corroborated by teacher ratings or objective observations. NFPP was not significantly better, and on a few outcomes significantly less effective, than HNC. ConclusionsThe results do not support the claim that NFPP addresses putative dysfunctions underlying ADHD, bringing about generalized change in ADHD, and its underpinning self-regulatory processes. The findings support documented difficulties in achieving generalization across nontargeted settings, and the importance of using blinded measures to provide meaningful assessments of treatment effects

Convex cocompact subgroups of mapping class groups

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1--> pi_1(S) --> L_G --> G -->1 we prove that if L_G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a "Schottky subgroup" of MCG, the converse is true as well; a semidirect product of pi_1(S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G=Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper5.abs.htm

Cosmological Time in (2+1) - Gravity

We consider maximal globally hyperbolic flat (2+1) spacetimes with compact space S of genus g>1. For any spacetime M of this type, the length of time that the events have been in existence is M defines a global time, called the cosmological time CT of M, which reveals deep intrinsic properties of spacetime. In particular, the past/future asymptotic states of the cosmological time recover and decouple the linear and the translational parts of the ISO(2,1)-valued holonomy of the flat spacetime. The initial singularity can be interpreted as an isometric action of the fundamental group of S on a suitable real tree. The initial singularity faithfully manifests itself as a lack of smoothness of the embedding of the CT level surfaces into the spacetime M. The cosmological time determines a real analytic curve in the Teichmuller space of Riemann surfaces of genus g, which connects an interior point (associated to the linear part of the holonomy) with a point on Thurston's natural boundary (associated to the initial singularity).Comment: Totally new version with strongly improved exposition. Clarifying examples and figures have been included. 21 pages, Latex, 9 figure

Cannon-Thurston Maps for Pared Manifolds of Bounded Geometry

Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M,P). Then there is a Cannon- Thurston map from the limit set of M_{gf} to that of N^h. Further, the limit set of N^h is locally connected. This answers in part a question attributed to Thurston.Comment: 57 pages, 4 figures, Final version incorporating referee's comments. To appear in Geometry and Topolog

Which type of parent training works best for preschoolers with comorbid ADHD and ODD? A secondary analysis of a randomized controlled trial comparing generic and specialized programs

The present study examined whether the presence of comorbid ODD differentially moderated the outcome of two Behavioral Parent Training (BPT) programs in a sample of preschoolers with ADHD: One designed specifically for ADHD (NFPP: New Forest Parenting Programme) and one designed primarily for ODD (HNC: Helping the Noncompliant Child). In a secondary analysis, 130 parents and their 3-4 year-old children diagnosed with ADHD were assigned to one of the two programs. 44.6 % of the children also met criteria for ODD. Significant interactions between treatment conditions (NFPP vs. HNC) and child ODD diagnosis (presence vs. absence) indicated that based on some parent and teacher reports, HNC was more effective with disruptive behaviors than NFPP but only when children had a comorbid diagnosis. Further, based on teacher report, NFPP was more effective with these behaviors when children had a diagnosis of only ADHD whereas HNC was equally effective across ADHD only and comorbid ODD diagnoses. Comorbidity profile did not interact with treatment program when parent or teacher reported ADHD symptoms served as the outcome. Implications for clinical interventions are discussed and directions for future work are provided

Analytic Continuation for Asymptotically AdS 3D Gravity

We have previously proposed that asymptotically AdS 3D wormholes and black holes can be analytically continued to the Euclidean signature. The analytic continuation procedure was described for non-rotating spacetimes, for which a plane t=0 of time symmetry exists. The resulting Euclidean manifolds turned out to be handlebodies whose boundary is the Schottky double of the geometry of the t=0 plane. In the present paper we generalize this analytic continuation map to the case of rotating wormholes. The Euclidean manifolds we obtain are quotients of the hyperbolic space by a certain quasi-Fuchsian group. The group is the Fenchel-Nielsen deformation of the group of the non-rotating spacetime. The angular velocity of an asymptotic region is shown to be related to the Fenchel-Nielsen twist. This solves the problem of classification of rotating black holes and wormholes in 2+1 dimensions: the spacetimes are parametrized by the moduli of the boundary of the corresponding Euclidean spaces. We also comment on the thermodynamics of the wormhole spacetimes.Comment: 28 pages, 14 figure

Boundary Conformal Field Theories, Limit Sets of Kleinian Groups and Holography

In this paper,based on the available mathematical works on geometry and topology of hyperbolic manifolds and discrete groups, some results of Freedman et al (hep-th/9804058) are reproduced and broadly generalized. Among many new results the possibility of extension of work of Belavin, Polyakov and Zamolodchikov to higher dimensions is investigated. Known in physical literature objections against such extension are removed and the possibility of an extension is convincingly demonstrated.Comment: 62 pages, 5 figure

Thurston's pullback map on the augmented Teichm\"uller space and applications

Let $f$ be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map $\sigma_f$ of a finite-dimensional Teichm\"uller space. We prove that this map extends continuously to the augmented Teichm\"uller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston's pullback map near invariant strata of the boundary of the augmented Teichm\"uller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston's pullback map. Our approach also yields new proofs of Thurston's theorem and Pilgrim's Canonical Obstruction theorem.Comment: revised version, 28 page