Topological Designs

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.Comment: 14 p., 4 Figures. To appear in Geometriae Dedicat

A Combination Theorem for Metric Bundles

We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added. Characterization of convex cocompact subgroups of mapping class groups of surfaces with punctures in terms of relative hyperbolicity given v4: Final version incorporating referee comments: 63 pages 5 figures. To appear in Geom. Funct. Ana

942-41 Dysplasia of the Atrioventricular Nodal Artery in Patients with Mitral Valve Prolapse and Sudden Cardiac Death

Patients with isolated mitral valve prolapse (MVP) without significant mitral regurgitation are at a slightly increased risk of sudden cardiac death compared to the normal population. The mechanism(s) of sudden cardiac death in isolated MVP are uncertain. We studied gross hearts and histologic sections of the atrioventricular (AV) node in 21 patients with isolated MVP dying suddenly without apparent cause other than MVP (Group 1, mean age 36±8 years, mean heart weight 420±89 grams). No patient in Group 1 had a history of or treatment for mitral regurgitation, and no left atrial dilatation was present at autopsy. Sudden cardiac death MVP cases were compared to 15 control hearts from trauma victims without cardiac disease (Group 2, mean age 30±7 years, mean heart weight 350±96 grams). Sections of the artery to the AV node within the atrial septum (mean 7 levels in each group) were stained for elastic tissue and proteoglycans. Dysplasia was defined as focal deposition of proteoglycans within the media and partial disruption of the internal elastic lamina. The ratio of lumen area to arterial area was determined by computerized planimetry at the level of greatest luminal narrowing and measured (mean±SEM) 0.33±0.04 for Group 1 and 0.52±0.04 for Group 2 (p=0.001) this ratio was independent of heart weight, age, or sex of patient by stepwise regression. Dysplasia of the AV nodal artery was present in 9/21 (43%) hearts in Group 1 versus 1/15 (7%) hearts in group 2 (p<0.02) We conclude that narrowing of the AV nodal artery and arterial dysplasia are more prevalent in patients with MVP and sudden cardiac death than controls and may represent a more generalized disorder of proteoglycan deposition. The relationship between these findings and sudden cardiac death needs to be further studied

Peripheral fillings of relatively hyperbolic groups

A group theoretic version of Dehn surgery is studied. Starting with an arbitrary relatively hyperbolic group $G$ we define a peripheral filling procedure, which produces quotients of $G$ by imitating the effect of the Dehn filling of a complete finite volume hyperbolic 3--manifold $M$ on the fundamental group $\pi_1(M)$. The main result of the paper is an algebraic counterpart of Thurston's hyperbolic Dehn surgery theorem. We also show that peripheral subgroups of $G$ 'almost' have the Congruence Extension Property and the group $G$ is approximated (in an algebraic sense) by its quotients obtained by peripheral fillings. Various applications of these results are discussed.Comment: The difference with the previous version is that Proposition 3.2 is proved for quasi--geodesics instead of geodesics. This allows to simplify the exposition in the last section. To appear in Invent. Mat

Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures", we provide a general algorithm that recognizes the class of groups that are hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in Israel J. Math, and Bull. London Math. Soc. respectivel

On the number of excursion sets of planar Gaussian fields

37 pages, 14 figures37 pages, 14 figures37 pages, 14 figure

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t = −1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added