Separable subgroups of mapping class groups

We investigate separability questions for the mapping class group of a surface. While this group is not subgroup separable in general, we prove a large family of interesting subgroups are separable. This includes many classically studied subgroups such as solvable subgroups, Heegaard and Handlebody groups, geometric subgroups, and all the terms in the Johnson filtration.     

Quantitative measurements of recombinant HIV surface glycoprotein 120 binding to several glycosphingolipids expressed in planar supported lipid bilayers

The interaction of recombinant HIV-1 surface glycoprotein gp120 (rgp120) with natural isolates of lactosylceramide (LacCer), glucosylceramide (GlcCer), and galactosylceramide (GalCer) has been quantitatively measured under equilibrium conditions using total internal reflection fluorescence (TIRF) spectroscopy. The binding affinity (K(a)) of rgp120 to these glycosphingolipids (GSLs), reconstituted at 5 mol % in supported planar lipid bilayers composed of 95 mol % POPC, is ca. 10(6) M(-1) for dissolved rgp120 concentrations greater than 25 nM. In contrast, at concentrations of rgp120 between 0.2 and 15 nM, rgp120 does not bind significantly to LacCer and GlcCer, but has a high affinity for GalCer with a measured K(a) value of 1.6 x 10(9) M(-1). However, protein surface coverage measurements show that this strong binding process accounts for very little of the total protein adsorbed over the entire concentration range studied. At a protein concentration of ca. 20 nM, the surface coverage is only 3% of that achieved at apparent saturation (i.e., when the protein concentration is ca. 220 nM). Thus the "high affinity" binding sites comprise only a small fraction of the total number of binding sites. Several other variables were investigated. Rgp120 binding behavior at membranes doped with alpha-hydroxygalactosylceramide (alpha-GalCer) was very similar to that observed with GalCer, showing that the presence/absence of an alpha-hydroxy moiety does not significantly affect galactosylceramide recognition. Phase segregation of GalCer, which occurs when the mole fraction of this GSL in a POPC bilayer exceeds ca. 0.1, was also investigated and showed no effect on binding affinity at low rgp120 concentrations. To investigate the influence of fatty acid chain length, GSLs with monodisperse C(18) and C(24) chain lengths, both with and without an alpha-hydroxy moiety, were synthesized, and their binding affinity to rgp120 was examined. Relative to the natural isolates (which contain a mixture of chain lengths), minimal differences were observed; thus among the compounds tested, fatty acid chain length does not affect GSL recognition. The results of this work should aid efforts to design anti-HIV-1 agents based on membrane-tethered, carbohydrate-based receptors for rgp120.     

Ring-Closing Metathesis Strategies to P -heterocycles

A general approach to the synthesis of P -heterocycles using the ring-closing metathesis reaction is described. We have developed a new method utilizing phosphorus nuclei as suitable temporary tethers for the coupling of non-racemic allylic amines. This approach allows for the generation of symmetric and unsymmetric 1,4-diamines. Subsequent coupling of these 1,4-diamines with various nuclei produces an array of 7-membered heterocycles.     

Finite Subgroups of Arithmetic Lattices in U (2, 1)

The principle result of this article is the determination of the possible finite subgroups of arithmetic lattices in U (2, 1).     

Arithmetic cusp shapes are dense

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n-manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n + 1)-orbifold is dense in the space of similarity classes of flat metrics on M. The set used for density is precisely the set of those classes which arise in arithmetic orbifolds.