Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra

We give explicit formulae for certain elements occurring in the Bernstein presentation of an affine Hecke algebra, in terms of the usual Iwahori- Matsumoto generators. We utilize certain minimal expressions for said elements and we give a sheaf-theoretic interpretation for the existence of these minimal expressions.Comment: To appear, J. of Algebr

On the Andreadakis-Johnson filtration of the automorphism group of a free group

The Johnson filtration of the automorphism group of a free group is composed of those automorphisms which act trivially on nilpotent quotients of the free group. We compute cohomology classes as follows: (i) we analyze analogous classes for a subgroup of the pure symmetric automorphism group of a free group, and (ii) we analyze features of these classes which are preserved by the Johnson homomorphism. One consequence is that the ranks of the cohomology groups in any fixed dimension between 1 and n-1 increase without bound for terms deep in the Johnson filtraton.Comment: Corrections; revisions to proof of main theore

Relative shapes of thick subsets of moduli space

A closed hyperbolic surface of genus g ≥ 2 can be decomposed into pairs of pants along shortest closed geodesics and if these curves are sufficiently short (and with lengths uniformly bounded away from 0), then the geometry of the surface is essentially determined by the combinatorics of the pants decomposition. These combinatorics are determined by a trivalent graph, so we call such surfaces trivalent. In this paper, in a first attempt to understand the "shape" of the subset Xg of moduli space consisting of surfaces whose systoles fill, we compare it metrically, asymptotically in g, with the set Yg of trivalent surfaces. As our main result, we find that the set Xg∩Yg is metrically "sparse" in Xg (where we equip Mg with either the Thurston or the Teichmuller metric)