Thursday, March 30, 2006

[math/0603675] The lower central series and pseudo-Anosov dilatations

[math/0603675] The lower central series and pseudo-Anosov dilatations

The lower central series and pseudo-Anosov dilatations


Authors:
Benson Farb,
Christopher J. Leininger,
Dan Margalit


Comments: 26 pages, 6 figures

Subj-class: Geometric Topology; Dynamical Systems


MSC-class: 37E30 (Primary) 57M60, 37B40 (Secondary)


The theme of this paper is that algebraic complexity implies dynamical
complexity for pseudo-Anosov homeomorphisms of a closed surface S_g of genus g.
Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov
homeomorphism of S_g tends to zero at the rate 1/g. We consider here the
smallest dilatation of any pseudo-Anosov homeomorphism of S_g acting trivially
on Gamma/Gamma_k, the quotient of Gamma = pi_1(S_g) by the k-th term of its
lower central series, k > 0. In contrast to Penner's asymptotics, we prove that
this minimal dilatation is bounded above and below, independently of g, with
bounds tending to infinity with k. For example, in the case of the Torelli
group I(S_g), we prove that L(I(S_g)), the logarithm of the minimal dilatation
in I(S_g), satisfies .197 < L(I(S_g))< 4.127. In contrast, we find
pseudo-Anosov mapping classes acting trivially on Gamma/Gamma_k whose
asymptotic translation lengths on the complex of curves tend to 0 as g tends
toward infinity.

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[math/0603669] All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

[math/0603669] All generating sets of all property T von Neumann algebras have free entropy dimension $\leq 1$

All generating sets of all property T von Neumann algebras have free
entropy dimension $\leq 1$


Authors:
Kenley Jung,
Dimitri Shlyakhtenko


Comments: 6 pages

Subj-class: Operator Algebras


MSC-class: 46L54; 52C17


Suppose $N$ is a diffuse, property T von Neumann algebra and X is an
arbitrary finite generating set of selfadjoint elements for N. By using
rigidity/deformation arguments applied to representations of N in full matrix
algebras, we deduce that the microstate spaces of X are asymptotically discrete
up to unitary conjugacy. We use this description to show that the free entropy
dimension of X, $\delta_0(X)$, is less than or equal to 1. It follows that when
N embeds into the ultraproduct of the hyperfinite $\mathrm{II}_1$-factor, then
$\delta_0(X)=1$ and otherwise, $\delta_0(X)=-\infinity$. This generalizes the
earlier results of Voiculescu, and Ge, Shen pertaining to $SL_n(\mathbb Z)$ as
well as the results of Connes, Shlyakhtenko pertaining to group generators of
arbitrary property T algebras.

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Friday, March 03, 2006

Interesting Mathematics

Interesting Mathematics

Here is a paper by Ursula Hamenstadt proving exactness of the mapping class group. The method is interesting and is related to ideas of Kaimanovich. It is math.GR/0510116. One can look on her web page for other papers as well. http://www.math.uni-bonn.de/people/ursula/papers.html

Thursday, March 02, 2006

[math/0603018] On the space of metrics with invertible Dirac operator

[math/0603018] On the space of metrics with invertible Dirac operator

This paper by Mattias Dahl shows that several geometric constructions, eg codim-3 surgery, which one knows how to do in the category of positive scalar curvature manifolds, can in fact be done in the category of spin manifolds with invertible Dirac operator.