Infinite Expanders

In a note published at

http://www.wisdom.weizmann.ac.il/~itai/infexp.ps

it is asked (by Binjamini I think), "Is there an infinite expander?".

By definition an infinite expander is an infinite connected bounded geometry graph with the following property: there exists a positive constant, call it c, such that for any set S of vertices (whether finite or not) and any ball B, less than half of whose points are in S, the ratio

(size of boundary S intersect B)/(size of S intersect B)

is greater than c.


The conjecture is that no such "infinite expander" exists.

QUESTIONS:

(a) What would it take for the graph of a group to be an infinite expander?
(b) Relate to the coarse property T problem.

[math/0606120] Maximal rank maps between Riemannian manifolds with bounded geometry

[math/0606120] Maximal rank maps between Riemannian manifolds with bounded geometry

This paper by Abreu-Suzuki gives a condition under which a coarse submersion between Riemannian manifolds is coarsely a product.

Interesting not only in itself but for its references - which are to another author (Kumeu) who came up with the basic coarse ideas, in the context of bg Riemannian manifolds, in the middle 1980s. I had not been aware of this before.

GT Monographs: Volume 9

GT Monographs: Volume 9

This is the proceedings of an Oberwolfach conference on "exotic" homology manifolds. (Roughly speaking, these are manifolds for which the "zero'th Pontrjagin class" is not equal to 1.)

[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.

Full-text: PostScript, PDF, or Other formats

[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.

Full-text: PostScript, PDF, or Other formats

Which authors of this paper are endorsers?

[math/0603621] Property A, partial translation structures and uniform embeddings in groups

[math/0603621] Property A, partial translation structures and uniform embeddings in groups

Brodzki, Niblo, Wright. - I blogged this before, but now there is an explicit invariant.

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

[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.

[math/0601700] Representations of residually finite groups by isometries of the Urysohn space

[math/0601700] Representations of residually finite groups by isometries of the Urysohn space: "Representations of residually finite groups by isometries of the Urysohn space
Authors: Vladimir G. Pestov, Vladimir V. Uspenskij
Comments: 12 pages, LaTeX 2e
Subj-class: Representation Theory
MSC-class: 43A65; 20C99; 22A05; 22F05; 22F50; 54E50

As a consequence of Kirchberg's work, Connes Embedding Conjecture is equivalent to the property that every homomorphism of the group $F_\infty\times F_\infty$ into the unitary group $U(\ell^2)$ with the strong topology is pointwise approximated by homomorphisms with a precompact range. In this form, the property (which we call Kirchberg's property) makes sense for an arbitrary topological group. We establish the validity of the Kirchberg property for the isometry group $Iso(U)$ of the universal Urysohn metric space $U$ as a consequence of a stronger result: every representation of a residually finite group by isometries of $U$ can be pointwise approximated by representations with a finite range. This brings up the natural question of which other concrete infinite-dimensional groups satisfy the Kirchberg property."

[math/0512592] Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

[math/0512592] Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

Here's an interesting paper from the arXiv by Behrstock, Drutu and Mosher

[math/0302310] Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

[math/0302310] Hyperbolic group $C^*$-algebras and free-product $C^*$-algebras as compact quantum metric spaces

An interesting paper on the metric geometry of the "dual absolute value of Dirac"

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

[math/0512040] A pairing between super Lie-Rinehart and periodic cyclic homology

From the arXiv

Authors: Tomasz Maszczyk,
Comments: 11 pages,
Subj-class: K-Theory and Homology; Mathematical Physics,
MSC-class: Primary 16E40, 17B35, 19K56, Secondary 46L87

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology.

Piotr Nowak's Homepage

Piotr Nowak

I was going to reference just a couple of Piotr's articles but then I thought that I might as well point to his entire home page. Everything is good here! In particular he shows that coarse embeddability into a Hilbert space (or into ell-one) is not the same as property A. The example is devastatingly simple: take the disjoint union of n-fold products of copies of some finite group (e.g. the group of order 2). Notice that the spaces here are quasi-isometric to cubes in R^n with the ell-one metric. If one took the ell-two (Euclidean) metric instead, wouldn't one get Yu's old counterexample to coarse Baum-Connes? Something interesting seems to be going on here.

[Of course these aren't bg spaces. Is there a bg space with the same property?]

[math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

[math/0511098] K-Theory of pseudodifferential operators with semi-periodic symbols

Authors: Severino T. Melo, Cintia C. Silva
Subj-class: Operator Algebras; K-Theory and Homology
MSC-class: 46L80; 47G30

Let A denote the C*algebra of bounded operators on L2(R) generated by: (i) all multiplications a(M) by functions a\in C[-\infty,+\infty], (ii) all multiplications by 2\pi-periodic continuous functions and (iii) all Fourier multipliers F^{-1}b(M)F, where F denotes the Fourier transform and b is in C[-\infty,+\infty]. The Fredholm property for operators in A is governed by two symbols, the principal symbol \sigma and an operator-valued symbol \gamma. We give two proofs of the fact that K0(A) and K1(A) are isomorphic, respectively, to Z and Z\oplus Z: by computing the connecting mappings in the standard K-theory six-term exact sequences associated to \sigma and to \gamma. For the second computation, we prove that the image of \gamma is isomorphic to the direct sum of two copies of the crossed product of C[-\infty,+\infty] by an automorphism, and then use the Pimsner-Voiculescu exact sequence to compute its K-theory.

Another Coarse Geometry correction

See this link for another correction to LCG, this time in the statement and proof of Rosenblatt's theorem in chapter 3. Thanks to Steve Ferrt and his students for spotting this one.

Best wishes

John

[math/0509526] An analogue of the Novikov Conjecture in complex algebraic geometry

[math/0509526] An analogue of the Novikov Conjecture in complex algebraic geometry

We're familiar with the idea that statements about positive scalar curvature metrics (Gromov-Lawson-Rosenberg conjecture) and statements about higher signatures (Novikov conjecture) have certain parallels - ultimately they involve the higher index theorem applied to different elliptic operators, Dirac in the first case and signature in the second.

In this new paper Jonathan Rosenberg proposes a further family of statements involving higher index theory for the Dolbeault operator. These are statements in complex algebraic geometry about "higher Todd genera" for varieties.

This could be a whole new playground for higher index theorists.

Discrete Morse theory and graph braid groups

AGT 5 (2005) Paper 44 (Abstract)

Discrete Morse theory and graph braid groups
Daniel Farley, Lucas Sabalka
Abstract. If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).

Subelliptic spinc Dirac operators

Charlie Epstein kindly gave me permission to publish this paper on index theory for generalizations of the d-bar Neumann problem and its relationship to index theory for Fourier integral operators.

[math/0505622] Free construction of CAT(1) spaces

[math/0505622] Free construction of CAT(1) spaces

[math/0508135] On the generalized Nielsen realization problem

[math/0508135] On the generalized Nielsen realization problem

Authors: Jonathan Block, Shmuel Weinberger
Subj-class: Geometric Topology
MSC-class: 57N

The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds $M$. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group $\pi$ on Euclidean space extends to an action of $\Pi$, a group containing $\pi$ as a subgroup of finite index, even when all the torsion of $\Pi$ lives in $\pi$.

GT Vol 9 (2005) Paper 34 Hadamard spaces with isolated flats

GT Vol 9 (2005) Paper 34 (Abstract)

By Hruska and Kleiner with Hindawi

[math/0507542] A new kind of index theorem

[math/0507542] A new kind of index theorem

This is Ron Douglas' talk at the Kaminker retirement conference.

[hep-th/0507206] Dixmier traces on noncompact isospectral deformations

[hep-th/0507206] Dixmier traces on noncompact isospectral deformations