[math/0506547] Coarse dimensions and partitions of unity

More development of "coarse dimension theory" by professional dimension-theorists.

## Wednesday, June 29, 2005

## Tuesday, June 28, 2005

### [math/0506544] Isometries, rigidity, and universal covers

[math/0506544] Isometries, rigidity, and universal covers

Farb and Weinberger classify aspherical Riemannian manifolds whose universal cover has "extra symmetry"

Farb and Weinberger classify aspherical Riemannian manifolds whose universal cover has "extra symmetry"

## Tuesday, June 21, 2005

### Coarse Homotopy - Trung's Thesis

This is a small advertising memo about part of Viet-Trung Luu's forthcoming thesis, which contains a very elegant unified treatment of various theorems about the 'coarse homotopy' or 'Lipschitz homotopy' invariance of the K-theory of C*(X). It formalizes the basic idea of the paper by Nigel and me in the Transactions, found here. One needs "C*(X) with coefficients", defined using Hilbert modules. Trung's approach has a geometric and an analytic part. The geometric part is to show that a coarse homotopy gives an element of C*(X; C[0,1]). The analytic part is to show that there is a natural pairing between the K-theory of C*(X;D) and the K-homology of D, with values in the K-theory of C*(X). Now one uses the known homotopy invariance of K-homology - the fact that the inclusions of 0 and 1 into [0,1] induce isomorphisms on K-homology - to conclude the coarse homotopy invariance of K_*(C*(X)).

### [math/0506361] Property (T) and rigidity for actions on Banach spaces

[math/0506361] Property (T) and rigidity for actions on Banach spaces

Here the authors study variations of property T for actions on Banach spaces, especially Lp. When p is not 2 the equivalence between the original version of property T (isolation of the trivial representation) and the fixed point version F (affine isometric actions) does not hold.

They prove that every property T group has T(L^p) for all p, and F(L^p) for p<2+\epsilon.

Here the authors study variations of property T for actions on Banach spaces, especially Lp. When p is not 2 the equivalence between the original version of property T (isolation of the trivial representation) and the fixed point version F (affine isometric actions) does not hold.

They prove that every property T group has T(L^p) for all p, and F(L^p) for p<2+\epsilon.

## Friday, June 17, 2005

### Corrections to "Lectures on Coarse Geometry"

Corrections to "Lectures on Coarse Geometry"

Bernd Grave pointed out a couple of errors in Lectures on Coarse Geometry, explained here.

One is a foolish mistake I made with Urysohn's lemma. Given two disjoint closed subsets of a compact Hausdorff space, Urysohn's lemma allows one to find a continuous function equal to zero on one set, one on the other. But in general the set of points where said function is zero will be *larger* than the originally given closed set. Only in metric spaces can one expect to arrange equality.

Bernd Grave pointed out a couple of errors in Lectures on Coarse Geometry, explained here.

One is a foolish mistake I made with Urysohn's lemma. Given two disjoint closed subsets of a compact Hausdorff space, Urysohn's lemma allows one to find a continuous function equal to zero on one set, one on the other. But in general the set of points where said function is zero will be *larger* than the originally given closed set. Only in metric spaces can one expect to arrange equality.

## Wednesday, June 15, 2005

### Property A and nuclearity of Roe algebras (application/pdf Object)

propAnuclear.pdf (application/pdf Object)

By Brodzki, Niblo and Wright. One knows that for a discrete

Comments: Gromov's counterexample construction gives a way of building an 'almost' injective embedding of a more-or-less arbitrary coarse space into a discrete group. Relate that to this paper!

By Brodzki, Niblo and Wright. One knows that for a discrete

*group*G, property A is equivalent to exactness/nuclearity of its translation algebra. But the same thing is not known for a general (uniformly discrete) coarse*space*. In this paper is a measure of the 'grouplikeness' of a coarse space in terms of "partial translation structures". If X is uniformly embeddable in a discrete group, then it has a "free partial translation structure" and the expected equivalence holds.Comments: Gromov's counterexample construction gives a way of building an 'almost' injective embedding of a more-or-less arbitrary coarse space into a discrete group. Relate that to this paper!

### [math/0403059] A K-Theoretic Proof of Boutet de Monvel's Index Theorem

[math/0403059] A K-Theoretic Proof of Boutet de Monvel's Index Theorem

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory

0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory

0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

## Tuesday, June 14, 2005

### wall.pdf (application/pdf Object)

wall.pdf (application/pdf Object)

Cambridge lecture notes on differential topology (ca. 1960) by C.T.C. Wall.

Detailed treatment of foundational theorems.

Cambridge lecture notes on differential topology (ca. 1960) by C.T.C. Wall.

Detailed treatment of foundational theorems.

### hgw.pdf (application/pdf Object)

hgw.pdf (application/pdf Object)

Guentner, Higson and Weinberger prove the Novikov conjecture for all linear groups. The proof goes via coarse embeddings into Hilbert space.

Guentner, Higson and Weinberger prove the Novikov conjecture for all linear groups. The proof goes via coarse embeddings into Hilbert space.

### cime.pdf (application/pdf Object)

cime.pdf (application/pdf Object)

Nigel Higson's notes on C*-algebras, K-theory and the Baum-Connes conjecture. Includes discussion of the "spectral picture" of K-theory; the E-theory approach to the Baum Connes conjecture; some counter-example stuff.

Nigel Higson's notes on C*-algebras, K-theory and the Baum-Connes conjecture. Includes discussion of the "spectral picture" of K-theory; the E-theory approach to the Baum Connes conjecture; some counter-example stuff.

### Mendel-Naor - cotype

Metric Cotype

[This gives the definitive answer to the question when Lp is uniformly (coarsely) embeddable into Lq.]

Title: Metric Cotype

Authors: Manor Mendel, Assaf Naor

Comments: 42 pages

Subj-class: Functional Analysis; Metric Geometry

MSC-class: 46B20; 51F99

\\

We introduce the notion of cotype of a metric space, and prove that

for Banach spaces it coincides with the classical notion of Rademacher

cotype. This yields a concrete version of Ribe's theorem, settling a

long standing open problem in the non-linear theory of Banach spaces.

We apply our results to several problems in metric geometry. Namely,

we use metric cotype in the study of uniform and coarse embeddings,

settling in particular the problem of classifying when L_p coarsely or

uniformly embeds into L_q. We also prove a non-linear analog of the

Maurey-Pisier theorem, and use it to answer a question posed by Arora,

Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain

quantitative bounds in a metric Ramsey theorem due to Matousek.

[This gives the definitive answer to the question when Lp is uniformly (coarsely) embeddable into Lq.]

Title: Metric Cotype

Authors: Manor Mendel, Assaf Naor

Comments: 42 pages

Subj-class: Functional Analysis; Metric Geometry

MSC-class: 46B20; 51F99

\\

We introduce the notion of cotype of a metric space, and prove that

for Banach spaces it coincides with the classical notion of Rademacher

cotype. This yields a concrete version of Ribe's theorem, settling a

long standing open problem in the non-linear theory of Banach spaces.

We apply our results to several problems in metric geometry. Namely,

we use metric cotype in the study of uniform and coarse embeddings,

settling in particular the problem of classifying when L_p coarsely or

uniformly embeds into L_q. We also prove a non-linear analog of the

Maurey-Pisier theorem, and use it to answer a question posed by Arora,

Lovasz, Newman, Rabani, Rabinovich and Vempala, and to obtain

quantitative bounds in a metric Ramsey theorem due to Matousek.

### Pseudocharacters

Geometry and Topology, Volume 9 (2005) Paper no. 26, pages 1147--1185

Geometry of pseudocharacters

Author(s):

Jason Fox Manning

Abstract:

If G is a group, a pseudocharacter f: G-->R is a function which is

"almost" a homomorphism. [in a coarse-geometric sense: JR]

If G admits a nontrivial pseudocharacter f,

we define the space of ends of G relative to f and show that if the

space of ends is complicated enough, then G contains a nonabelian free

group. We also construct a quasi-action by G on a tree whose space of

ends contains the space of ends of G relative to f. This construction

gives rise to examples of "exotic" quasi-actions on trees.

Geometry of pseudocharacters

Author(s):

Jason Fox Manning

Abstract:

If G is a group, a pseudocharacter f: G-->R is a function which is

"almost" a homomorphism. [in a coarse-geometric sense: JR]

If G admits a nontrivial pseudocharacter f,

we define the space of ends of G relative to f and show that if the

space of ends is complicated enough, then G contains a nonabelian free

group. We also construct a quasi-action by G on a tree whose space of

ends contains the space of ends of G relative to f. This construction

gives rise to examples of "exotic" quasi-actions on trees.

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