Tag Archives: amenable

Ghostbusting and Property A

Let \(X\) be a bounded geometry discrete metric space.  Guoliang Yu defined a ghost to be an element of the Roe algebra \( C^*_u(X) \) that is given by a matrix \(T_{xy}\) whose entries tend to zero as \(x,y\to\infty\).

The original counterexamples of Higson to the coarse Baum-Connes conjecture were noncompact ghost projections on box spaces derived from property T groups.  On the other hand, all ghost operators on a property A space are compact.

In Ghostbusting and Property A, Rufus Willett and I show that all ghosts on \(X\) are compact if and only if \(X\) has property A.  (Appropriately enough, on a space without property A we construct ghosts using the spectral theorem.) The paper will appear in the Journal of Functional Analysis.

Question: To what characterization of ordinary amenability does this correspond?

 

Around soficity

Andreas Thom just posted the article [1005.0823] About the metric approximation of Higman’s group on the arXiv today. It is quite short with a specific result about Higman’s group, but the introduction was most helpful to me in learning a bit about the ideas related to “soficity” of groups. It refers to another interesting paper: Elek, Gábor, and Endre Szabó. “Hyperlinearity, essentially free actions and L2-invariants. The sofic property.” Mathematische Annalen 332, no. 2 (4, 2005): 421-441.

It seems that these authors use some words like “hyperlinear” and “amenable action” in a sense different to that which is common to us in Baum-Connes land. for instance, for Elek-Szabo, the trivial action of a group on a point is *always* amenable.

More about characterizations of exactness

Following up an earlier post with some notes on the three papers below:

All of these papers focus on the question of characterizing in “homological” terms what it is for a discrete group \( G \) to be exact (or, more generally, to act amenably on some compact space — it is known that exactness is equivalent to the amenability of the action of \( G \) on its Stone-Cech compactification \( \beta G \)).

Continue reading

Various characterizations of exactness

This post is a place to list a number of papers that have recently appeared on the arXiv which reformulate the notion of exactness for groups (or property A for spaces or amenable actions) in different ways:

I’ll post more later about the relations between these.

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

\[ \frac{\# (\partial S \cap B)}{\# (S\cap B)} \]

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.