# 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 $$\beta G$$).

# 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.