# 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.