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.