Infinite Expanders

In a note published at

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.


(a) What would it take for the graph of a group to be an infinite expander?
(b) Relate to the coarse property T problem.

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