# Geometric characterization of property (T) and the Haagerup property

Are there geometric characterizations of property (T) and the Haagerup property, i.e., such that can be stated in terms of a Cayley graph, in analogy to the Folner set characterization of amenability?

• In principle yes, because f.g. groups $G_1,G_2$ with the same Cayley graph satisfy: $G_1$ has T iff $G_2$ has T, and the same for Haagerup. But there is no known definition in these terms. And unlike amenability there is no coarse characterization, because these are not QI-invariants (for Haagerup it's a recent result of Carette).
– YCor
May 23 '14 at 17:25
• Have you seen this question? mathoverflow.net/q/154431/1345 May 24 '14 at 3:50
• @Ian I don't see any close links about these questions. Vladimir's question is whether Property T can be defined purely in terms of the (unlabeled) Cayley graph. Ozawa's question is about algorithmic recognition of Property T from a presentation. That the set of presentations of Prop T groups is enumerable is not what I'd call a geometric characterization of Cayley graphs.
– YCor
May 24 '14 at 8:58
• @YvesCornulier: I assumed that Vladimir meant the labeled Cayley graph. My understanding of the Folner condition is that it implicitly uses the group multiplication, so I assumed that he meant to incorporate the group action. There is a graph-theoretic intrinsic characterization due to Brooks, e.g. in terms of the Cheeger constant or smallest eigenvalue $=0$, but this is not mentioned. May 24 '14 at 17:47
• @Ian: the Følner condition can be made purely metric, since: if $F$ is the Følner set (that is $|SF-F|/|F|$ is small where $S$ is the generating set), then the 1-neighbourhood of $F^{-1}$ in the Cayley graph is small with respect to $|F|$. This condition (existence of Følner sets) is actually a QI-invariant for connected graphs of bounded valency.
– YCor
May 24 '14 at 18:20