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?

6$\begingroup$ 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 QIinvariants (for Haagerup it's a recent result of Carette). $\endgroup$ – YCor May 23 '14 at 17:25

$\begingroup$ Have you seen this question? mathoverflow.net/q/154431/1345 $\endgroup$ – Ian Agol May 24 '14 at 3:50

$\begingroup$ @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. $\endgroup$ – YCor May 24 '14 at 8:58

$\begingroup$ @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 graphtheoretic intrinsic characterization due to Brooks, e.g. in terms of the Cheeger constant or smallest eigenvalue $=0$, but this is not mentioned. $\endgroup$ – Ian Agol May 24 '14 at 17:47

$\begingroup$ @Ian: the Følner condition can be made purely metric, since: if $F$ is the Følner set (that is $SFF/F$ is small where $S$ is the generating set), then the 1neighbourhood of $F^{1}$ in the Cayley graph is small with respect to $F$. This condition (existence of Følner sets) is actually a QIinvariant for connected graphs of bounded valency. $\endgroup$ – YCor May 24 '14 at 18:20