Timeline for Geometric characterization of property (T) and the Haagerup property
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| May 27, 2014 at 8:43 | comment | added | YCor | @Vladimir: the Bekka-Harpe-Valette book. The examples are one cocompact lattice in $G\times\mathbf{R}$ and one cocompact lattice in $\tilde{G}$, where $G$ is a simple lie group of rank $\ge 2$ and infinite fundamental group, e.g. $Sp_{2n\ge 4}(\mathbf{R})$ or $SO(2,n\ge 3)$. | |
| May 27, 2014 at 3:48 | comment | added | Ian Agol | @Vladimir: see the answer to this question: mathoverflow.net/a/98727/1345 | |
| May 27, 2014 at 2:35 | comment | added | Vladimir | @YvesCornulier: could you please direct me to the reference showing that property (T) is not QI-invariant? Thanks! | |
| May 27, 2014 at 2:35 | comment | added | Vladimir | Thanks everyone - this is a very interesting discussion. @Ian: I didn't know about this - thanks a lot! | |
| May 24, 2014 at 18:20 | comment | added | YCor | @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. | |
| May 24, 2014 at 17:47 | comment | added | Ian Agol | @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, 2014 at 8:58 | comment | added | YCor | @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. | |
| May 24, 2014 at 3:50 | comment | added | Ian Agol | Have you seen this question? mathoverflow.net/q/154431/1345 | |
| May 23, 2014 at 17:25 | comment | added | YCor | 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). | |
| May 23, 2014 at 17:15 | history | asked | Vladimir | CC BY-SA 3.0 |