Notes of Koji Fujiwara’s Cambridge lecture 09-03-2017

Computing Kazhdan constants by computer

Joint with Yuichi Kabaya.

Experimental mathematics!

1. Mathematical problem

Let {\Gamma} be a group generated by a finite set {S}. Define {\kappa(\Gamma,S)} as the infimum over all unitary representations {\pi} of {\Gamma} without invariant vectors of displacement

\displaystyle  \begin{array}{rcl}  \kappa(\Gamma,S,\pi)=\inf_{|\xi|=1}\max_{s\in S}|\pi(s)\xi-\xi|. \end{array}

Let {\Delta=|S|-\sum_s\in S s\in{\mathbb R}[\Gamma]} denote the Laplacian of {S}. It satisfies {\Delta^*=\Delta}. For every unitary representation {\pi} of {\Gamma}, {\pi(\Delta)} becomes a bounded self-adjoint operator. Let {\epsilon} denote infimum of spectral gaps of {\pi(\Delta)} over all unitary representations {\pi} of {\Gamma} without invariant vectors. Then

\displaystyle  \begin{array}{rcl}  \sqrt{\frac{2\epsilon}{|S|}}\leq \kappa(\Gamma,S). \end{array}

Ozawa has given a useful characterization of property (T): {\Gamma} has Kazhdan’s property iff there exist {b_1,\ldots,b_n\in{\mathbb R}[\Gamma]} such that

\displaystyle  \begin{array}{rcl}  \Delta^2-\epsilon\Delta=\sum_{i=1}^n b_i^* b_i. \end{array}

The optimal {\epsilon} will be denoted by {\epsilon(\Gamma,S)}. Ozawa also shows that {b_i}‘s can be chosen in {{\mathbb Q}[\Gamma]} provided {\sum_{i=1}^n b_i^* b_i} is replaced with

\displaystyle  \begin{array}{rcl}  \sum_{i=1}^n r_i b_i^* b_i. \end{array}

with positive rationals {r_i}.

Finding a solution {(b_1,\ldots,b_n)} amounts to solving a semidefinite programming (SDP) problem. There exist software for this purpose, but they are limited to matrices of size {10^4 \times 10^4}. This limits the size of the supports of candidates {b_i} to a ball of radius 4 if {|S|=10}. Note that 10 is small. The classical generating system of {SL(3,{\mathbb Z})} has 12 elements.

Fortunately, the SDP algorithm does not require a solution of the word problem in {\Gamma}. It provides an approximate solution. A lemma due to Netzer-Thom guarantees that if an approximate solution exists, then a true solution exists.

2. Experiments

I have played with {\tilde A_2} buildings. The smallest one has links with 14 vertices (incidence graph of projective plane over {F_q}, {q=2}).

A classical theorem asserts that discrete cocompact isometry groups of such buildings have property (T). The proof in Bekka-de la Harpe-Valette’s proof gives a generating system {S} and an explicit lower bound on {\kappa(\Gamma,S)}.

The exact value of the Kazhdan’s constant has been obtained by Cartwright-Mlotowski-Steger.

{\tilde A_2} groups for {q=2} and {3} acting transitively on vertices have been classified (Cartwright-Mantero-Steger-Zappa). For {q=2}, there are 9 examples. I did run the software, and it produced figures equal to the CMS bound up to the 6th digit (applying Ntzer-Thom, I could guarantee only the first 4 digits). It also works with the first 8 {q=3} groups.

Ronan has exhibited 4 more groups with 1 triangle quotients, whose Kazhdan constant is unknown. These are triangles of groups with trivial face group, {{\mathbb Z}/3{\mathbb Z}}-edge groups, and order 21 Frobenius vertex groups. They are automatic (Gersten-Short), and they have exactly the same rational growth function (Floyd). Two of them are linear (Koehler-Meixner-Wester), the third has an index 3 normal subgroup which is linear, the fourth one is not linear (Bader-Caprace-Lecureux). On these 4 examples, the software gives the same value of the Kazhdan constant, which seems to be very close to {(\sqrt{2}-1)/\sqrt{3}}. Is this exactly true ?

For {Sl(3,{\mathbb Z})} and {Sl(4,{\mathbb Z})}, we get a slightly better bound that Netzer-Thom’s, roughly a third of Zuk’s upper bound. Unexpectedly, our bound for {Sl(4,{\mathbb Z})} is larger than that for {Sl(3,{\mathbb Z})}.

For finite Coxeter groups and finite complex reflection groups

Is the spectral gap achieved (its a sup)? Answer is yes for {\tilde A_2} groups.

Vdovina: how does spectral gap or Kazhdan constant behave when passing to an index 2 subgroup ? This would help for Ronan groups, which are commensurable to certain {\tilde A_2} groups. There seems to be no simple sharp mathematical answer, but perhaps the algorithmic method.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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