## 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.