Computing Kazhdan constants by computer
Joint with Yuichi Kabaya.
1. Mathematical problem
Let be a group generated by a finite set . Define as the infimum over all unitary representations of without invariant vectors of displacement
Let denote the Laplacian of . It satisfies . For every unitary representation of , becomes a bounded self-adjoint operator. Let denote infimum of spectral gaps of over all unitary representations of without invariant vectors. Then
Ozawa has given a useful characterization of property (T): has Kazhdan’s property iff there exist such that
The optimal will be denoted by . Ozawa also shows that ‘s can be chosen in provided is replaced with
with positive rationals .
Finding a solution amounts to solving a semidefinite programming (SDP) problem. There exist software for this purpose, but they are limited to matrices of size . This limits the size of the supports of candidates to a ball of radius 4 if . Note that 10 is small. The classical generating system of has 12 elements.
Fortunately, the SDP algorithm does not require a solution of the word problem in . It provides an approximate solution. A lemma due to Netzer-Thom guarantees that if an approximate solution exists, then a true solution exists.
I have played with buildings. The smallest one has links with 14 vertices (incidence graph of projective plane over , ).
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 and an explicit lower bound on .
The exact value of the Kazhdan’s constant has been obtained by Cartwright-Mlotowski-Steger.
groups for and acting transitively on vertices have been classified (Cartwright-Mantero-Steger-Zappa). For , 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 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, -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 . Is this exactly true ?
For and , we get a slightly better bound that Netzer-Thom’s, roughly a third of Zuk’s upper bound. Unexpectedly, our bound for is larger than that for .
For finite Coxeter groups and finite complex reflection groups
Is the spectral gap achieved (its a sup)? Answer is yes for 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 groups. There seems to be no simple sharp mathematical answer, but perhaps the algorithmic method.