## Notes of Emmanuel Breuillard’s informal talk in Cambridge 21-02-2017

Informal discussion on spectral gaps for isometric group actions

1. Finite Kazhdan data

1.1. Banach Kazhdan data ?

Cornelia Drutu wants a finite subset ${S}$ of a Lie group ${G}$ which is a Kazhdan set for every affine isometric action of ${G}$ on a Banach space from a certain class. This means there exists ${\epsilon>0}$ such that if there exists a point ${v}$ such that ${\max_S d(sv,v)<\epsilon}$, then there is a fixed point.

Shalom provides one for ${Sl(n,{\mathbb R})}$ which does the job for Hilbert spaces. Cornulier generalized this to all Kazhdan Lie groups.

Bader-Furman-Gelander-Monod establish ${F_{L^p}}$ for products of higher rank simple Lie groups, and this passes to lattices.

Here is how examples of lattices are obtained. Let ${\mathbb{G}}$ be an algebraic group defined over a number field ${K}$. Let ${K_i}$ be the Archimedean completions of ${K}$. ${\mathbb{G}(K_i)}$ can be compact or not. For instance, if ${\mathbb{G}=SO(q)}$, ${q}$ a quadratic form, then ${\mathbb{G}(K_i)}$ is compact iff ${q}$ is ${K_i}$-anisotropic, i.e. does not represent 0 over ${K_i}$.

Then (Borel-Harish Chandra) ${\Gamma=\mathbb{G}(\mathcal{O}_K)}$ is a lattice in the real Lie group

$\displaystyle \begin{array}{rcl} G=\prod_i \mathbb{G}(K_i) \end{array}$

and ${\Gamma}$ is cocompact iff one of the factors is compact. Remove the compact factors. This is a source of lattices. One can arrange that each remaining factor has higher rank. Then BFGM applies, a finite generating set of ${\Gamma}$ does the job for the family of ${L^p}$ spaces.

For a specific example, let ${K={\mathbb Q}[\sqrt{2}]}$ and

$\displaystyle \begin{array}{rcl} q(x_1,\ldots,x_4)=x_1^2+x_2^2+\sqrt{2}x_3^2+\sqrt{2}x_4^2. \end{array}$

Then ${\Gamma=SO({\mathbb Z}[\sqrt{2}])}$, up to finite index, is a lattice of rank 2 Lie group ${SO(2,2)}$.

PP: aren’t we making our life harder? If ${S}$ generates a dense subgroup, it should be easier ?

AV: Some of it remains for lattices in products of rank one groups ? Property ${\tau}$ is sometimes known.

1.2. Search for expanders

EB: In other words, what is looked for is a source of especially strong expanders.

CD: From Goulnara Arzhantseva’s point of view, a good expander should also have a large girth, proportional to diameter. This is required for counterexamples to Baum-Connes conjecture. Such graphs can be embedded into groups, by Gromov’s ansatz.

EB: Cayley graphs of finite simple groups are expected to behave in this way, for suitable generating sets. Babai’s conjecture states that for all generating sets, the diameter of the alternating group ${A_n}$ grows at most polynomially. On the other hand, for generic generating sets, girth is at least ${\log(n!)\sim n\log n}$. Babai’s conjecture is not even known for generic generating sets (best result by Helfgott and Seres is not far).

Babai’s conjecture for finite simple groups is

$\displaystyle \begin{array}{rcl} \sup_{S\, \mathrm{finite\,subset}}\mathrm{diameter}(Cay(G,S))=O(\log|G|^C). \end{array}$

For bounded rank, much better is expected,

$\displaystyle \begin{array}{rcl} \sup_{S\, \mathrm{finite\,subset}}\mathrm{diameter}(Cay(G,S))\leq \mathrm{Const}(\mathrm{rank}).\,\log|G|. \end{array}$

Known for ${Sl(2,p)}$. Furthermore, in this case, for random generators, both diameter and girth ${\sim\log p}$.

EB: Akhmedov shows that if ${\Gamma}$ is a linear group, not virtually solvable, then there exist finite generating sets with arbitrarily large girth.

Conjecture. For ${PSl(2,q)}$, for all generating sets, the Cayley graphs have a uniform lower bound on ${\lambda_1}$.

It is known for ${q}$ prime with a small family of exceptions.

Conjecture. ${Sl(3,{\mathbb Z})}$ has uniform property (T).

This would imply trivially a lot of recent results for ${Sl(3,p)}$, ${Sl(3,n)}$,… So it is pretty strong and should not generalize much. For instance, lattices in product cannot have this property, since they map onto dense subgroups of factors, which can be generated by small elements, and thus violate any uniformity.

It is hard since an arbitrary generating set need not contain unipotents, which are crucial to classical arguments (Shalom,…).

Fixed point properties. Is there a strong property ${\tau}$ ? Yes, see Lafforgue.

2. Finite simple groups

EB: This emerged from Lubotzky’s 1,2,3 question. 1 and 2 is classical, 3 is due to Bourgain-Gamburd. Historically, this goes back to Pimsker and Margulis. This question showed how few tools one had to obtain spectral gaps. Property (T), Selberg’s ${\frac{3}{16}}$ theorem (applies to congruence covers of the modular surface, but converts into an estimate for finite Cayley graphs, see my survey in Groups St Andrews 2013). Zig-zag products. Random graphs. Zuk’s criterion and applications by Dymara-Januskiewicz, Kassabov.

The Bourgain-Gamburd method was a revolution. It starts from the exponential mixing interpretation, i.e. behaviour of random walks. In short time, girth tells that things behave like on a tree. At larger time scales, additive combinatorics enters. The ${\ell^2}$ norm of the convolution product of measures must decay fast unless measures charge approximate subgroups. To conclude, use Sarnak-Xue’s multiplicity trick: because of symmetry, eigenspaces are representations, their dimensions are quite large.

The spectral gap for compact Lie groups was present in people’s minds, since Lubotzky-Philips-Sarnak, see Sarnak’s book. There is a motivation from quantum computation. One need to explore all of ${SU(2)}$ by combining a finite number of gates, i.e. multiplying elements from of fixed finite set. To get a ${\delta}$-dense subset of ${SU(2)}$, one needs words of length polylog in ${\delta}$, see the book of …