## Notes of Alina Vdovina’s Cambridge lecture 31-05-2017

Expanders, Beauville surfaces and buildings

1. Expanders

For a graph, Cheeger’s constant enters in the isoperimetric inequality

$\displaystyle \begin{array}{rcl} \min\{|A|,|V\setminus A|\}\leq h\,|A|. \end{array}$

Example. For a ${n}$-regular tree, ${h=n-2}$.

Definition 1 An expander is a sequence of finite bounded degree graphs whose Cheeger constants are bounded below.

I will stick to ${p}$-regular graphs.

1.1. Examples

In 1973, Pinsker showed that most ${p}$-regular graphs with enough vertices have Cheeger cobstant ${h\geq 1/2}$.

Margulis gave explicit examples of 8-regular expander graphs as Cayley graphs of finite groups.

Note that Cheeger constant is not very robust. Small changes affect it. Therefore direct computation is uneasy.

1.2. Spectral properties

Let ${A}$ be the 0-1 adjacency matrix. If graph is ${p}$-regular, all its eigenvalues belong to ${[-p,p]}$, and ${p=\lambda_1}$ is an eigenvalue. The second eigenvalue ${\lambda_2 iff graph is connected.

The spectral gap is ${p-\lambda_2}$. Then a sequence of ${p}$-regular graphs is an expander iff the spectral gaps are bounded below (Dodziuk 1984, Alon-Milman 1985, Alon 1986).

2. New examples

Ramanujan graphs, zig-zag products, finite simple groups, Lubotzky-Samuels-Vishne, Sarveniazi: link to Euclidean buildings.

The following examples arise from group actions on buildings.

Theorem 2 (Peyerimhof-Vdovina) We give an explicit sequence ${\Gamma_n=\langle x,y|r_1,r_2,r_3,[y,x,\ldots,x]\rangle}$ of finite groups whose Cayley graphs on ${\{x,y\}}$ form an expander. The pro-2 completion ${\Gamma_0}$ satisfies Golod-Shafarevitch’s inequality.

We give an explicit sequence ${N_k}$ of finite ${2}$-groups and generators ${\{x_k,y_k\}}$, forming a tower, whose Cayley graphs on ${\{x_k,y_k\}}$ form an expander.

2.1. The construction

Start with a group ${\Gamma}$ with 7 generators and 7 relators, whose Cayley graph is the 1-skeleton of a thick ${\tilde A_2}$ building whose links are projective planes over ${F_2}$. It appears first in Edjvet-Howie 1989 in the context of small cancellation.

2.2. SQ-universality of small cancellation groups

The star graph of a group presentation ${\langle X|R\rangle}$ has vertex set ${X\cup X^{-1}}$ and as many edges between ${x}$ and ${y}$ as words ${w}$ such that ${x^{-1}yw}$ is a cyclic conjugate of an element of ${R\cup R^{-1}}$.

The previous example has star graph the projective plane over ${F_2}$.

Say a presentation is ${(m,k)}$-special if relators have length ${k}$ and its star graph is a generalized ${m}$-gon.

Question (Jim Howie 1989). Are groups with ${(3,3)}$-special presentations SQ-universal?

Martin Edjvet and I proved that the answer is negative. Indeed, we show that all of them act on ${\tilde A_2}$ buildings, and apply Margulis’ finiteness theorem: all their quotients are finite (attributed to Shalom-Steger but unpublished for non-Bruhat-Tits buildings).

Nicolas Radu recently found a group with ${(3,3)}$-special presentation which acts on a building with non-classical links.

2.3. Back to construction

The subgroup ${\Gamma_0}$ generated by the first two generators has index 2. It has a linear representation in ${Gl(9,2)}$. We turn them into finite band infinite Toeplitz matrices, getting elements in ${Gl(9,F_2[1/y])}$.

We conjecture that the lower exponent-2 central series of ${\Gamma_0}$ givens rise to finite abelian quotients or orders 8,8,4,… periodically.

3. Beauville surfaces

A colleague in Newcastle once observed that a 256-element group arising in our construction also appeared in a paper by Bauer-Catanese-Grunewald on Beauville surfaces.

A Beauville surface is a complex algebraic surface of the form ${(C_1\times C_2)/G}$ where ${C_1}$ and ${C_2}$ are non-singular projective curves of genera ${\geq 2}$ and ${G}$ a finite group acting freely and holomorphically.

Beauville’s original example consists of two Fermat curves of degree 5 with ${G={\mathbb Z}/5{\mathbb Z}\times {\mathbb Z}/5{\mathbb Z}}$. It is a fake ${P^1\times P^1}$ (same numerical invariants).

Theorem 3 (Barker-Boston-Peyerimhoff-Vdovina) Consider the lower exponent-2 central series ${(H_k)}$ of ${\Gamma}$ (resp. of ${\Gamma_0}$). The quotients ${\Gamma/H_k}$ induce Beauville surfaces if ${k}$ is not a power of 2.

The Riemann surfaces form a tower of Belyi branched coverings (coverings of ${P^1}$ branched over 3 points ${a,b,c}$). The etale covering over the punctured Riemann sphere is induced by the morphism

$\displaystyle \pi_1(P^1\setminus\{a,b,c\})\rightarrow\Gamma/H_k$

mapping the three generators (with trivial product) to three generators of ${\Gamma}$ (with trivial product).