** Expanders, Beauville surfaces and buildings **

**1. Expanders **

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

**Example**. For a -regular tree, .

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

I will stick to -regular graphs.

** 1.1. Examples **

In 1973, Pinsker showed that most -regular graphs with enough vertices have Cheeger cobstant .

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 be the 0-1 adjacency matrix. If graph is -regular, all its eigenvalues belong to , and is an eigenvalue. The second eigenvalue iff graph is connected.

The *spectral gap* is . Then a sequence of -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 of finite groups whose Cayley graphs on form an expander. The pro-2 completion satisfies Golod-Shafarevitch’s inequality.

We give an explicit sequence of finite -groups and generators , forming a tower, whose Cayley graphs on form an expander.

** 2.1. The construction **

Start with a group with 7 generators and 7 relators, whose Cayley graph is the 1-skeleton of a thick building whose links are projective planes over . 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 has vertex set and as many edges between and as words such that is a cyclic conjugate of an element of .

The previous example has star graph the projective plane over .

Say a presentation is -special if relators have length and its star graph is a generalized -gon.

**Question (Jim Howie 1989)**. Are groups with -special presentations SQ-universal?

Martin Edjvet and I proved that the answer is negative. Indeed, we show that all of them act on buildings, and apply Margulis’ finiteness theorem: all their quotients are finite (attributed to Shalom-Steger but unpublished for non-Bruhat-Tits buildings).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 -special presentation which acts on a building with non-classical links.

** 2.3. Back to construction **

The subgroup generated by the first two generators has index 2. It has a linear representation in . We turn them into finite band infinite Toeplitz matrices, getting elements in .

We conjecture that the lower exponent-2 central series of 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 where and are non-singular projective curves of genera and a finite group acting freely and holomorphically.

Beauville’s original example consists of two Fermat curves of degree 5 with . It is a fake (same numerical invariants).

Theorem 3 (Barker-Boston-Peyerimhoff-Vdovina)Consider the lower exponent-2 central series of (resp. of ). The quotients induce Beauville surfaces if is not a power of 2.

The Riemann surfaces form a tower of Belyi branched coverings (coverings of branched over 3 points ). The etale covering over the punctured Riemann sphere is induced by the morphism

mapping the three generators (with trivial product) to three generators of (with trivial product).