Expanders, Beauville surfaces and buildings
For a graph, Cheeger’s constant enters in the isoperimetric inequality
Example. For a -regular tree, .
Definition 1 An expander is a sequence of finite bounded degree graphs whose Cheeger constants are bounded below.
I will stick to -regular graphs.
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on buildings, and apply Margulis’ finiteness theorem: all their quotients are finite (attributed to Shalom-Steger but unpublished for non-Bruhat-Tits buildings).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).