** Local testability in group theory, II **

Joint work with Alex Lubotzky.

Positive results for amenable groups. We stick to finitely presented groups.

**1. Translation into graphs **

A -tuple of size permutations defines a -vertex directed graph of outgoing degree . Each edge has a label in . Conversely, such graphs determine permutations.

Fix a set of relators. Let be the set of vertices fixed by relators. Say graph is -GOOD if this is the proportion of GOOD vertices. Group is testable if -GOOD graphs can be made 1-GOOD by changing an -fraction of edges.

Think of an -GOOD graph as a challenge, and changing an -fraction of edges as a solution to the challenge.

Amenable groups come with plenty of challenges: pick a Folner set, the correponding piece of the Cayley graph, add edges to make its outgoing degree equal to everywhere. The resulting -GOOD graph is called a Cayley challenge.

**2. Residually finiteness **

Arzhantseva-Paunescu: Cayley challenges are solvable iff is residually finite.

I explain how residual finiteness allows to solve Cayley challenges. This relies on Ornstein-Weiss’ quasi-tiling, in Elek-Szabo style.

Fix a radius . Start with a -GOOD Cayley challenge . Then almost all -balls of embed in the Cayley graph. Following Ornstein-Weiss, place a Folner set inside every such -ball.These cover most of , but with a lot of overlap. Extract a maximal -disjoint subcover. The error is necessary.

**Fact**. Such a subcover covers a constant fraction of the cover.

Then iterate: cover the remaining uncovered part with smaller Folner sets, get an -cover of all of .

If each Folner set is almost isomorphic to a finite Schreier graph of , we are done. This works for residually finite groups (Weiss).

This solves Cayley challenges.

** 2.1. What about other challenges? **

For instance, Schreier challenges, coming from Schreier graphs of .

In the abelian case, Schreier graphs are Cayley graphs of quotients, and the previous quasi-tiling method applies. However, the constants may depend on quotient.

In general, non-normal subgroups pose an extra difficulty.

**3. Local testability systems **

This is our technical tool. Given a general challenge , every -ball embeds in some Schreier graph of , but different Schreier graphs may occur. Folner sets are used to tile such balls. The local testability system provides such Folner sets. It takes as arguments a radius and a pointed set with a transitive action whose stabilizers are finitely generated, and returns a finite subsets of .

Locality. Such a map is local if the finite subset only depends on the combinatorics of the ball in and not all of .

Folner condition. The orbit of the marked point under is -Folner.

Transversality. is a transversal for the cosets of some finite inex subgroup of .

Local evenness. This is used to ensure that a uniform fraction of challenges will be covered.

** 3.1. Existence of LTS **

We are able to construct LTS for abelian groups, nilpotent groups of class 2 (in fact, merely a weaker version of LTS), for . This relies on understanding all Schreier graphs.

I explain the construction in the abelian case, in a way that hints how to do for larger classes of groups.

Say , . We need to produce a tile which is Folner and almost isomorphic to a finite Schreier graph. We use Weiss’s general existence theorem, and construct in a greedy manner. Start with the tile of (i.e. ) provided by Weiss. If it fits, let us take it. Otherwise, there exist short elements in stabilizer , let be the subgroup they generate. use Weiss’s tile for .

**4. Questions **

Arzhantseva: can use it for groups approximable by amenable groups? I doubt it.

Fisher: what about the 3-dimensional SOL group? In the nilpotent case, things go wild after 3 steps, so we are stuck with SOL as well.