Wall structures and coarse embeddings
Lots of spaces have Property A, great! This provides many coarse embeddings in Hilbert space. Nevertheless, there exist spaces which coarsely embed but do not satisfy Property A. I will describe the best known obstruction to Property A, expansion. This also prevents Hilbertian embeddings. Then I will describe an other mechanism for the construction if embeddings, spaces with walls.
1. Is Property A equivalent to coarse embeddability in Hilbert space ?
A sequence of finite graphs is called an expander if
- Size tends to infinity.
- Valency stays uniformly bounded.
- Cheeger constant stays bounded below.
Note the forthcoming conference in Neuchatel (Switzerland), december 1-5: Expanders everywhere!
Given such a sequence, form a metric space, their disjoint union, by requiring that distance between distinct graphs be larger than their diameters.
Fact. An expander does not have Property A and does not coarsely embed in Hilbert space.
1.2. A space which does not have Property A but does coarsely embed in Hilbert space
The following example is due to P. Nowak (2002). One deals with embeddings in (from the point of view of coarse embeddings, this is equivalent to ).
Let be a finite group, with generating set itself, embed it bi-Lispchitaly in . Consider direct sum with obvious generating system. This embeds in with the same distorsion (direct sum of embeddings), so the disjoint union embeds as well.
When , we are dealing with Hamming cubes, we shall see below an alternating embedding in .
Nowak verifies that Property A fails. This requires some effort (not included in my notes).
2. Wall spaces
Let be a graph. A wall on is a subset of edges whose removal yields exactly two connected components. A wall structure on is a set of walls such that each edge is contained in exactly one wall. Notation :
The wall (pseudo) metric is defined by
When is a metric, it embeds isometrically in as follows. Fix an origin . Map vertex to indicator of .
Example. The graph metric on a tree is a wall metric. Each single edge is a wall. Take all of them.
Example. Hamming cube. The set of edges in one direction is a wall. Take all of them. The resulting metric equals the Hamming metric.
2.2. Bounded geometry
Ultimately, we would like to provide examples of finitely generated groups without property A (but which embed coarsely in . For this, we would like to proide a graph counterexample and then embed it is a group. This requires the graph to have bounded geometry.
Nowak’s example does not have this property (valency tends to infinity), so we turn to different constructions.
3. Residual finiteness
Say a group is residually finite if every non trivial element has a non trivial image under some homomorphism to some finite group. Equivalently, the intersection of all finite index normal subgroups of is .
This often happens.
Examples. Free groups, abelian groups, linear groups are residually finite. It is an open question wether all hyperbolic groups are residually finite.
Definition 1 Given a sequence of nested finite index normal subgroups
the box space is the coarse disjoint union of , where is the projection to of a fixed generating system of .
Theorem 2 (Guentner) The box space of a group has property A if and only if is amenable.
Indeed, F\o lner sets can be used for the box space. Conversely, the F\o lner sets of the box space can be averaged to provide F\o lner sets for .
3.1. Derived series
Given a finitely generated group , the derived -series is defined recursively by and
The idea is to kill commutators and -powers, i.e. to provide the largest quotient which is abelian with order . Then is a direct sum of cyclic groups .
Theorem 3 (Arzhantseva-Guentner-Spakula 2012) Let be a non abelian free group. Let be its derived 2-series. Let be the corresponding box space. Then embeds coarsely in .
View as a cover of . This will provide us with a wall structure on whose wall metric is close to the origin metric.
The covering map is constructed explicitely by lifting a maximal subtree of en specifying how the copies are glued together. The fibers are cubes, which are spaces with walls.
The walls of are inverse images, under the covering map, of edges of . The wall metric is smaller than the graph metric, and they coincide below the girth. Since girth tends to infinity, ther wall metric gets closer and closer to the graph metric. This yields the embedding.
What we really use is the existence of embeddings for the disjoint union of quotient groups, not that much the wall structure.