Notes on the Abért–Nikolov theorem on rank gradient and cost (notes by Holger Kammeyer after his own lecture)

These are notes of a talk on a theorem of Abért–Nikolov: Let be a finitely generated group and let be a chain of finite index subgroups. Assume that the action of on the boundary of the coset tree of is essentially free. Then the rank gradient of with respect to the chain equals the cost of the action of on .

The coset tree

Let be a countable, discrete group and let be a chain of finite index subgroups. For the moment we do not make any further assumptions on the chain . So the subgroups must neither be normal, nor are they required to have trivial total intersection.

Definition: The (right) coset tree of has vertex set and an edge from to if and only if and .

Not that has the canonical root , so that the -th level vertex set is just , and each node in has precisely children.


A typical coset tree.

Definition: The boundary of the coset tree consits of all infinite rays starting at the root .

Thus in mathematical terms we have and this description makes sense not only in the category of sets but also in the category of topological spaces and of measure spaces. Each carries the discrete topology and the uniform probability measure. Thus as a space, is compact, totally disconneted and Hausdorff. It has a basis of the topology given by shadows where a shadow consists of all rays going through the vertex . The Borel probability measure on is determined by the values . The group permutes the cosets in by right multiplication preserving the child–parent relation. Thus acts on from the right by tree automorphisms and we obtain an induced probability measure preserving right action of on by homeomorphisms.

Lemma: The action of on is ergodic.

Proof: If the chain stabilizes, the boundary is finite and the action is transitive so that the assertion is clear. Otherwise, Kuratowski’s theorem says that is Borel-isomorphic to the unit interval with Lebesgue measure so that Lebesgue’s theorem applies: any measurable is almost everywhere dense. In particular, if , then for all there is some with


If is moreover -invariant, then the same must hold for all . Adding up all these inequalities gives

which implies .

Lemma : Suppose each is normal in and that the total intersection is trivial. Then the action of on is free.

Proof: Given any nontrivial there is such that . Since is a normal subgroup of , the element permutes the set without fixed points. Thus moves all rays in .

In view of this lemma the following concept is a natural generalization of the common assumptions on a chain given in the Lemma.

Definition: A chain of subgroups of is called Farber if the action of on is essentially free.

Here, as usual, « essentially free » means the set of points in with nontrivial stabilizer has measure zero.

Cost and groupoid cost

Cost

Let be a standard Borel space and let act on preserving a probability measure, with finitely many ergodic components. « Lying in the same orbit » defines an equivalence relation on . So is a measurable subset of which we can picture as (the edge set of) a directed graph. Reflexivity, symmetry and transitivity say that the connected components of are complete as directed gaphs: each vertex carries a loop and any two distinct vertices in the same component are joined by precisely two edges, one in each direction. Since is countable, so are the complete directed graphs given by each component.

Apparently, we can reconstruct if we only know which pairs of points in are connected by some finite path along edges in , regardless of their direction. This information, in turn, is captured by any measurable subset with the property that two points are joined by an edge in if and only if there is a path between and along edges in .

Definition:

  1. A measurable subset is called a subgraph of .
  2. The -th power of a subgraph is defined by requiring if and only if there is an undirected path from to in of length with .
  3. We say that a subgraph spans if . In this case we write .

Note that is the set of loop edges in , regardless of what is: each point is joined to itself by the empty path of length zero. The rules of the game are to find spanning subgraphs of which are as « small » as possible. It makes sense to quantify « small » by the average number of edges starting in a vertex of . The next definition makes this idea precise.

Definition: The edge measure of a subgraph is given by

Note that can be infinite. The largest lower bound of the edge measures of spanning subgraphs of can be thought of measuring how much effort we must invest to generate the graph and thus the equivalence relation which the action of on defines.

Definition: The cost of the action of on is given by

As an example, say is finitely generated by . Then


clearly defines a spanning subgraph of . So we always have

where is the minimal number of generators of , also known as the rank of . Note that if the action of on is essentially free, almost every connected component of this subgraph looks like the Cayley graph of with respect to the generators but with no distinguished base point whatsoever.

The idea comes to mind that one can improve upon a spanning subgraph as above by picking a maximal tree in each component. The problem is that depending on there might be no measurable such choice which already hints at cost being quite a delicate invariant of the action. Let us mention that the cost of an action immediately yields an invariant for the group .

Definition: We define as the infimum of the numbers running over all essentially free, ergodic, probability measure preserving actions of on a standard Borel space .

Groupoid cost

In addition to examining (sub-)graphs, meaning measurable subsets of , it is useful to also study graphings by which we mean any measurable subset . We find it convenient to picture an element as an « arrow » in pointing from to . Note that (almost all) these arrows are determined by their initial and final point if and only if the action of on is (essentially) free. We can sort the arrows in the subset either by initial point or by direction: either by the - or by the -coordinate. So interchangeably we think of as a family of subsets


parametrized by group elements , or as a family of subsets

parametrized by points . Guided by what we did above, we define the -th power of by all the arrows we obtain by composing up to arrows from regardless of their direction. In more mathematical terms this means if and only if there is and a decomposition in such that for all either

Note that , regardless of what is.

Definition: We say that a graphing spans if we have . In this case we write .

Let be the measure on given by the product of and the counting measure on .

Definition: The groupoid cost of the action of on is given by

To explain the terminology, we observe that the set has a groupoid structure: two arrows can be composed if and only if , meaning the first arrow points to the initial point of the second, and in that case their composition is . So a graphing spans if and only if it generates as groupoid.

Proposition: We have


with equality if the action is essentially free.

Proof: A graphing defines a subgraph of the orbit relation by setting


Clearly so that preserves the spanning property. We have with equality almost everywhere if the action of on is essentially free. Integrating over gives the inequality. To obtain equality for essentially free actions one still has to show that each spanning subgraph can be obtained from a spanning graphing via ; we skip this argument which needs some technical care but no unusual ideas.

Rank gradient and cost of the coset tree boundary

Let us now assume that is finitely generated so that there is an epimorphism from the free group on a finite number of letters. If is a subgroup of finite index, then by the Schreier theorem is likewise free. It has index in so that an Euler characteristic argument reveals that the rank of is given by . For the ranks of the groups in our chain this means that the numbers


form a monotone non-increasing sequence of positive numbers.

Definition: The rank gradient of with respect to is given by

This notion is due to Lackenby and here is its dynamic interpretation.

Theorem (Abért–Nikolov, 2012): For any chain we have

Before we come to the proof let us combine this theorem with the proposition.

Corollary: For any Farber chain we have

Proof of Theorem: We first show . For all we find some with . Thus the integer


gives an upper bound for . Say is generated by and let be a system of representatives for . We define a graphing by setting

and otherwise. We claim that . Indeed, let and let and be the representatives from the list for which and . Hence we can write as a word of the form . With respect to this factorization of one easily verifies the criterion to conclude proving the claim. By definition equals for precisely elements in and is empty otherwise. Hence the graphing has measure

It follows that

The reverse inequality is somewhat harder. Given , there exists a graphing which spans and has measure . The first thing to do now is to construct yet another graphing which is close to in the sense that and has the convenient property that each is a finite union of shadows which is nonempty only for finitely many . Since the shadows form a countable basis of the topology of , it is conceivable that such a « finite approximation » to exists. So we shall allow ourselves to skip the precise technical construction. Since is made up from only finitely many shadows altogether, there exists a large enough such that each is in fact a finite union of level- shadows of the form .

We define a finite, directed, labeled graph as follows. The vertex set is and for each we connect with by an edge of label if and only if . The graph has the canonical base point . This data clearly defines a homomorphism of groups


which multiplies the labels along a loop of edges, inverting the label whenever we travel through an edge in reverse direction. We claim that the image of the homomorphism is precisely . Indeed, for each we have by the construction of the graph . Thus . Let be any element and pick some ray . Since spans , there is a factorization with such that for all either

For let be the level- vertex in the coset tree through which the ray passes. Then we have and for each either is connected to by an edge in with label or is connected to by an edge of label . Hence these edges form a loop with proving the claim.

Thus is a quotient group of . The latter group is free of rank where is the Euler characterisitc of the graph . Note that

so that is the number of edges in while the number of vertices in is of course . Putting pieces together we obtain

By subadditivity of the measure applied to we conclude

References

  • Miklós Abért, Nikolay Nikolov, Rank gradient, cost of groups and the rank versus Heegaard genus, J. Eur. Math. Soc.