Continuous and Borel Dynamics of Countable Borel Equivalence Relations

Continuous and Borel Dynamics of Countable Borel Equivalence Relations S. Jackson (joint with S. Gao, E. Krohne, and B. Seward) Department of Mathematics University of North Texas June 9, 2015 BLAST 2015, UNT

In recent years there has been progress in the theory of countable Borel equivalence relations. In particular, there has been an infusion of new techniques from several different areas. New techniques involve ideas from logic, set theory, recursion theory, and dynamics. This has resulted in the solution to several problems that had been around for a while, and points to possible future progress.

Some examples: Adams and Kechris build on the Zimmer superrigidity theory to construct many new examples of complicated structure within the context of countable Borel equivalence relations. These arguments were extended and simplified by Hjorth, Kechris, and Miller. A. Marks uses methods relating to Borel determinacy to prove new results about actions of free products of groups. These methods answered several open combinatorial questions about these actions. Kechris and Marks have developed a theory of these group actions.

S. Thomas has used Martin s conjecture on the degrees to obtain new fundamental results about some basic equivalence relations such as T. Marks, Slaman, Steel have used methods from recursion theory to about arithmetic equivalence and restricted form of universality. The current authors have introduced new dynamical methods and methods from logic (forcing) to answer questions about actions of abelian groups (and some more generally).

Basic Concepts If X is a Polish space, E an equivalence relation on X: We say E is Borel if E X X is Borel. We say E is countable if each E-class is countable. [x] E = {y : xey} If Γ X is an action of Γ on X, the orbit equivalence relation is defined by E Γ X = {(x, y) X X : g Γ (g x = y)}

By Feldman-Moore every countable Borel equivalence relation E on X is of the form E = E Γ for some countable group Γ. X Thus, we can view the theory as the study of continuous or Borel actions of countable groups Γ. It makes sense to study them group-by-group. Even for the simplest groups, e.g. Γ = Z n, there are many interesting problems, and only recently have some of them been approachable. On the other hand, some of the recent results of Marks and Kechris-Marks have interesting parallels to these results in the case Γ = F n.

The basic notion in the theory is that of a reduction of one relation (X, E) to another (Y, F). This can be viewed as saying the effective cardinality of the quotient space X/E is less than or equal to that of Y/F. Definition If E is on X and F is on Y, we say ϕ: X Y is a reduction of E to F if: x E y ϕ(x) F ϕ(y). We write E F. If E F and F E we say E, F are bi-reducible and write E F. We write E F if the reduction from E to F is one-to-one. Note that if E F then X/E and Y/F are in bijection by a Borel function.

Some terminology concerning group actions Γ X denotes a (continuous or Borel) action of Γ on X. If Γ X, Γ Y, we say ϕ: X Y is a Γ-equivariant map if ϕ(g x) = g ϕ(x). If ϕ is also one-to-one we say it is an equivariant embedding or a Γ-embedding. The action Γ X is free if for all g 1 Γ, g x x for all x X.

The (left) shift-action of Γ on 2 Γ is the action: (g x)(h) = x(g 1 h) The shift action of Γ on 2 Γ is important because it is essentially universal. Theorem (Becker-Kechris) The action Γ 2 Γ ω given by (g x)(h, n) = x(g 1 h, n) is a universal Borel Γ-action. That is, every Γ action Borel equivariantly embeds into the shift action. If X is 0-dimensional and the action is continuous, then the embedding is continuous.

hyperfiniteness The simplest case is Γ = Z. Definition E is hyperfinite if E = n E n, an increasing union, with each E n finite. Theorem (Slaman-Steel) E is hyperfinite iff E is induced by a Borel action of Z. For x, y 2 ω, let xe 0 y n m n (x(m) = y(m)) Then E is hyperfinite iff E E 0. All non-smooth hyperfinite E are bi-reducible with E 0.

New Methods Three new methods have been developed recently which have been used to answer questions about continuous and Borel action of Γ = Z n, Γ abelian, and in some cases general Γ. New dynamical methods for introducing marker structures in actions by Γ, in particular the orthogonal marker construction. The development of certain combinatorial constructions on groups, in particular the notion of a 2-coloring of a group Γ. The use of certain forcing constructions to prove results about general Borel actions of certain groups. The three techniques are inter-related. Working together, they can sometimes obtain the sharpest positive and negative results for a particular problem.

Orthogonal Markers Introduction A marker set M X refers to a Borel complete, co-complete section of X, that is, [x] M and [x] M c for all x X. Usually we are interested in marker sets M or sequences of marker sets M n with a regular structure or prescribed behavior. Theorem (Kechris-Miller) For any countable aperiodic Borel equivalence relation E on X, there is a sequence M n of Borel markers such that n M n =. We call these Slaman-Steel Markers.

Marker sequences are frequently used to do positive continuous or Borel constructions. One application is to the hyperfiniteness problem. Question (Kechris) Is the Borel action of every amenable group hyperfinite? Some Results: Slaman-Steel proof that 2 Z is hyperfinite uses markers M n with n M n =. Weiss proof that 2 Zn is hyperfinite uses markers M n with bounded geometry. Gao-J proof that all abelian actions are hyperfinite uses orthogonal markers. Seward-Schneider showed that all nilpotent actions are hyperfinite using a generalization of orthogonal markers.

Some related applications: (Boykin-J) There is a continuous embedding of 2 Zn into E 0. This extends (Gao-J) to the free part of 2 Z<ω. (Gao-J) There is a continuous 4-coloring of F(2 Zn ). The last result is an instance of the chromatic number problem. Problem Let Γ = x 1,..., x m R 1, R 2,... be a finitely generated group. Determine the continuous and Borel chromatic numbers of the actions Γ X. In particular, determine the continuous and Borel chromatic numbers χ c (F(2 Γ )), χ b (F(2 Γ )) of F(2 Γ ).

Marker Regions By marker regions R we mean a subequivalence relation of E with finite classes. We say the regions are clopen if the relation is clopen. A(x, g) g x R x Question What kinds of marker regions can we get for 2 Γ?

We can define the shape of a marker region (an equivalence class of a subset of Γ under translation). Namely, if x is in the marker region R, then the shape S x of R relative to x is {g Γ: g x R x}. If g 0 x R x, then the shape relative to g 0 x is S x g 1 0. If G is an abelian group, it makes sense to speak of the geometrical shape of a marker region, e.g., being a rectangle, etc. Seward-Schneider use the fact that if G is nilpotent then we can speak of the approximate shape of certain marker regions.

Many fundamental problems in the theory of marker structures are illustrated in the case of G = Z n. The following is the basic marker lemma. Fact For any d > 0 there is a relatively clopen M d F(2 Zn ) satisfying: 1. For any x y M d, ρ(x, y) > d. 2. For any x F(2 Zd ), there is a y M d with ρ(x, y) d. Given d 1 < d 2 <, we can get a sequence M d1 M d2 of relatively clopen marker sets in F(2 Zn ) satisfying the above with d i (1 ɛ) and d i (1 + ɛ).

On each class [x] we have [x] i M di 1. We can modify these markers in a Borel way to get i M di =, that is, to get a Borel set of Slaman-Steel markers. This can be used to get the Slaman-Steel result that there is a Borel embedding of 2 Z into E 0. The corrresponding marker regions (Voronoi regions) were usde by Weiss to show that 2 Zn is Hyperfinite.

We can improve the marker regions to have a more regular geometric shape. Theorem (Gao-J) For any d > 1 there is a clopen rectangular marker structure R on F(2 Zn ) such that each R class R is a rectangle with side lengths in {d, d + 1}. Gao-J used this to get an upper-bound for the continuous chromatic number (for n = 1 the exact number is easily 3): χ c (F(2 Zn )) 4

Orthogonal Markers Introduction There are limitations to what the basic rectangular marker regions can do. Fact A clopen decreasing marker sequence in F(2 Z ) cannot be a Slaman-Steel marker sequence, so cannot directly witness hyperfiniteness as in the Slaman-Steel proof. Fact There does not exists a Borel sequence of rectangular marker regions in F(2 Zn ) with bounded geometry and such that lim n ρ(x, R n ) =.

Fix d 1 d 2 d 3. Theorem (Orthogonal Markers) There is a clopen sequence of marker regions M i F(2 Zn ) satisfying the following. 1. Each R M i is a rectangle on scale d i. For j < i, each R M i on scale d j is a rectangular polyhedron with edge lengths d j. 2. For i 1, i 2 j and R 1 M i1, R 2 M i2, if e 1 is an edge of R 1 at scale d j, and e 2 an edge of R 2 at scale d j, and e 1 is parallel to e 2, then ρ(e 1, e 2 ) > c n d j for some positive constant c n. Remark So at scale d j, if an edge e 1 from R i1 and e 2 from R i2 intersect, they must be orthogonal.

Figure: Adjusting from R j+1 i to R j i

Figure: Adjusting from R j+1 i to R j i

It follows that if R i are orthogonal marker regions, then lim n ρ(x, R i ) = for all x F(2 Zn ). This gives a continuous embedding from 2 Zn into E 0. This is the basis for the proof of the hyperfiniteness of abelian group actions. B. Seward found ways to extend the orthogonal marker arguments. Theorem The is a Borel chromatic 3-coloring of F(2 Zn ). Theorem There is a Borel perfect matching of F(2 Zn ), n 2.

Let Γ X. Definition An element x X is hyperaperiodic if [x] F(X). That is, x is hyperaperiodic if x is contained in a subflow (closed, invariant set) contained in the free part F(X). When X = 2 Γ, then [x] is compact when x is hyperaperiodic.

Definition x 2 Γ is a 2-coloring if for all nonidentity s Γ there is a T Γ <ω such that: g Γ t T (x(gt) x(gst)) Fact (GJ) x 2 Γ is hyperaperiodic iff x is a 2-coloring. Theorem (GJS) For every countable group Γ there is a 2-coloring x 2 Γ.

Corollary There does not exist a clopen set of Slaman-Steel markers for F(2 Zn ). By constructing the 2-coloring more carefully, we can rule out various continuous structures. E. Krohne was able to use these techniques to get the following. Theorem For n 2, there is no continuous 3-coloring of F(2 Zn ). Corollary χ b (2 Z ) = χ c (2 Z ) = 3 3 = χ b (2 Zn ) < χ c (2 Zn ) = 4 for n 2

Another application: Theorem There does not exists a continuous perfect matching of F(2 Zn ). Remark E. Krohne has developed an exact condition for there to be a continuous graph homomorphism from F(2 Zn ) to a graph G. The condition is that for all sufficiently large co-prime p, q that there are homomorphisms from 12 tiles T p,q 1,..., T p,q to G. 12

sketch of proof Construct a 2-coloring x F(2 Z2 ) with the following structure. p 2 p 1 Figure: A special 2-coloring, the p i are odd

Let K = [x] F(2 Z2 ), so K is compact. Let n be such that for y K, y [ n, n] 2 determines the point y is matched with. Let p i n. Then on [x] we have a p i p i square which is perfectly matched. This is a contradiction as p 2 is odd. i

Continuous homomorphisms to graphs Using a special case of the 12 tiles theorem (for the negative direction) and the orthgonal marker construction (for the positive direction) we get positive and negative conditions. Let Γ be a graph. Let H = H v (Γ) be the homotopy group of Γ with base point v V(Γ). Let be the normal subgroup of H generated by the 4-cycles in Γ. Let G = G(Γ) = H/.

Negative Condition Introduction Let γ be a countable group, G = G(Γ) = H/. Negative Condition: co-prime p, q p cycles γ in Γ [γ q is not a p power in G(Γ)] (p, q not necessarily primes here). Weighting Condition: homomorphism ϕ: G(Γ) A torsion-free abelian, p p-cycles γ in Γ [ϕ(γ) is not a p power in A]

Positive Condition Introduction Positive Condition: odd γ G(Γ) of finite order in G(Γ)

Figure: Satisfies weighting condition using ±1.

i j a k b c h g f d e Figure: Grötzsch graph, satisfies the positive condition.

Proof of positive condition: Let γ = (a, e, i, c, g, a), a 5-cycle in the Grötzsch graph Γ. We have γ 2 = e in G(Γ) as follows. (abcdea) (agcdea) (agciea) = γ 1. (abcdea) (ajihgfea) (aeihgfea) (aeicgfea) (aeicgaea) (aeicga) = γ. So, γ = γ 1 in G(Γ).

Orbit Forcing First we extend the methods to the Borel context. The compactness arguments using are replaced by various forcing and minimality arguments. Definition (Orbit forcing) Let Γ X, and x X. Conditions in the orbit-forcing P x are non-empty open sets U X with U [x]. These are ordered by inclusion: U V iff U V.

As an abstract forcing, P x is isomorphic to Cohen forcing, but the nature of x plays a large role in the properties of the forcing vis-à-vis the equivalence relation. Theorem Let Γ be a countable group, X a compact Polish space, and Γ X a continuous action giving rise to the orbit equivalence relation E. Let A n be finite subsets of Γ such that every finite subset of Γ is contained in some A n. Let {S n } be a sequence of Borel complete sections of E. Then there is an x X such that for infinitely many n we have A n x S n. Remark In practice, we usually apply this to K = [x] where x is a 2-coloring in 2 Γ (perhaps with extra properties).

Corollary Let f : ω ω be such that lim sup n f(n) = +. Let {S n } be a sequence of Borel complete sections of F(2 Zd ). Then there is an x F(2 Zd ) such that for infinitely many n we have ρ(x, S n ) < f(n). Corollary There does not exist a Borel layered toast structure on F(2 Z2 ).

Figure: (a) layered toast (b) general toast