Interplay with other subjects ◮ Any countable Borel group action Γ � X can be turned into a continuous one by replacing the Polish topology on X . — Enables topological tools such as Baire category. ◮ Borel group actions and graphs naturally occur on measure spaces ( X , µ ). — Enables measure-theoretic tools such as the Borel–Cantelli lemma and much much more. ◮ All these together has created extremely active two-way traffic between the study of CBERs and ergodic theory (ergodic theorems, mixing, entropy, . . . ) measured group theory (measure equivalence, cost, ℓ 2 -(co)homology, . . . ) graph combinatorics (colorings, matchings, flows, . . . ) geometric group theory (tree constructions, growth,. . . ) percolation theory (graph/action constructions, amenability barriers, . . . ) probabilistic combinatorics (Lov´ asz Local Lemma, concentration bounds, . . . ) topological dynamics (subshifts, topological entropy, compact actions, . . . ) von Neumann algebras (spectral gap, superrigidity, vN-dimension, . . . )
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking?
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking? ◮ We think pointwise, analyzing the local combinatorics at a point, whereas analysts analyze the space through the prism of functions on it.
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking? ◮ We think pointwise, analyzing the local combinatorics at a point, whereas analysts analyze the space through the prism of functions on it. ◮ What allows us to do this is the Luzin–Novikov uniformization theorem!
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking? ◮ We think pointwise, analyzing the local combinatorics at a point, whereas analysts analyze the space through the prism of functions on it. ◮ What allows us to do this is the Luzin–Novikov uniformization theorem! — Every Borel set B ⊆ X × Y with countable X -fibers is a countable union of Borel partial functions X ⇀ Y .
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking? ◮ We think pointwise, analyzing the local combinatorics at a point, whereas analysts analyze the space through the prism of functions on it. ◮ What allows us to do this is the Luzin–Novikov uniformization theorem! — Every Borel set B ⊆ X × Y with countable X -fibers is a countable union of Borel partial functions X ⇀ Y . — Allows each x ∈ X to quantify over its (countable) equivalence class
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking? ◮ We think pointwise, analyzing the local combinatorics at a point, whereas analysts analyze the space through the prism of functions on it. ◮ What allows us to do this is the Luzin–Novikov uniformization theorem! — Every Borel set B ⊆ X × Y with countable X -fibers is a countable union of Borel partial functions X ⇀ Y . — Allows each x ∈ X to quantify over its (countable) equivalence class and give (Borel) names to the other guys in its class.
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking? ◮ We think pointwise, analyzing the local combinatorics at a point, whereas analysts analyze the space through the prism of functions on it. ◮ What allows us to do this is the Luzin–Novikov uniformization theorem! — Every Borel set B ⊆ X × Y with countable X -fibers is a countable union of Borel partial functions X ⇀ Y . — Allows each x ∈ X to quantify over its (countable) equivalence class and give (Borel) names to the other guys in its class. ◮ We are not allowed to choose a point from each class! (Measure Theory 101)
Our secret weapon But what’s our contribution? What’s the difference between descriptive set theoretic and analytic thinking? ◮ We think pointwise, analyzing the local combinatorics at a point, whereas analysts analyze the space through the prism of functions on it. ◮ What allows us to do this is the Luzin–Novikov uniformization theorem! — Every Borel set B ⊆ X × Y with countable X -fibers is a countable union of Borel partial functions X ⇀ Y . — Allows each x ∈ X to quantify over its (countable) equivalence class and give (Borel) names to the other guys in its class. ◮ We are not allowed to choose a point from each class! (Measure Theory 101) ◮ Thus, our way of thinking is best described as originless combinatorics.
pmp actions of groups and ergodicity As an example, today we’ll discuss ergodic theorems.
pmp actions of groups and ergodicity As an example, today we’ll discuss ergodic theorems. ◮ ( X , µ ) — a standard probability space, e.g. [0 , 1] with Lebesgue measure
pmp actions of groups and ergodicity As an example, today we’ll discuss ergodic theorems. ◮ ( X , µ ) — a standard probability space, e.g. [0 , 1] with Lebesgue measure ◮ Γ — a countable (discrete) group
pmp actions of groups and ergodicity As an example, today we’ll discuss ergodic theorems. ◮ ( X , µ ) — a standard probability space, e.g. [0 , 1] with Lebesgue measure ◮ Γ — a countable (discrete) group ◮ An action Γ � ( X , µ ) is ergodic if every invariant measurable subset is null or conull.
pmp actions of groups and ergodicity As an example, today we’ll discuss ergodic theorems. ◮ ( X , µ ) — a standard probability space, e.g. [0 , 1] with Lebesgue measure ◮ Γ — a countable (discrete) group ◮ An action Γ � ( X , µ ) is ergodic if every invariant measurable subset is null or conull. ◮ A probability measure preserving (pmp) action of Γ is a Borel action Γ � ( X , µ ) such that µ ( γ · A ) = µ ( A ), for each γ ∈ Γ and A ⊆ X .
pmp actions of groups and ergodicity As an example, today we’ll discuss ergodic theorems. ◮ ( X , µ ) — a standard probability space, e.g. [0 , 1] with Lebesgue measure ◮ Γ — a countable (discrete) group ◮ An action Γ � ( X , µ ) is ergodic if every invariant measurable subset is null or conull. ◮ A probability measure preserving (pmp) action of Γ is a Borel action Γ � ( X , µ ) such that µ ( γ · A ) = µ ( A ), for each γ ∈ Γ and A ⊆ X . ◮ Dating back to Birkhoff, pointwise ergodic theorems for pmp actions Γ � ( X , µ ) are bridges between the global condition of ergodicity and the a.e. local combinatorics of the action.
Pointwise ergodic theorems for pmp actions Theorem (Pointwise ergodic, Birkhoff 1931) A pmp action Z � α ( X , µ ) is ergodic if and only if for each f ∈ L 1 ( X , µ ) and for a.e. x ∈ X, � limit of local averages n →∞ A f [ F n · α x ] = lim f d µ global average X where F n . . = [0 , n ] ⊆ Z and A f [ F n · α x ] is the average of f over F n · α x.
Pointwise ergodic theorems for pmp actions Theorem (Pointwise ergodic, Birkhoff 1931) A pmp action Z � α ( X , µ ) is ergodic if and only if for each f ∈ L 1 ( X , µ ) and for a.e. x ∈ X, � limit of local averages n →∞ A f [ F n · α x ] = lim f d µ global average X where F n . . = [0 , n ] ⊆ Z and A f [ F n · α x ] is the average of f over F n · α x. ◮ This was generalized to amenable groups by Lindenstrauss (2001).
Pointwise ergodic theorems for pmp actions Theorem (Pointwise ergodic, Birkhoff 1931) A pmp action Z � α ( X , µ ) is ergodic if and only if for each f ∈ L 1 ( X , µ ) and for a.e. x ∈ X, � limit of local averages n →∞ A f [ F n · α x ] = lim f d µ global average X where F n . . = [0 , n ] ⊆ Z and A f [ F n · α x ] is the average of f over F n · α x. ◮ This was generalized to amenable groups by Lindenstrauss (2001). ◮ Analogous statements for free groups have been proven by Grigorchuk (1987), Nevo (1994), Nevo–Stein (1994), Bufetov (2002), and Bowen–Nevo (2013).
Pointwise ergodic theorems for pmp actions Theorem (Pointwise ergodic, Birkhoff 1931) A pmp action Z � α ( X , µ ) is ergodic if and only if for each f ∈ L 1 ( X , µ ) and for a.e. x ∈ X, � limit of local averages n →∞ A f [ F n · α x ] = lim f d µ global average X where F n . . = [0 , n ] ⊆ Z and A f [ F n · α x ] is the average of f over F n · α x. ◮ This was generalized to amenable groups by Lindenstrauss (2001). ◮ Analogous statements for free groups have been proven by Grigorchuk (1987), Nevo (1994), Nevo–Stein (1994), Bufetov (2002), and Bowen–Nevo (2013). ◮ Bowen–Nevo (2013) also have pointwise ergodic theorems for other nonamenable groups, but for special kinds of actions.
Pointwise ergodic theorems for quasi-pmp actions ◮ An action Γ � ( X , µ ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e. maps nonnull sets to nonnull sets (possibly changing their measure).
Pointwise ergodic theorems for quasi-pmp actions ◮ An action Γ � ( X , µ ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e. maps nonnull sets to nonnull sets (possibly changing their measure). ◮ Why would one consider quasi-pmp actions?
Pointwise ergodic theorems for quasi-pmp actions ◮ An action Γ � ( X , µ ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e. maps nonnull sets to nonnull sets (possibly changing their measure). ◮ Why would one consider quasi-pmp actions? Theorem (Ratio Pointwise Ergodic, Hopf 1937) Let Z � α ( X , µ ) be an ergodic pmp action. For each f , g ∈ L 1 ( X , µ ) with g > 0 and for a.e. x ∈ X, � A f [ F n · α x ] X f d µ lim A g [ F n · α x ] = X gd µ. � n →∞
Pointwise ergodic theorems for quasi-pmp actions ◮ An action Γ � ( X , µ ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e. maps nonnull sets to nonnull sets (possibly changing their measure). ◮ Why would one consider quasi-pmp actions? Theorem (Ratio Pointwise Ergodic, Hopf 1937) Let Z � α ( X , µ ) be an ergodic pmp action. For each f , g ∈ L 1 ( X , µ ) with g > 0 and for a.e. x ∈ X, � A f [ F n · α x ] X f d µ lim A g [ F n · α x ] = X gd µ. � n →∞ ◮ Replacing the measure µ with d µ g . . = gd µ makes the action Γ � ( X , µ g ) quasi-pmp and turns the ratio ergodic theorem into a usual one.
Pointwise ergodic theorems for quasi-pmp actions ◮ An action Γ � ( X , µ ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e. maps nonnull sets to nonnull sets (possibly changing their measure). ◮ Why would one consider quasi-pmp actions? Theorem (Ratio Pointwise Ergodic, Hopf 1937) Let Z � α ( X , µ ) be an ergodic pmp action. For each f , g ∈ L 1 ( X , µ ) with g > 0 and for a.e. x ∈ X, � A f [ F n · α x ] X f d µ lim A g [ F n · α x ] = X gd µ. � n →∞ ◮ Replacing the measure µ with d µ g . . = gd µ makes the action Γ � ( X , µ g ) quasi-pmp and turns the ratio ergodic theorem into a usual one. ◮ A quasi-pmp ergodic theorem for Z d was proven by Feldman (2007) and generalized by Hochman (2010) and Dooley–Jarrett (2016).
Pointwise ergodic theorems for quasi-pmp actions ◮ An action Γ � ( X , µ ) is quasi-pmp if every γ ∈ Γ is nonsingular, i.e. maps nonnull sets to nonnull sets (possibly changing their measure). ◮ Why would one consider quasi-pmp actions? Theorem (Ratio Pointwise Ergodic, Hopf 1937) Let Z � α ( X , µ ) be an ergodic pmp action. For each f , g ∈ L 1 ( X , µ ) with g > 0 and for a.e. x ∈ X, � A f [ F n · α x ] X f d µ lim A g [ F n · α x ] = X gd µ. � n →∞ ◮ Replacing the measure µ with d µ g . . = gd µ makes the action Γ � ( X , µ g ) quasi-pmp and turns the ratio ergodic theorem into a usual one. ◮ A quasi-pmp ergodic theorem for Z d was proven by Feldman (2007) and generalized by Hochman (2010) and Dooley–Jarrett (2016). ◮ Also known for lattice actions on homogeneous spaces (Nevo and co.), a weaker form for groups of polynomial growth (Hochman 2013), an indirect version for free groups (Bowen–Nevo 2014).
Failures of pointwise ergodic for some quasi-pmp actions ◮ However, Hochman also showed (2012) that the ratio ergodic theorem is false for Γ . . = � n ∈ N Z along any sequence ( F n ) of window frames F n ⊆ Γ.
Failures of pointwise ergodic for some quasi-pmp actions ◮ However, Hochman also showed (2012) that the ratio ergodic theorem is false for Γ . . = � n ∈ N Z along any sequence ( F n ) of window frames F n ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any subsequence of balls.
Failures of pointwise ergodic for some quasi-pmp actions ◮ However, Hochman also showed (2012) that the ratio ergodic theorem is false for Γ . . = � n ∈ N Z along any sequence ( F n ) of window frames F n ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any subsequence of balls. ◮ Thus, the pointwise ergodic theorem is false for the quasi-pmp actions of � n ∈ N Z and the nonablian free groups.
Failures of pointwise ergodic for some quasi-pmp actions ◮ However, Hochman also showed (2012) that the ratio ergodic theorem is false for Γ . . = � n ∈ N Z along any sequence ( F n ) of window frames F n ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any subsequence of balls. ◮ Thus, the pointwise ergodic theorem is false for the quasi-pmp actions of � n ∈ N Z and the nonablian free groups. ◮ Still, is there any pointwise ergodic statement for these actions?
Failures of pointwise ergodic for some quasi-pmp actions ◮ However, Hochman also showed (2012) that the ratio ergodic theorem is false for Γ . . = � n ∈ N Z along any sequence ( F n ) of window frames F n ⊆ Γ. ◮ Moreover, it is also false for the nonabelian free groups along any subsequence of balls. ◮ Thus, the pointwise ergodic theorem is false for the quasi-pmp actions of � n ∈ N Z and the nonablian free groups. ◮ Still, is there any pointwise ergodic statement for these actions? ◮ Let’s abandon the action and look at the induced Cayley–Schreier graph.
The Cayley–Schreier graph and uniform test windows ◮ A Borel action of a countable group Γ = � S � on ( X , µ ) induces a locally countable Borel graph G S on X — its Cayley–Schreier graph: xG S y . . ⇔ σ · x = y for some σ ∈ S .
The Cayley–Schreier graph and uniform test windows ◮ A Borel action of a countable group Γ = � S � on ( X , µ ) induces a locally countable Borel graph G S on X — its Cayley–Schreier graph: xG S y . . ⇔ σ · x = y for some σ ∈ S . ◮ Pointwise ergodic theorems extract a sequence ( F n ) of subsets of Γ (window frames)
The Cayley–Schreier graph and uniform test windows ◮ A Borel action of a countable group Γ = � S � on ( X , µ ) induces a locally countable Borel graph G S on X — its Cayley–Schreier graph: xG S y . . ⇔ σ · x = y for some σ ∈ S . ◮ Pointwise ergodic theorems extract a sequence ( F n ) of subsets of Γ (window frames) and for a.e. x ∈ X , assert that the local averages A f [ F n · x ] of an f ∈ L 1 ( X , µ ) over the test windows F n · x converge to � the global average X f d µ .
The Cayley–Schreier graph and uniform test windows ◮ A Borel action of a countable group Γ = � S � on ( X , µ ) induces a locally countable Borel graph G S on X — its Cayley–Schreier graph: xG S y . . ⇔ σ · x = y for some σ ∈ S . ◮ Pointwise ergodic theorems extract a sequence ( F n ) of subsets of Γ (window frames) and for a.e. x ∈ X , assert that the local averages A f [ F n · x ] of an f ∈ L 1 ( X , µ ) over the test windows F n · x converge to � the global average X f d µ . ◮ Let’s consider graphs in general.
More generally: pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ).
More generally: pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ E G — the connectedness equivalence relation of G .
More generally: pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ E G — the connectedness equivalence relation of G . ◮ G is ergodic if each E G -invariant measurable subset is null or conull.
More generally: pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ E G — the connectedness equivalence relation of G . ◮ G is ergodic if each E G -invariant measurable subset is null or conull. ◮ G is pmp if every Borel automorphism γ of X that permutes every G -connected component is measure preserving.
More generally: pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ E G — the connectedness equivalence relation of G . ◮ G is ergodic if each E G -invariant measurable subset is null or conull. ◮ G is pmp if every Borel automorphism γ of X that permutes every G -connected component is measure preserving. ◮ In other words, the points in the same G -connected component have equal mass.
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ).
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every G -connected component is nonsingular (i.e. null preserving).
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every G -connected component is nonsingular (i.e. null preserving). ◮ Think: the points in the same G -connected component have possibly different nonzero masses.
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every G -connected component is nonsingular (i.e. null preserving). ◮ Think: the points in the same G -connected component have possibly different nonzero masses. ◮ This difference is given by a Borel cocycle ρ : E G → R + , which satisfies ρ ( x , y ) ρ ( y , z ) = ρ ( x , z ) .
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every G -connected component is nonsingular (i.e. null preserving). ◮ Think: the points in the same G -connected component have possibly different nonzero masses. ◮ This difference is given by a Borel cocycle ρ : E G → R + , which satisfies ρ ( x , y ) ρ ( y , z ) = ρ ( x , z ) . ◮ Thinking of ρ ( x , y ) as mass ( x ) / mass ( y ), write ρ y ( x ) instead.
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every G -connected component is nonsingular (i.e. null preserving). ◮ Think: the points in the same G -connected component have possibly different nonzero masses. ◮ This difference is given by a Borel cocycle ρ : E G → R + , which satisfies ρ ( x , y ) ρ ( y , z ) = ρ ( x , z ) . ◮ Thinking of ρ ( x , y ) as mass ( x ) / mass ( y ), write ρ y ( x ) instead. ◮ It makes µ ρ -invariant,
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every G -connected component is nonsingular (i.e. null preserving). ◮ Think: the points in the same G -connected component have possibly different nonzero masses. ◮ This difference is given by a Borel cocycle ρ : E G → R + , which satisfies ρ ( x , y ) ρ ( y , z ) = ρ ( x , z ) . ◮ Thinking of ρ ( x , y ) as mass ( x ) / mass ( y ), write ρ y ( x ) instead. ◮ It makes µ ρ -invariant, i.e. for any γ as above, � ρ x ( γ x ) d µ ( x ) . µ ( γ B ) = B
Even more generally: quasi-pmp graphs ◮ G — a locally countable Borel graph on ( X , µ ). ◮ G is quasi-pmp if every Borel automorphism γ of X that permutes every G -connected component is nonsingular (i.e. null preserving). ◮ Think: the points in the same G -connected component have possibly different nonzero masses. ◮ This difference is given by a Borel cocycle ρ : E G → R + , which satisfies ρ ( x , y ) ρ ( y , z ) = ρ ( x , z ) . ◮ Thinking of ρ ( x , y ) as mass ( x ) / mass ( y ), write ρ y ( x ) instead. ◮ It makes µ ρ -invariant, i.e. for any γ as above, � ρ x ( γ x ) d µ ( x ) . µ ( γ B ) = B ◮ The ρ -weighted average of a function f over a finite G -connected U ⊆ X : u ∈ U f ( u ) ρ x ( u ) � f [ U ] . A ρ . = , u ∈ U ρ x ( u ) � where x is any/some point in the G -connected component of U .
A pointwise ergodic theorem for quasi-pmp graphs ◮ Let G be a locally countable quasi-pmp ergodic graph on ( X , µ ) and let ρ : E G → R + be the corresponding cocycle.
A pointwise ergodic theorem for quasi-pmp graphs ◮ Let G be a locally countable quasi-pmp ergodic graph on ( X , µ ) and let ρ : E G → R + be the corresponding cocycle. ◮ Even if G came from a group action (it always does), Hochman’s results show that uniform/deterministic window frames taken from the group may not yield correct averages.
A pointwise ergodic theorem for quasi-pmp graphs ◮ Let G be a locally countable quasi-pmp ergodic graph on ( X , µ ) and let ρ : E G → R + be the corresponding cocycle. ◮ Even if G came from a group action (it always does), Hochman’s results show that uniform/deterministic window frames taken from the group may not yield correct averages. ◮ Nevertheless, we can obtain a Borel pairwise disjoint collection of such windows in X (i.e., a finite Borel equivalence relation on X ) ensuring that each window is G -connected — the main challenge.
A pointwise ergodic theorem for quasi-pmp graphs ◮ Let G be a locally countable quasi-pmp ergodic graph on ( X , µ ) and let ρ : E G → R + be the corresponding cocycle. ◮ Even if G came from a group action (it always does), Hochman’s results show that uniform/deterministic window frames taken from the group may not yield correct averages. ◮ Nevertheless, we can obtain a Borel pairwise disjoint collection of such windows in X (i.e., a finite Borel equivalence relation on X ) ensuring that each window is G -connected — the main challenge. In other words, Theorem ( Ծ . 2018) There is an increasing sequence ( F n ) of G-connected finite Borel equivalence relations on X such that for every f ∈ L 1 ( X , µ ) , � n →∞ A ρ lim f [ x ] F n = f d µ, for a.e. x ∈ X . X Here, A ρ f [ x ] F n is the ρ -weighted average of f over the F n -class [ x ] F n of x .
Deterministic vs. random test windows
Credits and applications The main theorem in the pmp case was first proven by Tucker-Drob: Theorem (Tucker-Drob 2016) The pointwise ergodic theorem is true for locally countable ergodic pmp graphs.
Credits and applications The main theorem in the pmp case was first proven by Tucker-Drob: Theorem (Tucker-Drob 2016) The pointwise ergodic theorem is true for locally countable ergodic pmp graphs. Applications Answer to Bowen’s question: Every pmp ergodic countable treeable Borel equivalence relation admits an ergodic hyperfinite free factor. Ergodic 1% lemma (Tucker-Drob and Conley–Gaboriau–Marks–Tucker-Drob): Every locally countable ergodic nonhyperfinite pmp Borel graph G admits Borel sets A ⊆ X of arbitrarily small measure such that G | A is still ergodic and nonhyperfinite. Could have been used in Ergodic Hjorth’s lemma for cost attained (Miller– Ծ . 2017): If a countable pmp ergodic Borel equivalence relation E is treeable and has cost n ∈ N ∪ {∞} , then it is induced by an a.e. free action of F n such that each of the n standard generators of F n alone acts ergodically.
Credits and applications The main theorem in the pmp case was first proven by Tucker-Drob: Theorem (Tucker-Drob 2016) The pointwise ergodic theorem is true for locally countable ergodic pmp graphs. Applications Answer to Bowen’s question: Every ✘✘ pmp ergodic countable treeable Borel ❳❳ ✘ ❳ equivalence relation admits an ergodic hyperfinite free factor. Ergodic 1% lemma (Tucker-Drob and Conley–Gaboriau–Marks–Tucker-Drob): Every locally countable ergodic nonhyperfinite pmp Borel graph G admits Borel sets A ⊆ X of arbitrarily small measure such that G | A is still ergodic and nonhyperfinite. Could have been used in Ergodic Hjorth’s lemma for cost attained (Miller– Ծ . 2017): If a countable pmp ergodic Borel equivalence relation E is treeable and has cost n ∈ N ∪ {∞} , then it is induced by an a.e. free action of F n such that each of the n standard generators of F n alone acts ergodically.
Two different proofs ◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest.
Two different proofs ◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest. ◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of other probabilistic techniques: (Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem.
Two different proofs ◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest. ◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of other probabilistic techniques: (Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem. ◮ All these techniques are inherently pmp.
Two different proofs ◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest. ◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of other probabilistic techniques: (Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem. ◮ All these techniques are inherently pmp. ◮ A year later, I found a combinatorial/purely descriptive set theoretic proof of Tucker-Drob’s theorem (for pmp graphs).
Two different proofs ◮ Tucker-Drob’s proof of his theorem is based on a rather deep result in percolation theory by Hutchcroft and Nachmias (2015) on indistinguishability of the Wired Uniform Spanning Forest. ◮ Furthermore, to derive his theorem from this, Tucker-Drob makes use of other probabilistic techniques: (Gaboriau–Lyons) Wilson’s algorithm rooted at infinity (essentially Hatami–Lov´ asz–Szegedy) an analogue for graphs of the Abert–Weiss theorem. ◮ All these techniques are inherently pmp. ◮ A year later, I found a combinatorial/purely descriptive set theoretic proof of Tucker-Drob’s theorem (for pmp graphs). ◮ Another half a year later, the argument became applicable to quasi-pmp graphs, yielding a generalization.
The main theorem (again) and what’s involved Diagonalizing through a dense sequence in L 1 ( X , µ ) reduces the main theorem to: Theorem Let G be a locally countable quasi-pmp ergodic Borel graph on ( X , µ ) and let ρ : E G → R + be the cocycle. For every f ∈ L 1 ( X , µ ) and ε > 0 , there is a G-connected finite Borel equivalence relation F on X such that � A ρ f [ x ] F ≈ ε f d µ X for all but ε -measure-many x ∈ X.
The main theorem (again) and what’s involved Diagonalizing through a dense sequence in L 1 ( X , µ ) reduces the main theorem to: Theorem Let G be a locally countable quasi-pmp ergodic Borel graph on ( X , µ ) and let ρ : E G → R + be the cocycle. For every f ∈ L 1 ( X , µ ) and ε > 0 , there is a G-connected finite Borel equivalence relation F on X such that � A ρ f [ x ] F ≈ ε f d µ X for all but ε -measure-many x ∈ X. The proof required new tools: asymptotic averages along a graph finitizing vertex-cuts (exploits nonhyperfiniteness) saturated and packed prepartitions an iteration technique via measure-compactness for a cocycle ρ on E G , notions of ρ -ratio and ( G , ρ )-visibility.
The main theorem (again) and what’s involved Diagonalizing through a dense sequence in L 1 ( X , µ ) reduces the main theorem to: Theorem Let G be a locally countable quasi-pmp ergodic Borel graph on ( X , µ ) and let ρ : E G → R + be the cocycle. For every f ∈ L 1 ( X , µ ) and ε > 0 , there is a G-connected finite Borel equivalence relation F on X such that � A ρ f [ x ] F ≈ ε f d µ X for all but ε -measure-many x ∈ X. The proof required new tools: asymptotic averages along a graph finitizing vertex-cuts (exploits nonhyperfiniteness) saturated and packed prepartitions an iteration technique via measure-compactness for a cocycle ρ on E G , notions of ρ -ratio and ( G , ρ )-visibility. In the remaining time, I’ll discuss the difficulty of getting each F -class G -connected.
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp.
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L ∞ ( X , µ ) and assume � X f d µ = 0.
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L ∞ ( X , µ ) and assume � X f d µ = 0. ◮ Building a G -connected finite Borel equivalence relation amounts to constructing a Borel collection P of pairwise disjoint finite G -connected subsets of X — call it a G -prepartition. . = � P to be all of X . — We do not require the domain dom( P ) .
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L ∞ ( X , µ ) and assume � X f d µ = 0. ◮ Building a G -connected finite Borel equivalence relation amounts to constructing a Borel collection P of pairwise disjoint finite G -connected subsets of X — call it a G -prepartition. . = � P to be all of X . — We do not require the domain dom( P ) . ◮ To prove the theorem, we need to build a Borel G -prepartition P with 1 − ε domain whose cells U ∈ P satisfy A f [ U ] ≈ ε 0 — call these U good.
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L ∞ ( X , µ ) and assume � X f d µ = 0. ◮ Building a G -connected finite Borel equivalence relation amounts to constructing a Borel collection P of pairwise disjoint finite G -connected subsets of X — call it a G -prepartition. . = � P to be all of X . — We do not require the domain dom( P ) . ◮ To prove the theorem, we need to build a Borel G -prepartition P with 1 − ε domain whose cells U ∈ P satisfy A f [ U ] ≈ ε 0 — call these U good. ◮ By Kechris–Miller, there is always a Borel maximal such prepartition P .
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L ∞ ( X , µ ) and assume � X f d µ = 0. ◮ Building a G -connected finite Borel equivalence relation amounts to constructing a Borel collection P of pairwise disjoint finite G -connected subsets of X — call it a G -prepartition. . = � P to be all of X . — We do not require the domain dom( P ) . ◮ To prove the theorem, we need to build a Borel G -prepartition P with 1 − ε domain whose cells U ∈ P satisfy A f [ U ] ≈ ε 0 — call these U good. ◮ By Kechris–Miller, there is always a Borel maximal such prepartition P . ◮ How much land does this maximal P conquer? That is: what’s the measure of dom( P )?
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L ∞ ( X , µ ) and assume � X f d µ = 0. ◮ Building a G -connected finite Borel equivalence relation amounts to constructing a Borel collection P of pairwise disjoint finite G -connected subsets of X — call it a G -prepartition. . = � P to be all of X . — We do not require the domain dom( P ) . ◮ To prove the theorem, we need to build a Borel G -prepartition P with 1 − ε domain whose cells U ∈ P satisfy A f [ U ] ≈ ε 0 — call these U good. ◮ By Kechris–Miller, there is always a Borel maximal such prepartition P . ◮ How much land does this maximal P conquer? That is: what’s the measure of dom( P )? ◮ The first step in the proof is showing that dom( P ) meets every G -connected component.
Borel G -prepartitions ◮ For the sake of the talk, assume G is pmp. ◮ It suffices to fix an f ∈ L ∞ ( X , µ ) and assume � X f d µ = 0. ◮ Building a G -connected finite Borel equivalence relation amounts to constructing a Borel collection P of pairwise disjoint finite G -connected subsets of X — call it a G -prepartition. . = � P to be all of X . — We do not require the domain dom( P ) . ◮ To prove the theorem, we need to build a Borel G -prepartition P with 1 − ε domain whose cells U ∈ P satisfy A f [ U ] ≈ ε 0 — call these U good. ◮ By Kechris–Miller, there is always a Borel maximal such prepartition P . ◮ How much land does this maximal P conquer? That is: what’s the measure of dom( P )? ◮ The first step in the proof is showing that dom( P ) meets every G -connected component. ◮ Thus, dom ( P ) has positive measure, but this measure could be arbitrarily small.
Maximal prepartition — not good enough P — a Borel maximal G -prepartition into good cells U , i.e. A f [ U ] ≈ ε 0.
Maximal prepartition — not good enough P — a Borel maximal G -prepartition into good cells U , i.e. A f [ U ] ≈ ε 0. ◮ To prove the theorem, we need the measure of dom( P ) to be 1 − ε .
Maximal prepartition — not good enough P — a Borel maximal G -prepartition into good cells U , i.e. A f [ U ] ≈ ε 0. ◮ To prove the theorem, we need the measure of dom( P ) to be 1 − ε . ◮ However, dom( P ) may leave out infinite G -connected components.
Maximal prepartition — not good enough P — a Borel maximal G -prepartition into good cells U , i.e. A f [ U ] ≈ ε 0. ◮ To prove the theorem, we need the measure of dom( P ) to be 1 − ε . ◮ However, dom( P ) may leave out infinite G -connected components. ◮ How to get rid of these infinite clusters?
Maximal prepartition — not good enough P — a Borel maximal G -prepartition into good cells U , i.e. A f [ U ] ≈ ε 0. ◮ To prove the theorem, we need the measure of dom( P ) to be 1 − ε . ◮ However, dom( P ) may leave out infinite G -connected components. ◮ How to get rid of these infinite clusters? ◮ Need a stronger notion of maximality — packed prepartitions!
Maximal prepartition — not good enough P — a Borel maximal G -prepartition into good cells U , i.e. A f [ U ] ≈ ε 0. ◮ To prove the theorem, we need the measure of dom( P ) to be 1 − ε . ◮ However, dom( P ) may leave out infinite G -connected components. ◮ How to get rid of these infinite clusters? ◮ Need a stronger notion of maximality — packed prepartitions! ◮ Going into these however is too much for a morning talk :)
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