Amenable equivalence relations, Kesten's property, and measurable lamplighters

This paper characterizes the amenability of countable Borel equivalence relations via the uniform Liouville property, investigates Kesten's property for return probabilities on topological groups, and constructs an amenable contractible Polish group lacking this property by linking it to anti-concentration inequalities in measurable lamplighter groups.

The Gist

Imagine you are a detective trying to solve a mystery about chaos versus order in mathematics. The "suspects" in this story are mathematical structures called groups (collections of objects that can be combined) and equivalence relations (ways of sorting things into piles).

The main question the detectives (the authors) are asking is: "Is this system 'amenable'?"

In plain English, amenability is a fancy word for "well-behaved" or "orderly." An amenable system is one where you can't get lost in an infinite loop of chaos; it has a kind of internal balance. A non-amenable system is like a wild, untamed beast that can spiral into infinite complexity.

Here is the breakdown of their investigation, using simple metaphors:

1. The "Liouville" Test: The Drunkard's Walk

Imagine a drunk person wandering through a city (the mathematical space).

• The Rule: If the city is "amenable" (orderly), no matter how the drunk person wanders, they will eventually settle into a pattern where their location becomes predictable. They can't hide in a corner that is fundamentally different from everywhere else.
• The Discovery: The authors proved a new rule for a specific type of city (called a countable Borel equivalence relation). They showed that if you can find one specific way for the drunk person to walk (a specific probability measure) such that they eventually become predictable on almost every street corner, then the whole city is "amenable."
• The Twist: They found a counter-example where a city looks orderly on every single street, but if you try to find one single walking rule that works for all streets at once, you can't. It's like a city where every neighborhood is calm, but the city as a whole is too chaotic to have a single "universal walking guide."

2. Kesten's Property: The "Return to Base" Test

This is the second major theme, inspired by a famous 1950s mathematician named Kesten.

• The Metaphor: Imagine a random walker starting at a base camp. Every step, they flip a coin to decide where to go.
• The Property: Kesten's property asks: "If the system is amenable, does the walker have a decent chance of returning to the base camp (or staying close to it) after a long time?"
• The Old Rule: For simple, finite, or "locally compact" groups (like a grid or a standard shape), the answer is YES. If the system is amenable, the walker stays close to home.
• The New Discovery: The authors looked at "topological groups" (shapes that are infinitely complex and stretchy). They found that for a specific class of these shapes (those with "Small Invariant Neighborhoods" or SIN), the old rule still holds: Amenable = Walker stays close.
• The Shock: But then, they built a monster. They constructed a specific, incredibly complex, "amenable" shape (a Measurable Lamplighter) where the walker does not stay close to home, even though the shape is technically "orderly."
  • Analogy: Imagine a "Lamplighter" who walks down a street turning lamps on and off. In the authors' new "Measurable Lamplighter," the street is so infinitely long and the lamps are so interconnected that even though the system is orderly, the lamplighter gets lost in the infinite dark. The "return probability" vanishes.

3. The "Inverted Orbit": The Shadow of the Past

To understand why the Lamplighter gets lost, the authors looked at something called Inverted Orbits.

• The Metaphor: Imagine the walker leaves a trail of breadcrumbs. An "Inverted Orbit" is like looking at the trail from the end of the walk backwards to the start. It asks: "How many unique places did the walker visit if we trace the path in reverse?"
• The Connection: They found a deep link between the "Lamplighter" getting lost and the "Inverted Orbit" growing too fast. If the orbit grows too fast (the walker visits too many unique spots), the system fails Kesten's property.
• The Result: They used this link to prove that their "Measurable Lamplighter" is a contractible shape (you can shrink it down to a single point without tearing it) and is amenable, yet it still fails the "Return to Base" test. This is a mathematical paradox that breaks the intuition that "orderly" always means "predictable return."

Summary of the Plot

1. The Goal: To understand when a complex mathematical system is "orderly" (amenable).
2. The First Clue: They proved that for certain systems, "orderly" is the same as "predictable walking patterns" (The Liouville Property).
3. The Second Clue: They tested an old rule (Kesten's Theorem) that says "orderly systems keep their walkers close to home."
4. The Climax: They built a new type of mathematical creature (the Measurable Lamplighter) that is orderly (amenable) but loses its walker (fails Kesten's property).
5. The Conclusion: This proves that the old rule doesn't work for all types of mathematical shapes. It opens a door to understanding how randomness behaves in infinitely complex, "stretchy" spaces.

In a nutshell: The paper shows that in the world of infinite, complex mathematics, a system can be perfectly "well-behaved" (amenable) but still be so vast and interconnected that a random traveler will inevitably get lost and never return home. They found a new way to measure this "lostness" using the shadows of past paths.

Technical Summary

Here is a detailed technical summary of the paper "Amenable Equivalence Relations, Kesten's Property, and Measurable Lamplighters" by Chaudhuri, Juschenko, and Schneider.

1. Problem Statement and Context

The paper addresses two central themes in the intersection of geometric group theory, ergodic theory, and probability:

1. The Liouville Property for Equivalence Relations: While the classical Kaimanovich–Vershik theorem characterizes the amenability of a discrete group $G$ via the Liouville property of its action on itself, it remains unclear how this extends to actions of countable groups on standard probability spaces where the induced orbit equivalence relation is amenable. Specifically, the authors investigate whether the amenability of an equivalence relation implies the existence of a single probability measure on the acting group that makes the action Liouville on almost every orbit.
2. Generalization of Kesten's Theorem to Topological Groups: Kesten's theorem states that a finitely generated discrete group is amenable if and only if the spectral radius of a symmetric random walk is 1. The authors seek to generalize this to general topological groups (specifically Polish groups) by defining "Kesten's property" (related to the decay rate of return probabilities to neighborhoods of the identity). They aim to determine if amenability of a topological group implies this property, and conversely, if the property implies amenability.

2. Methodology

The authors employ a blend of techniques from:

• Ergodic Theory and Equivalence Relations: Utilizing the theory of countable Borel equivalence relations (CBER), full groups, and the Connes–Feldman–Weiss theorem.
• Probabilistic Group Theory: Analyzing random walks, inverted orbits, and return probabilities.
• Topological Group Theory: Studying properties like Small Invariant Neighborhoods (SIN), contractibility, and the structure of Polish groups.
• Combinatorial and Analytic Tools: Using Følner sequences, spectral radius estimates (Mohar's inequality), and anti-concentration inequalities for inverted orbits.

A key methodological innovation is the construction of Measurable Lamplighter Groups. These are semidirect products $C_\mu(R) \rtimes [R]$, where $[R]$ is the full group of a CBER $R$, and $C_\mu(R)$ is a group of finite-measure subsets of $R$ (analogous to the lamp configuration space in discrete lamplighters). This construction allows the authors to lift discrete amenability problems to a topological setting.

3. Key Contributions and Results

A. Uniform Liouville Property for Equivalence Relations (Theorem A)

The authors prove a characterization of the amenability of a countable Borel equivalence relation $R$ on a standard measure space $(X, \mu)$.

• Result: $R$ is $\mu$-amenable if and only if there exists a symmetric non-degenerate probability measure $\nu$ on a dense countable subgroup $G \subset [R]$ such that the action of $G$ on almost every orbit in $X$ is $\nu$-Liouville.
• Significance: This extends the Kaimanovich–Vershik theorem to the setting of equivalence relations. It implies that if a group admits an action where no single measure makes all orbits Liouville, the group cannot be amenable.
• Counterexample: They construct a non-amenable group $G$ acting on a space such that for each orbit, there exists a measure making that specific orbit Liouville, but no single measure works for all orbits simultaneously.

B. Kesten's Property for Topological Groups (Theorem B)

The authors define the Kesten property for a topological group $G$: for every open neighborhood $U$ of the identity and every symmetric measure $\nu$ with countable support, $\limsup_{n \to \infty} \nu^n(U^n)^{1/n} = 1$.

• Result: Every amenable topological group with Small Invariant Neighborhoods (SIN) possesses the Kesten property.
• Context: This generalizes Kesten's theorem to a broad class of non-locally compact groups (e.g., groups with bi-invariant metrics).
• Limitation: The SIN condition is crucial. The authors show that without it, the implication fails.

C. Measurable Lamplighter Groups and Counterexamples (Theorem C & Corollary 6.5)

The authors define the measurable lamplighter group $L = C_\mu(R) \rtimes [R]$.

• Structural Results (Theorem C):
  • If $R$ is $\mu$-amenable, $L$ is an amenable Polish group.
  • If $\mu$ is $R$-ergodic, $L$ is contractible (and thus locally generated) but does not have SIN.
• Main Counterexample (Corollary 6.5): By choosing $R$ to be the orbit equivalence relation of the free group $F_2$ acting on its Poisson boundary (which is amenable as a relation but generated by a non-amenable group), they construct a group $L$ that is:
  1. Amenable (as a topological group).
  2. Contractible (and thus locally generated).
  3. Does not satisfy Kesten's property.
• Mechanism: The failure of Kesten's property in this group is linked to the exponential decay of the inverted orbits of the random walk on the orbits of $F_2$. Since $F_2$ is non-amenable, its inverted orbits grow exponentially, preventing the return probabilities in the lamplighter group from decaying subexponentially.

D. Connection to Extensive Amenability

The paper establishes a bridge between Kesten's property and extensive amenability (a property of group actions introduced by Juschenko and Monod).

• Theorem 6.3: For measurable lamplighters, the Kesten property is equivalent to the subexponential decay of the expected size of inverted orbits (specifically, $\int E[2^{-|O_n(x)|}] d\mu(x)$).
• Implication: If the Kesten property held for all amenable measurable lamplighters, it would imply that actions of amenable groups on orbits of amenable equivalence relations are extensively amenable. This connects to the open problem of the amenability of the group of interval exchange transformations (IET).

4. Significance and Impact

1. Resolution of a Topological Question: The paper provides the first known example of an amenable, contractible, locally generated Polish group that fails Kesten's property. This demonstrates that the connection between amenability and return probabilities (Kesten's theorem) is fragile in the topological setting and requires specific structural assumptions like SIN.
2. New Characterization of Equivalence Relations: Theorem A provides a powerful new tool for studying the amenability of equivalence relations via the Liouville property of group actions, offering a potential pathway to resolve open amenability problems for groups like Thompson's group $F$ or IET groups.
3. Probabilistic Insight: The work deepens the understanding of inverted orbits. It shows that the behavior of inverted orbits in the "measurable" setting (averaged over the space) is governed by the Kesten property of the associated lamplighter group.
4. Open Problems: The paper leaves open the question of whether Kesten's property holds for measurable lamplighters associated with measure-preserving (invariant) amenable equivalence relations. A positive answer here would likely resolve the amenability of the IET group.

In summary, the paper successfully generalizes classical amenability criteria to topological and measurable settings, identifies the precise structural conditions (SIN) under which these generalizations hold, and constructs a sophisticated counterexample using measurable lamplighters to delineate the boundaries of Kesten's theorem in infinite-dimensional topology.