The boundary action of a sofic random subgroup Jan Cannizzo - - PowerPoint PPT Presentation

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The boundary action of a sofic random subgroup Jan Cannizzo - - PowerPoint PPT Presentation

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The boundary action of a sofic random subgroup Jan Cannizzo University of Ottawa May 29, 2013 Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 1 / 19 The boundary action of a subgroup of the free group Consider the

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The boundary action of a sofic random subgroup

Jan Cannizzo

University of Ottawa

May 29, 2013

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 1 / 19

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The boundary action of a subgroup of the free group

Consider the free group: Fn = a1, . . . , an.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 2 / 19

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The boundary action of a subgroup of the free group

Consider the free group: Fn = a1, . . . , an. Has a natural boundary, denoted ∂Fn, which comes with a natural uniform measure, denoted m.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 2 / 19

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The boundary action of a subgroup of the free group

Consider the free group: Fn = a1, . . . , an. Has a natural boundary, denoted ∂Fn, which comes with a natural uniform measure, denoted m. If H ≤ Fn is any subgroup, then there is a natural boundary action H (∂Fn, m), which goes like this: (h, ω) → hω.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 2 / 19

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The free group with boundary

a b ∂F2 . . . . . . . . . . . .

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 3 / 19

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Our main question

Question

What are the ergodic properties of the action H (∂Fn, m)?

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 4 / 19

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Our main question

Question

What are the ergodic properties of the action H (∂Fn, m)? Recently studied by Grigorchuk, Kaimanovich, and Nagnibeda

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 4 / 19

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Our main question

Question

What are the ergodic properties of the action H (∂Fn, m)? Recently studied by Grigorchuk, Kaimanovich, and Nagnibeda Analogous to the action of a Fuchsian group on the boundary of the hyperbolic plane ∂H2 ∼ = S1 equipped with Lebesgue measure

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 4 / 19

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Our main question

Question

What are the ergodic properties of the action H (∂Fn, m)? Recently studied by Grigorchuk, Kaimanovich, and Nagnibeda Analogous to the action of a Fuchsian group on the boundary of the hyperbolic plane ∂H2 ∼ = S1 equipped with Lebesgue measure

The question that interests us:

What happens when H is an invariant random subgroup?

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 4 / 19

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Invariant random subgroups

Definition

An invariant random subgroup is a conjugation-invariant probability measure on L(Fn), the lattice of subgroups of Fn.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 5 / 19

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Invariant random subgroups

Definition

An invariant random subgroup is a conjugation-invariant probability measure on L(Fn), the lattice of subgroups of Fn. In other words: a probability measure µ on L(Fn) invariant under the action (g, H) → gHg−1, for any g ∈ Fn.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 5 / 19

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Invariant random subgroups

Definition

An invariant random subgroup is a conjugation-invariant probability measure on L(Fn), the lattice of subgroups of Fn. In other words: a probability measure µ on L(Fn) invariant under the action (g, H) → gHg−1, for any g ∈ Fn. Invariant random subgroups have recently attracted a great deal of attention (cf. Vershik, Bowen, Grigorchuck, Ab´ ert, Glasner, and Vir´ ag among others), but much remains unknown.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 5 / 19

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Schreier graphs

There is a nice geometric interpretation!

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 6 / 19

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Schreier graphs

There is a nice geometric interpretation! Given a subgroup H ≤ Fn, consider its Schreier graph: Hg Hg′ = (Hg)ai ai

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 6 / 19

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Schreier graphs

There is a nice geometric interpretation! Given a subgroup H ≤ Fn, consider its Schreier graph: Hg Hg′ = (Hg)ai ai Via the correspondence H → (Γ, H), may just as well talk about invariant random Schreier graphs

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 6 / 19

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Conjugation in terms of Schreier graphs

. . . . . . . . . . . . Conjugating a Schreier graph of F2 = a, b by the element ba2

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 7 / 19

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First examples: Cayley graphs

. . . . . . . . . . . .

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 8 / 19

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Normal subgroups vs. invariant random subgroups

Normal subgroups are themselves invariant under conjugation: they are spatially homogenous.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 9 / 19

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Normal subgroups vs. invariant random subgroups

Normal subgroups are themselves invariant under conjugation: they are spatially homogenous. Invariant random subgroups are stochastically homogenous. Morally speaking, they should behave like normal subgroups.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 9 / 19

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Normal subgroups vs. invariant random subgroups

Normal subgroups are themselves invariant under conjugation: they are spatially homogenous. Invariant random subgroups are stochastically homogenous. Morally speaking, they should behave like normal subgroups. But do they?

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 9 / 19

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Normal subgroups vs. invariant random subgroups

Normal subgroups are themselves invariant under conjugation: they are spatially homogenous. Invariant random subgroups are stochastically homogenous. Morally speaking, they should behave like normal subgroups. But do they?

Theorem (Kaimanovich)

The boundary action N (∂Fn, m) of a normal subgroup N Fn is conservative.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 9 / 19

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Conservativity

Definition

An action G (X, µ) is conservative if every subset E ⊆ X is recurrent, i.e. contained in the union of its translates gE, where g ∈ G\{e}.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 10 / 19

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Conservativity

Definition

An action G (X, µ) is conservative if every subset E ⊆ X is recurrent, i.e. contained in the union of its translates gE, where g ∈ G\{e}. (X, µ) x gx ( )

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 10 / 19

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Conservativity

Definition

An action G (X, µ) is conservative if every subset E ⊆ X is recurrent, i.e. contained in the union of its translates gE, where g ∈ G\{e}. (X, µ) x gx ( )

Theorem (Grigorchuk, Kaimanovich, and Nagnibeda)

The bounday action H (∂Fn, m) is conservative if and only if lim

r→∞

|Br(Γ, H)| |Br(Fn, e)| = 0.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 10 / 19

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An observation about the size of r-neighborhoods

With positive probability, the root of an (interesting) invariant random Schreier graph will belong to a cycle of length k (for some k ∈ N). Denote the set of such graphs by A.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 11 / 19

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An observation about the size of r-neighborhoods

With positive probability, the root of an (interesting) invariant random Schreier graph will belong to a cycle of length k (for some k ∈ N). Denote the set of such graphs by A. Cycles attached to the root of a Schreier graph cause neighborhoods of the root to shrink in size relative to |Br(Fn, e)|.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 11 / 19

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An observation about the size of r-neighborhoods

With positive probability, the root of an (interesting) invariant random Schreier graph will belong to a cycle of length k (for some k ∈ N). Denote the set of such graphs by A. Cycles attached to the root of a Schreier graph cause neighborhoods of the root to shrink in size relative to |Br(Fn, e)|. Now suppose that the proportion of vertices in Br(Γ, H) attached to a k-cycle—that is, the density of the set A inside of Br(Γ, H)—is bounded away from zero.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 11 / 19

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An observation about the size of r-neighborhoods

With positive probability, the root of an (interesting) invariant random Schreier graph will belong to a cycle of length k (for some k ∈ N). Denote the set of such graphs by A. Cycles attached to the root of a Schreier graph cause neighborhoods of the root to shrink in size relative to |Br(Fn, e)|. Now suppose that the proportion of vertices in Br(Γ, H) attached to a k-cycle—that is, the density of the set A inside of Br(Γ, H)—is bounded away from zero. Then we can show that the ratio of neighborhood sizes |Br(Γ, H)|/|Br(Fn, e)| does indeed shrink to zero.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 11 / 19

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The density of a given set inside of large neighborhoods

Question

Given an invariant random Schreier graph and a nontrivial subset A, must the average density of A inside of r-neighborhoods of the root actually be bounded away from zero?

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 12 / 19

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The density of a given set inside of large neighborhoods

Question

Given an invariant random Schreier graph and a nontrivial subset A, must the average density of A inside of r-neighborhoods of the root actually be bounded away from zero? We don’t know the answer!

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 12 / 19

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The density of a given set inside of large neighborhoods

Question

Given an invariant random Schreier graph and a nontrivial subset A, must the average density of A inside of r-neighborhoods of the root actually be bounded away from zero? We don’t know the answer! To investigate further, consider the function τr : Γ → Q given by τr(x) =

  • y∈Br(x)

1 |Br(y)|. Say that a (Schreier) graph Γ is relatively thin at a point x ∈ Γ if τr(x) < 1.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 12 / 19

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Relative thinness

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Relative thinness

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Relative thinness

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Relative thinness

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Relative thinness

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 13 / 19

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Density and relative thinness

Suppose Γ is a finite graph, and A ⊆ Γ a subset.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 14 / 19

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Density and relative thinness

Suppose Γ is a finite graph, and A ⊆ Γ a subset. Then the function τr and the density of the set A inside of r-neighborhoods of Γ—call this ρA,r—are related to one another:

Proposition, C.

  • Γ

ρA,r dµ =

  • A

τr dµ

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 14 / 19

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Density and relative thinness

Suppose Γ is a finite graph, and A ⊆ Γ a subset. Then the function τr and the density of the set A inside of r-neighborhoods of Γ—call this ρA,r—are related to one another:

Proposition, C.

  • Γ

ρA,r dµ =

  • A

τr dµ Thus, in order for E(ρA,r) to be small relative to µ(A), must be that A is concentrated at points where Γ is relatively thin

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 14 / 19

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Soficity

How can this be transferred over to infinite graphs?

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 15 / 19

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Soficity

How can this be transferred over to infinite graphs? The uniform measure on a finite Schreier graph determines an invariant random Schreier graph...so pass to a limit

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 15 / 19

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Soficity

How can this be transferred over to infinite graphs? The uniform measure on a finite Schreier graph determines an invariant random Schreier graph...so pass to a limit

Definition

An invariant random Schreier graph is sofic if it is the weak limit of invariant measures determined by finite Schreier graphs.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 15 / 19

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Soficity

How can this be transferred over to infinite graphs? The uniform measure on a finite Schreier graph determines an invariant random Schreier graph...so pass to a limit

Definition

An invariant random Schreier graph is sofic if it is the weak limit of invariant measures determined by finite Schreier graphs. As in the context of groups, the following is unknown:

Open problem

Is every invariant random Schreier graph sofic?

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 15 / 19

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What we can say

Suppose µ is a sofic random Schreier graph which is:

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 16 / 19

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What we can say

Suppose µ is a sofic random Schreier graph which is:

  • i. Ergodic

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 16 / 19

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What we can say

Suppose µ is a sofic random Schreier graph which is:

  • i. Ergodic
  • ii. Answers our question negatively: there is a set A such that E(ρA,r) is

not bounded away from zero

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 16 / 19

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What we can say

Suppose µ is a sofic random Schreier graph which is:

  • i. Ergodic
  • ii. Answers our question negatively: there is a set A such that E(ρA,r) is

not bounded away from zero

Theorem, C.

There exists a sequence of finite Schreier graphs (Γi, Ai, µi) with subsets Ai ⊆ Γi such that the Γi are a sofic approximation to µ, µi(Ai) → 1, and lim

i→∞ E(τi | Ai) = 0.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 16 / 19

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What we can say

Suppose µ is a sofic random Schreier graph which is:

  • i. Ergodic
  • ii. Answers our question negatively: there is a set A such that E(ρA,r) is

not bounded away from zero

Theorem, C.

There exists a sequence of finite Schreier graphs (Γi, Ai, µi) with subsets Ai ⊆ Γi such that the Γi are a sofic approximation to µ, µi(Ai) → 1, and lim

i→∞ E(τi | Ai) = 0.

Note that we can contract a Schreier graph “by a factor of r” by constructing it with respect to a bigger generating set, namely Br(Fn, e).

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 16 / 19

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What this means

A Γ\A

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What this means

A Γ\A

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What this means

A Γ\A

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What this means

A Γ\A

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What this means

A Γ\A

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 17 / 19

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Conclusion

If this happens (and it is unclear whether it can happen), then we still find that |Br(Γ, H)|/|Br(Fn, e)| → 0.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 18 / 19

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Conclusion

If this happens (and it is unclear whether it can happen), then we still find that |Br(Γ, H)|/|Br(Fn, e)| → 0. If it doesn’t happen, then we have our argument involving the density of short cycles.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 18 / 19

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Conclusion

If this happens (and it is unclear whether it can happen), then we still find that |Br(Γ, H)|/|Br(Fn, e)| → 0. If it doesn’t happen, then we have our argument involving the density of short cycles. Therefore:

Theorem, C.

The boundary action H (∂Fn, m) of a sofic random subgroup of the free group is conservative.

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 18 / 19

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Thank You!

Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 19 / 19