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 free group: F n = � a 1 , . . . , a n � . Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 2 / 19
The boundary action of a subgroup of the free group Consider the free group: F n = � a 1 , . . . , a n � . Has a natural boundary , denoted ∂ F n , which comes with a natural uniform measure , denoted m . Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 2 / 19
The boundary action of a subgroup of the free group Consider the free group: F n = � a 1 , . . . , a n � . Has a natural boundary , denoted ∂ F n , which comes with a natural uniform measure , denoted m . If H ≤ F n is any subgroup, then there is a natural boundary action H � ( ∂ F n , m ), which goes like this: ( h , ω ) �→ h ω. Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 2 / 19
The free group with boundary . . . b a . . . . . . ∂ F 2 . . . Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 3 / 19
Our main question Question What are the ergodic properties of the action H � ( ∂ F n , m )? Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 4 / 19
Our main question Question What are the ergodic properties of the action H � ( ∂ F n , m )? Recently studied by Grigorchuk, Kaimanovich, and Nagnibeda Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 4 / 19
Our main question Question What are the ergodic properties of the action H � ( ∂ F n , m )? Recently studied by Grigorchuk, Kaimanovich, and Nagnibeda Analogous to the action of a Fuchsian group on the boundary of the hyperbolic plane ∂ H 2 ∼ = S 1 equipped with Lebesgue measure Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 4 / 19
Our main question Question What are the ergodic properties of the action H � ( ∂ F n , m )? Recently studied by Grigorchuk, Kaimanovich, and Nagnibeda Analogous to the action of a Fuchsian group on the boundary of the hyperbolic plane ∂ H 2 ∼ = S 1 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
Invariant random subgroups Definition An invariant random subgroup is a conjugation-invariant probability measure on L ( F n ), the lattice of subgroups of F n . Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 5 / 19
Invariant random subgroups Definition An invariant random subgroup is a conjugation-invariant probability measure on L ( F n ), the lattice of subgroups of F n . In other words: a probability measure µ on L ( F n ) invariant under the action ( g , H ) �→ gHg − 1 , for any g ∈ F n . Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 5 / 19
Invariant random subgroups Definition An invariant random subgroup is a conjugation-invariant probability measure on L ( F n ), the lattice of subgroups of F n . In other words: a probability measure µ on L ( F n ) invariant under the action ( g , H ) �→ gHg − 1 , for any g ∈ F n . 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
Schreier graphs There is a nice geometric interpretation! Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 6 / 19
Schreier graphs There is a nice geometric interpretation! Given a subgroup H ≤ F n , consider its Schreier graph : Hg ′ = ( Hg ) a i a i Hg Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 6 / 19
Schreier graphs There is a nice geometric interpretation! Given a subgroup H ≤ F n , consider its Schreier graph : Hg ′ = ( Hg ) a i a i Hg 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
Conjugation in terms of Schreier graphs . . . . . . . . . . . . Conjugating a Schreier graph of F 2 = � a , b � by the element ba 2 Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 7 / 19
First examples: Cayley graphs . . . . . . . . . . . . Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 8 / 19
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
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
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
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 � ( ∂ F n , m ) of a normal subgroup N � F n is conservative. Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 9 / 19
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
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 gx ( X , µ ) Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 10 / 19
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 gx ( X , µ ) Theorem (Grigorchuk, Kaimanovich, and Nagnibeda) The bounday action H � ( ∂ F n , m ) is conservative if and only if | B r (Γ , H ) | lim | B r ( F n , e ) | = 0 . r →∞ Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 10 / 19
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
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 | B r ( F n , e ) | . Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 11 / 19
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 | B r ( F n , e ) | . Now suppose that the proportion of vertices in B r (Γ , H ) attached to a k -cycle—that is, the density of the set A inside of B r (Γ , H )—is bounded away from zero. Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 11 / 19
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 | B r ( F n , e ) | . Now suppose that the proportion of vertices in B r (Γ , H ) attached to a k -cycle—that is, the density of the set A inside of B r (Γ , H )—is bounded away from zero. Then we can show that the ratio of neighborhood sizes | B r (Γ , H ) | / | B r ( F n , e ) | does indeed shrink to zero. Jan Cannizzo (University of Ottawa) Sofic boundary actions May 29, 2013 11 / 19
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