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:
Fn = ha1 , . . . , an i.
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:
Fn = ha1 , . . . , an i.
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
The boundary action of a subgroup of the free group
Consider the free group:
Fn = ha1 , . . . , an i.
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) 7→ hω.
Jan Cannizzo (University of Ottawa)
Sofic boundary actions
May 29, 2013
2 / 19
The free group with boundary
..
.
b
...
a
..
.
Jan Cannizzo (University of Ottawa)
Sofic boundary actions
...
∂F2
May 29, 2013
3 / 19
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
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
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
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
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
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) 7→ gHg −1 ,
for any g ∈ Fn .
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(Fn ), the lattice of subgroups of Fn .
In other words: a probability measure µ on L(Fn ) invariant under the
action
(g , H) 7→ gHg −1 ,
for any g ∈ Fn .
Invariant random subgroups have recently attracted a great deal of
attention (cf. Vershik, Bowen, Grigorchuck, Abért, Glasner, and Virág
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 ≤ Fn , consider its Schreier graph:
ai
Hg 0 = (Hg )ai
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 ≤ Fn , consider its Schreier graph:
ai
Hg 0 = (Hg )ai
Hg
Via the correspondence H 7→ (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 F2 = ha, bi by the element ba2
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 (∂Fn , m) of a normal subgroup N E Fn 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
Jan Cannizzo (University of Ottawa)
gx
)
Sofic boundary actions
(X , µ)
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 (∂Fn , m) is conservative if and only if
lim
r →∞
Jan Cannizzo (University of Ottawa)
|Br (Γ, H)|
= 0.
|Br (Fn , e)|
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 |Br (Fn , 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 |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
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
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
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
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) =
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
Relative thinness
Relative thinness
Relative thinness
Relative thinness
Relative thinness
Jan Cannizzo (University of Ottawa)
Sofic boundary actions
May 29, 2013
13 / 19
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
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.
Z
Z
ρA,r dµ =
Γ
Jan Cannizzo (University of Ottawa)
τr dµ
A
Sofic boundary actions
May 29, 2013
14 / 19
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.
Z
Z
ρA,r dµ =
Γ
τr dµ
A
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
Soficity
How can this be transferred over to infinite graphs?
Jan Cannizzo (University of Ottawa)
Sofic boundary actions
May 29, 2013
15 / 19
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
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
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
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
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
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
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 E(τi | Ai ) = 0.
i→∞
Jan Cannizzo (University of Ottawa)
Sofic boundary actions
May 29, 2013
16 / 19
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 E(τi | Ai ) = 0.
i→∞
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
What this means
Γ\A
A
What this means
Γ\A
A
What this means
Γ\A
A
What this means
Γ\A
A
What this means
Γ\A
A
Jan Cannizzo (University of Ottawa)
Sofic boundary actions
May 29, 2013
17 / 19
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
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
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
Thank You!
Jan Cannizzo (University of Ottawa)
Sofic boundary actions
May 29, 2013
19 / 19