arXiv:1802.00751v4 [math.GR] 22 Feb 2019
CHOQUET-DENY GROUPS AND THE INFINITE
CONJUGACY CLASS PROPERTY
JOSHUA FRISCH, YAIR HARTMAN, OMER TAMUZ,
AND POOYA VAHIDI FERDOWSI
Abstract.
A countable discrete group
G
is called Choquet-Deny
if for every non-degenerate probability measure
μ
on
G
it holds
that all bounded
μ
-harmonic functions are constant. We show
that a finitely generated group
G
is Choquet-Deny if and only if
it is virtually nilpotent. For general countable discrete groups, we
show that
G
is Choquet-Deny if and only if none of its quotients
has the infinite conjugacy class property. Moreover, when
G
is
not Choquet-Deny, then this is witnessed by a symmetric, finite
entropy, non-degenerate measure.
1.
Introduction
Let
G
be a countable discrete group. A probability measure
μ
on
G
is
non-degenerate
if its support generates
G
as a semigroup.
1
A function
f
:
G
→
R
is
μ
-harmonic
if
f
(
k
) =
∑
g
∈
G
μ
(
g
)
f
(
kg
) for all
k
∈
G
.
We say that the
measured group
(
G, μ
) is
Liouville
if all the bounded
μ
-harmonic functions are constant; this is equivalent to the triviality
of the Poisson boundary Π(
G, μ
) [
12
–
14
] (also called the Furstenberg-
Poisson boundary; for formal definitions see also, e.g., Furstenbe
rg and
Glasner [
11
], Bader and Shalom [
1
], or a survey by Furman [
10
]).
When
G
is non-amenable, (
G, μ
) is not Liouville for every non-
degenerate
μ
[
14
]. Conversely, when
G
is amenable, then there ex-
ists some non-degenerate
μ
such that (
G, μ
) is Liouville, as shown by
Kaimanovich and Vershik [
21
] and Rosenblatt [
24
]. It is natural to
ask for which groups
G
it holds that (
G, μ
) is Liouville for
every
non-
degenerate
μ
. We call such groups
Choquet-Deny
groups; as we discuss
in
§
1.1
, there are a few variants of this definition (see, e.g., [
15
–
17
],
or [
19
]), which, however, we show to be equivalent.
The classical Choquet-Deny Theorem (which was first proved for
Z
d
by Blackwell [
3
]) states that abelian groups are Choquet-Deny [
4
]; the
same holds for virtually nilpotent groups [
6
]. There are many examples
Date
: February 25, 2019.
1
In the context of Markov chains such measures are called
irreducible
.
1
2
J. FRISCH, Y. HARTMAN, O. TAMUZ, AND P. VAHIDI FERDOWSI
of amenable groups that are not Choquet-Deny: first examples of
such
groups
2
are due to Kaimanovich [
20
] and Kaimanovich and Vershik [
21
],
they include locally finite groups; Erschler shows that finitely gener-
ated solvable groups that are not virtually nilpotent are not Choque
t-
Deny [
8
], and that even some groups of intermediate growth are not
Choquet-Deny [
7
]. Kaimanovich and Vershik [
21
, p. 466] conjecture
that “Given an exponential group G, there exists a symmetric (non
-
finitary, in general) measure with non-trivial boundary.” See Barth
oldi
and Erschler [
2
] for additional related results and further references and
discussion.
Our main result is a characterization of Choquet-Deny groups. We
say that
G
has the
infinite conjugacy class
property (ICC) if it is non-
trivial, and if each of its non-trivial elements has an infinite conjugac
y
class. We say that
μ
is
fully supported
if supp
μ
=
G
; obviously this
implies that
μ
is non-degenerate.
Theorem 1.
A countable discrete group
G
is Choquet-Deny if and
only if it has no ICC quotients. Moreover, when
G
does have an ICC
quotient, then there exists a fully supported, symmetric, fi
nite entropy
probability measure
μ
on
G
such that
(
G, μ
)
is not Liouville. In par-
ticular, if
G
is finitely generated, then it is Choquet-Deny if and only
if it is virtually nilpotent.
That a group with no ICC quotients is Choquet-Deny was shown by
Jaworski [
18
, Theorem 4.8].
3
Our contribution is therefore in the proof
of the converse, which appears in
§
2
.
Groups with no ICC quotients are known as FC-hypercentral (see
,
e.g., [
5
,
22
], or [
23
,
§
4.3]). This class is closed under forming subgroups,
quotients, direct products and finite index extensions, and include
s all
virtually nilpotent groups. Among finitely generated groups, virtu-
ally nilpotent groups are precisely those with no ICC quotients (see
[
22
, Theorem 2] and [
5
, Theorem 2]); this implies the result in The-
orem
1
for finitely generated groups. Since finitely generated groups
of exponential growth are not virtually nilpotent, Theorem
1
implies
that the above mentioned conjecture of Kaimanovich and Vershik [
21
]
is correct.
A very recent result by three of the authors of this paper shows t
hat
a countable discrete group is strongly amenable if and only if it has
no ICC quotients [
9
]. This implies that
G
is strongly amenable if and
only if (
G, μ
) is Liouville for every non-degenerate
μ
, paralleling the
2
In the Lie group setting, an example of an amenable group that is not
Choquet-
Deny was already known to Furstenberg [
12
].
3
In fact, Jaworski proves there a stronger statement; see the
discussion in
§
1.1
.
3
above mentioned characterization of amenability as equivalent to th
e
existence of a non-degenerate
μ
such that (
G, μ
) is Liouville. While
the proofs of these two similar results are different, it is natural to
ask
whether there is some deeper connection between strong amenab
ility
and the Choquet-Deny property.
1.1.
Different possible definitions of Choquet-Deny groups.
Our definition of Choquet-Deny groups is not the usual one, which
states that a group is Choquet-Deny if (
G, μ
) is Liouville for every
adapted
measure
μ
, where
μ
is called adapted if its support generates
G
as a
group
(rather than as a semigroup, as in the non-degenerate
case) [
15
–
17
]. Yet another definition used in the literature requires
that for
every
μ
, every bounded
μ
-harmonic function is constant on
the left cosets of
G
μ
, where
G
μ
is the subgroup of
G
generated by the
support of
μ
[
19
].
While a priori these are different definitions, they are equivalent, as
demonstrated by our result and by Jaworski’s Theorem 4.8 in [
18
]. Ja-
worski’s result shows that groups with no ICC quotients are Choque
t-
Deny according to any of these definitions. Since our construction
of
μ
with a non-trivial boundary yields measures that are supported on
all of
G
(hence non-degenerate, hence adapted), it shows that groups
with ICC quotients are not Choquet-Deny according to any of thes
e
definitions. Moreover, our result shows that the class of Choquet
-
Deny groups (whether defined with adapted or with non-degenera
te
measures) is closed under taking subgroups, which, to the best of
our
knowledge, was also not previously known.
Acknowledgments.
We would like to thank Anna Erschler and Vadim
Kaimanovich for many useful comments on the first draft of this pa
-
per. We thank Wojciech Jaworski for bringing a number of errors t
o
our attention and suggesting many improvements. We likewise thank
an anonymous referee for many helpful suggestions.
2.
Proofs
In this section we prove the main result of our paper, Theorem
1
.
Unless stated otherwise, we will assume that all groups are counta
ble
and discrete.
Recall that a probability measure
μ
on
G
is symmetric if
μ
(
g
) =
μ
(
g
−
1
) for all
g
∈
G
. Its Shannon entropy (or just entropy) is
H
(
μ
) =
−
∑
g
∈
G
μ
(
g
) log
μ
(
g
).
4
J. FRISCH, Y. HARTMAN, O. TAMUZ, AND P. VAHIDI FERDOWSI
Our Theorem
1
is a direct consequence of [
18
, Theorem 4.8], which
proves it for the case of groups with no ICC quotients, and of the f
ollow-
ing proposition, which handles the case of groups with ICC quotients
.
Proposition 2.1.
Let
G
be a group with an ICC quotient. Then there
exists a fully-supported, symmetric, finite entropy probab
ility measure
μ
on
G
such that
Π(
G, μ
)
is non-trivial.
The main technical effort in the proof of Proposition
2.1
is in the
proof of the following proposition.
Proposition 2.2.
Let
G
be an amenable ICC group. For every
h
∈
G
\
{
e
}
there exists a fully supported, symmetric, finite entropy pr
obability
measure
μ
such that
lim
m
→∞
k
hμ
∗
m
−
μ
∗
m
k
>
0
.
(2.1)
Here
μ
∗
m
is the
m
-fold convolution
μ
∗···∗
μ
. We will prove this
Proposition later, and now turn to the proof of Proposition
2.1
.
Proof of Proposition
2.1
.
The case of non-amenable
G
is known, so as-
sume that
G
is amenable and has an ICC quotient
Q
. Let
h
be a
non-identity element of
Q
. Applying Proposition
2.2
to
Q
and
h
yields
a finite entropy, symmetric measure ̄
μ
on
Q
that is fully supported,
and satisfies (
2.1
).
Since ̄
μ
has full support and satisfies (
2.1
), it follows from [
15
, The-
orem 2] that (
Q,
̄
μ
) has a non-trivial Poisson boundary. Let
μ
be any
symmetric, finite entropy non-degenerate probability measure on
G
that is projected to ̄
μ
; the existence of such a
μ
is straightforward.
Then (
G, μ
) has a non-trivial Poisson boundary.
2.1.
Switching Elements.
Here we introduce two notions: switching
elements and super-switching elements. We will use these notions in
the proof of Proposition
2.2
.
Definition 2.3.
Let
X
be a finite symmetric subset of a group
G
.
•
We call
g
∈
G
a
switching element for
X
if
X
∩
gXg
−
1
⊆{
e
}
.
•
We call
g
∈
G
a
super-switching element for
X
if
X
∩
(
gXg
∪
gXg
−
1
∪
g
−
1
Xg
∪
g
−
1
Xg
−
1
)
⊆{
e
}
.
Note that since
X
is symmetric,
g
∈
G
is a switching element for
X
if and only if
g
−
1
is a switching element for
X
.
5
Claim 2.4.
Let
X
be a finite symmetric subset of a group
G
and let
g
∈
G
be a super-switching element for
X
. If
g
w
1
xg
w
2
=
y
for
x, y
∈
X
and
w
1
, w
2
∈{−
1
,
+1
}
, then
x
=
y
=
e
.
Proof.
Let
g
w
1
xg
w
2
=
y
for
x, y
∈
X
and
w
1
, w
2
∈{−
1
,
+1
}
. Since
y
=
g
w
1
xg
w
2
∈
(
gXg
∪
gXg
−
1
∪
g
−
1
Xg
∪
g
−
1
Xg
−
1
)
and
y
∈
X
, it follows from the definition of a super-switching element
for
X
that
y
=
e
.
From
g
w
1
xg
w
2
=
y
, we get
g
−
w
1
yg
−
w
2
=
x
. So, by symmetry, the
same argument shows
x
=
e
.
Proposition 2.5.
Let
G
be a discrete (not necessarily countable) amenable
ICC group, and let
X
be a finite symmetric subset of
G
. The set of
super-switching elements for
X
is infinite.
Proof of Proposition
2.5
.
Fix an invariant finitely additive probability
measure
d
on
G
. For
A
⊆
G
, we call
d
(
A
) the density of
A
. We will
need the fact that infinite index subgroups have zero density, and
that
d
(
A
) = 0 for every finite subset
A
⊂
G
.
Let
C
G
(
x
) be the centralizer of a non-identity
x
∈
X
. Then, since
X
is finite, there is a finite set of cosets of
C
G
(
x
) that includes all
g
∈
G
such that
g
−
1
xg
∈
X
. So, non-switching elements for
X
are in the
union of finitely many cosets of subgroups with infinite index, since
G
is ICC. This means that the set of non-switching elements for
X
has
zero density, and so the set
S
of switching elements for
X
has density
one.
Let
T
be the set of all super-switching elements for
X
. Let
A
⊆
G
be the set of involutions
{
g
∈
G
|
g
2
=
e
}
.
If
d
(
A
)
>
0, then
d
(
A
∩
S
)
>
0. On the other hand, for any
g
∈
A
∩
S
,
since
g
is switching for
X
and
g
−
1
=
g
,
g
is super-switching for
X
.
Hence
A
∩
S
⊆
T
. This shows that if
d
(
A
)
>
0, then
d
(
T
)
≥
d
(
A
∩
S
)
>
0, and so we are done.
So, we can assume that
d
(
A
) = 0. For any
x, y
∈
X
, let
S
x,y
=
{
g
∈
S
|
gxg
=
y
}
. Note that
T
=
S
\
⋃
x,y
∈
X
(
x,y
)
6
=(
e,e
)
S
x,y
.
It is thus enough to be shown that each
S
x,y
has zero density when
(
x, y
)
6
= (
e, e
). So assume for the sake of contradiction that
d
(
S
x,y
)
>
0.
6
J. FRISCH, Y. HARTMAN, O. TAMUZ, AND P. VAHIDI FERDOWSI
Fix
g
∈
S
x,y
. We have the following for all
h
∈
g
−
1
S
x,y
.
gxg
=
y
=
ghxgh
=
⇒
(
xg
) =
h
(
xg
)
h
=
⇒
(
xg
)
−
1
h
−
1
(
xg
) =
h
=
⇒
h
= (
xg
)
−
1
h
−
1
(
xg
)
= (
xg
)
−
1
[(
xg
)
−
1
h
−
1
(
xg
)]
−
1
(
xg
)
= (
xg
)
−
2
h
(
xg
)
2
=
⇒
h
is in the centralizer of (
xg
)
2
.
So, the centralizer of (
xg
)
2
includes
g
−
1
S
x,y
, which has a positive den-
sity. So, the centralizer of (
xg
)
2
has finite index. This implies that
(
xg
)
2
=
e
, because in an ICC group only the identity can have a finite
index centralizer. Hence
xg
∈
A
for all
g
∈
S
x,y
. So
xS
x,y
⊆
A
. Hence
S
x,y
also has zero density, which is a contradiction.
2.2.
A Heavy-Tailed Probability Distribution on
N
.
Here we
state and prove a lemma about the existence of a probability distri-
bution on
N
=
{
1
,
2
, . . .
}
such that infinite i.i.d. samples from this
measure have certain properties. We will use this distribution in the
proof of Proposition
2.2
.
Lemma 2.6.
Let
p
be the following probability measure on
N
:
p
(
n
) =
cn
−
5
/
4
, where
1
/c
=
∑
∞
n
=1
n
−
5
/
4
. Then
p
has finite entropy and the
following property: for any
ε >
0
there exist constants
K
ε
, N
ε
∈
N
such
that for any natural number
m
≥
K
ε
there exists an
E
ε,m
⊆
N
m
such
that:
(1)
p
×
m
(
E
ε,m
)
≥
1
−
ε
, where
p
×
m
is the
m
-fold product measure
p
×···×
p
.
(2) For any
s
= (
s
1
, . . . , s
m
)
∈
E
ε,m
, the maximum of
{
s
1
, . . . , s
K
ε
}
is at most
N
ε
.
(3) For any
s
= (
s
1
, . . . , s
m
)
∈
E
ε,m
and for any
K
ε
≤
k
≤
m
, the
maximum of
{
s
1
, . . . , s
k
}
is at least
k
2
.
(4) For any
s
= (
s
1
, . . . , s
m
)
∈
E
ε,m
and for any
K
ε
≤
k
≤
m
, the
maximum of
{
s
1
, . . . , s
k
}
appears in
(
s
1
, . . . , s
k
)
only once.
Proof.
It is straightforward to see that
p
has finite entropy.
7
Let
s
= (
s
1
, s
2
, . . .
)
∈
N
∞
have distribution
p
×∞
; i.e.,
s
is a sequence
of i.d.d. random variables with distribution
p
. Since each
s
i
has distri-
bution
p
, for each
n
∈
N
we have:
P
[
s
i
≥
n
] =
∞
∑
m
=
n
p
(
m
) =
c
∞
∑
m
=
n
m
−
5
/
4
≥
c
∫
∞
n
x
−
5
/
4
d
x
= 4
cn
−
1
/
4
.
(2.2)
For
k
≥
1, let
M
k
:
= max
{
s
1
, . . . , s
k
}
,
and let
next(
k
)
:
= min
{
i > k
|
s
i
≥
M
k
}
.
In words, next(
k
) is the first index
i > k
for which
s
i
matches or exceeds
M
k
.
We first show that with probability one,
M
k
≥
k
2
for all
k
large
enough. To this end, let
A
k
be the event that
M
k
< k
2
. We have:
P
[
A
k
] =
P
[
s
i
< k
2
∀
i
∈{
1
, . . . , k
}
]
= (1
−
P
[
s
1
< k
2
]
)
k
≤
(1
−
4
c
(
k
2
)
−
1
/
4
)
k
≤
e
−
4
ck
1
/
2
.
Since the sum of these probabilities is finite, by Borel-Cantelli we get
that
P
[
A
k
infinitely often] = 0
.
Hence
M
k
≥
k
2
for all
k
large enough, almost surely. Furthermore, the
expectation of 1
/M
k
is small:
E
[
1
M
k
]
=
E
[
1
M
k
∣
∣
∣
∣
A
k
]
P
[
A
k
] +
E
[
1
M
k
∣
∣
∣
∣
¬
A
k
]
P
[
¬
A
k
]
≤
e
−
4
ck
1
/
2
+
1
k
2
·
(2.3)
Next, we show that, with probability one,
s
next(
k
)
> M
k
for all
k
large enough. That is, for large enough
k
, the first time that
M
k
is
matched or exceeded after index
k
, it is in fact exceeded.
Let
B
k
be the event that
s
next(
k
)
=
M
k
. We would like to show that
this occurs only finitely often. Note that
P
[
B
k
|
M
k
] =
P
[
s
next(
k
)
=
M
k
∣
∣
M
k
]
=
∞
∑
i
=
k
+1
P
[
s
i
=
M
k
,
next(
k
) =
i
|
M
k
]
.
8
J. FRISCH, Y. HARTMAN, O. TAMUZ, AND P. VAHIDI FERDOWSI
Applying the definition of next(
k
) yields
P
[
B
k
|
M
k
] =
∞
∑
i
=
k
+1
P
[
s
i
=
M
k
, s
k
+1
, . . . , s
i
−
1
< M
k
|
M
k
]
.
By the independence of the
s
i
’s we can write this as
P
[
B
k
|
M
k
] =
∞
∑
i
=
k
+1
P
[
s
i
=
M
k
|
M
k
]
i
−
(
k
+1)
∏
n
=1
P
[
s
k
+
n
< M
k
|
M
k
]
=
∞
∑
i
=
k
+1
c
M
5
/
4
k
P
[
s
k
+1
< M
k
|
M
k
]
i
−
(
k
+1)
.
By (
2.2
),
P
[
s
k
+1
< M
k
|
M
k
]
≤
1
−
4
cM
−
1
/
4
k
. Hence
P
[
B
k
|
M
k
]
≤
c
M
5
/
4
k
·
1
4
cM
−
1
/
4
k
=
1
4
M
k
·
Using (
2.3
) it follows that
P
[
B
k
] =
E
[
P
[
B
k
|
M
k
]]
≤
E
[
1
4
M
k
]
≤
1
4
e
−
4
ck
1
/
2
+
1
4
k
2
.
Hence
∑
k
P
[
B
k
]
<
∞
, and so by Borel-Cantelli
B
k
occurs only finitely
often.
Since
A
k
and
B
k
both occur for only finitely many
k
, the (random)
index ind
′
at which they stop occurring is almost surely finite, and is
given by
ind
′
= min
{
ℓ
∈
N
:
s
6∈
A
k
∪
B
k
for all
k
≥
ℓ
}
.
Let
ind = next(ind
′
)
.
Hence for
k
≥
ind,
M
k
≥
k
2
and
M
k
appears in (
s
1
, . . . , s
k
) only once.
Fix
ε >
0. Since ind is almost surely finite, then for large enough
constants
K
ε
∈
N
and
N
ε
∈
N
the event
E
ε
=
{
ind
≤
K
ε
and
M
K
ε
≤
N
ε
}
has probability at least 1
−
ε
, and additionally, conditioned on
E
ε
it
holds that
k
≥
ind for all
k
≥
K
ε
, and hence
M
k
≥
k
2
and
M
k
appears
in (
s
1
, . . . , s
k
) only once. Therefore, if for
m
≥
K
ε
we let
E
ε,m
be
the projection of
E
ε
to the first
m
coordinates, then
E
ε,m
satisfies the
desired properties.
9
2.3.
Proof of Proposition
2.2
.
Let
1
8
> ε >
0. Let
p
,
K
ε
∈
N
, N
ε
∈
N
, and
E
ε,m
⊆
N
m
be the probability measure, the constants, and the
events from Lemma
2.6
. To simplify notation let
N
=
N
ε
and
K
=
K
ε
.
Let
G
=
{
a
1
, a
2
, . . .
}
, where
a
1
=
a
2
=
···
=
a
N
=
e
. We define
(
g
n
)
n
, (
A
n
)
n
, (
B
n
)
n
and (
C
n
)
n
recursively. Given
g
1
, . . . , g
n
, let
A
n
=
{
g
n
, g
−
1
n
, a
n
, a
−
1
n
}
and
B
n
=
∪
i
≤
n
A
i
. Denote
C
n
=
B
n
∪{
h
−
1
, h
}
. Note
that
A
n
,
B
n
, and
C
n
are finite and symmetric for any
n
∈
N
. Let
g
1
=
g
2
=
. . .
=
g
N
=
e
. For
n
+ 1
> N
, given
C
n
, let
g
n
+1
∈
G
be a super-
switching element for (
C
n
)
2
n
+1
which is not in (
C
n
)
8
n
+1
. The existence
of such a super-switching element is guaranteed by Proposition
2.5
and the facts that (
C
n
)
2
n
+1
is a finite symmetric subset of
G
and that
(
C
n
)
8
n
+1
is finite.
For
n
∈
N
, define a symmetric probability measure
μ
n
on
A
n
by
μ
n
=
ε
2
−
n
(
1
2
δ
a
n
+
1
2
δ
a
−
1
n
) + (1
−
ε
2
−
n
)(
1
2
δ
g
n
+
1
2
δ
g
−
1
n
)
.
Here
δ
g
is the point mass on
g
∈
G
. Finally, let
μ
=
∞
∑
n
=1
p
(
n
)
μ
n
.
Obviously
μ
is symmetric and supp
μ
=
G
. Since
p
has finite entropy
and each
μ
n
has support of size at most 4, it follows easily that
μ
has
finite entropy.
We want to show that
lim
m
→∞
k
hμ
∗
m
−
μ
∗
m
k
>
0
.
Fix
m
∈
N
larger than
K
and
N
. For each
n
∈
N
define
f
n
:
{
1
,
2
,
3
,
4
}→
A
n
by
f
n
(1) =
a
n
, f
n
(2) =
a
−
1
n
, f
n
(3) =
g
n
, f
n
(4) =
g
−
1
n
,
and define
ν
n
:
{
1
,
2
,
3
,
4
}→
[0
,
1] by
ν
n
(1) =
ν
n
(2) =
1
2
ε
2
−
n
, ν
n
(3) =
ν
n
(4) =
1
2
(1
−
ε
2
−
n
)
.
Let
Ω =
{
(
s, w
)
|
s
∈
N
m
, w
∈{
1
,
2
,
3
,
4
}
m
}
.
We define the measure
η
on the countable set Ω by specifying its
values on the singletons:
η
(
{
(
s, w
)
}
) =
p
×
m
(
s
)
ν
s
1
(
w
1
)
ν
s
2
(
w
2
)
. . . ν
s
m
(
w
m
)
.
It follows immediately from this definition that
η
is a probability mea-
sure.
10
J. FRISCH, Y. HARTMAN, O. TAMUZ, AND P. VAHIDI FERDOWSI
Define
r
: Ω
→
G
by
r
(
s, w
) =
f
s
1
(
w
1
)
f
s
2
(
w
2
)
. . . f
s
m
(
w
m
)
.
It is not difficult to see that
r
∗
η
=
μ
∗
m
, and so we need to show that
k
hr
∗
η
−
r
∗
η
k
is uniformly bounded away from zero for
m
larger than
K
and
N
.
Recall that
E
ε,m
⊆
N
m
is the event given by Lemma
2.6
. Fix
s
∈
E
ε,m
. Define
i
s,
1
= min
{
j
∈{
1
. . . , m
} |
s
j
> N
}
,
i
s,
2
= min
{
j > i
s,
1
|
s
j
≥
s
i
s,
1
}
,
.
.
.
i
s,l
(
s
)
= min
{
j > i
s,l
(
s
)
−
1
|
s
j
≥
s
i
s,l
(
s
)
−
1
}
.
Note that by the second property of
E
ε,m
in Lemma
2.6
, we know that
K < i
s,
1
< i
s,
2
<
···
< i
s,l
(
s
)
,
and by the fourth property,
N < s
i
s,
1
< s
i
s,
2
<
···
< s
i
s,l
(
s
)
= max
{
s
1
, . . . , s
m
}
.
Let
W
s
ε
=
{
w
∈{
1
,
2
,
3
,
4
}
m
| ∀
k
≤
l
(
s
)
w
i
s,k
= 3
,
4
}
.
For
s
∈
N
m
let
η
s
be the measure
η
, conditioned on the first coordi-
nate equalling
s
. I.e., let
η
s
(
A
) =
η
(
A
∩
Ω
s
)
η
(Ω
s
)
,
where Ω
s
=
{
s
}×{
1
,
2
,
3
,
4
}
m
⊆
Ω.
Then
η
s
(
{
s
}×
W
s
ε
) = 1
−
η
s
(
{
w
i
s,
1
= 1
,
2; or
w
i
s,
2
= 1
,
2;
. . .
; or
w
i
s,l
(
s
)
= 1
,
2
}
)
≥
1
−
l
(
s
)
∑
k
=1
η
s
(
{
w
i
s,k
= 1
,
2
}
)
= 1
−
l
(
s
)
∑
k
=1
ε
2
−
s
i
s,k
≥
1
−
∞
∑
j
=1
ε
2
−
j
= 1
−
ε,
11
where the first inequality follows from the union bound, and the last
inequality holds since
s
i
s,
1
< s
i
s,
2
<
···
< s
i
s,l
(
s
)
.
Finally, let
Ω
ε
=
{
(
s, w
)
∈
Ω
|
s
∈
E
ε,m
, w
∈
W
s
ε
}
.
By the above, and since
η
(
E
ε,m
×{
1
,
2
,
3
,
4
}
m
)
≥
1
−
ε
by Lemma
2.6
,
we have shown that
η
(Ω
ε
)
≥
(1
−
ε
)(1
−
ε
)
>
1
−
2
ε.
Claim 2.7.
For any
α, β
∈
Ω
ε
, we have
hr
(
α
)
6
=
r
(
β
)
.
We prove this claim after we finish the proof of the Proposition.
Let
η
1
be equal to
η
conditioned on Ω
ε
, and
η
2
be equal to
η
condi-
tioned on the complement of Ω
ε
. We have
η
=
η
(Ω
ε
)
η
1
+(1
−
η
(Ω
ε
))
η
2
,
and by the above claim we know
k
hr
∗
η
1
−
r
∗
η
1
k
= 2. So for
m
larger
than
K
and
N
k
hμ
∗
m
−
μ
∗
m
k
=
k
hr
∗
η
−
r
∗
η
k
=
k
η
(Ω
ε
)(
hr
∗
η
1
−
r
∗
η
1
) + (1
−
η
(Ω
ε
))(
hr
∗
η
2
−
r
∗
η
2
)
k
≥
η
(Ω
ε
)
k
hr
∗
η
1
−
r
∗
η
1
k−
2(1
−
η
(Ω
ε
))
≥
2(1
−
2
ε
)
−
2(2
ε
) = 2
−
8
ε,
which is uniformly bounded away from zero since
ε <
1
8
. Since
k
hμ
∗
m
−
μ
∗
m
k
is a decreasing sequence, this completes the proof of Proposition
2.2
.
Proof of Claim
2.7
.
Let
α
= (
s, w
)
, β
= (
t, v
)
∈
Ω
ε
. Hence max
{
K, N
}
<
m
,
s
∈
E
ε,m
,
t
∈
E
ε,m
,
w
∈
W
s
ε
, and
v
∈
W
t
ε
. Assume that
hr
(
α
) =
r
(
β
). So, we have
hf
s
1
(
w
1
)
···
f
s
m
(
w
m
) =
f
t
1
(
v
1
)
···
f
t
m
(
v
m
)
.
Let
K < i
1
< i
2
<
···
< i
l
(
s
)
and
K < j
1
< j
2
<
···
< j
l
(
t
)
be
the indices we defined for
s
and
t
in the proof of Proposition
2.2
. We
remind the reader that the unique maximum of (
s
1
, . . . , s
m
) is attained
at
i
l
(
s
)
, with a corresponding statement for (
t
1
, . . . , t
m
) and
j
l
(
t
)
. So we
have
h
b
1
︷
︸︸
︷
f
s
1
(
w
1
)
···
f
s
i
l
(
s
)
−
1
(
w
i
l
(
s
)
−
1
)
f
s
i
l
(
s
)
(
w
i
l
(
s
)
)
b
2
︷
︸︸
︷
f
s
i
l
(
s
)
+1
(
w
i
l
(
s
)
+1
)
···
f
s
m
(
w
m
)
=
f
t
1
(
v
1
)
···
f
t
j
l
(
t
)
−
1
(
v
j
l
(
t
)
−
1
)
︸
︷︷
︸
c
1
f
t
j
l
(
t
)
(
v
j
l
(
t
)
)
f
t
j
l
(
t
)
+1
(
v
j
l
(
t
)
+1
)
···
f
t
m
(
v
m
)
︸
︷︷
︸
c
2
.
Let
p
=
s
i
l
(
s
)
= max
{
s
1
, . . . , s
m
}
and
q
=
t
j
l
(
t
)
= max
{
t
1
, . . . , t
m
}
. Since
w
∈
W
s
ε
and
v
∈
W
t
ε
, we know
f
s
i
l
(
s
)
(
w
i
l
(
s
)
) =
g
±
1
p
and
f
t
j
l
(
t
)
(
v
j
l
(
t
)
) =
12
J. FRISCH, Y. HARTMAN, O. TAMUZ, AND P. VAHIDI FERDOWSI
g
±
1
q
, so
hb
1
g
±
1
p
b
2
=
c
1
g
±
1
q
c
2
.
(2.4)
Since
p
= max
{
s
1
, . . . , s
m
}
, and since
m
≥
K
, we know that
m
≤
m
2
≤
p
. So
b
1
, b
2
∈
(
B
p
−
1
)
p
−
1
⊆
(
C
p
−
1
)
p
−
1
. Similarly
c
1
, c
2
∈
(
C
q
−
1
)
q
−
1
.
Consider the case that
p > q
. Then
c
1
, c
2
, g
±
1
q
∈
(
C
q
)
q
⊆
(
C
p
−
1
)
p
−
1
.
Hence
g
±
1
p
= [
b
−
1
1
]
h
−
1
[
c
1
g
±
1
q
c
2
b
−
1
2
] by (
2.4
), and so
g
p
∈
(
C
p
−
1
)
4(
p
−
1)
{
h, h
−
1
}
(
C
p
−
1
)
4(
p
−
1)
⊆
(
C
p
−
1
)
8(
p
−
1)+1
,
which is a contradiction with our choice of
g
p
, since
p > N
. Similarly,
if
p < q
, we get a contradiction. So we can assume that
p
=
q
.
If
p
=
q
, then by (
2.4
) we have
hb
1
g
±
1
p
b
2
=
c
1
g
±
1
p
c
2
,
and
c
1
, c
2
, b
1
, b
2
∈
(
C
p
−
1
)
p
−
1
. So, for
x
=
c
−
1
1
hb
1
∈
(
C
p
−
1
)
2(
p
−
1)+1
we
have
g
±
1
p
xg
±
1
p
=
c
2
b
−
1
2
∈
(
C
p
−
1
)
2(
p
−
1)
⊆
(
C
p
−
1
)
2(
p
−
1)+1
. By the fact that
g
p
is a super-switching element for (
C
p
−
1
)
2(
p
−
1)+1
and from Claim
2.4
,
we get that
x
is the identity.
So
hb
1
=
c
1
, i.e.
hf
s
1
(
w
1
)
···
f
s
i
l
(
s
)
−
1
(
w
i
l
(
s
)
−
1
) =
f
t
1
(
v
1
)
···
f
t
j
l
(
t
)
−
1
(
v
j
l
(
t
)
−
1
)
.
By the exact same argument, we can see this leads to a contradictio
n
unless
hf
s
1
(
w
1
)
···
f
s
i
l
(
s
)
−
1
−
1
(
w
i
l
(
s
)
−
1
−
1
) =
f
t
1
(
v
1
)
···
f
t
j
l
(
t
)
−
1
−
1
(
v
j
l
(
t
)
−
1
−
1
)
.
And again, this leads to a contradiction unless
hf
s
1
(
w
1
)
···
f
s
i
l
(
s
)
−
2
−
1
(
w
i
l
(
s
)
−
2
−
1
) =
f
t
1
(
v
1
)
···
f
t
j
l
(
t
)
−
2
−
1
(
v
j
l
(
t
)
−
2
−
1
)
.
Note that if
l
(
s
)
6
=
l
(
t
), at some point in this process we get that
either all the
s
i
’s or all the
t
i
’s are at most
N
while the other string
has characters strictly greater than
N
. This leads to a contradiction
similar to the case
p
6
=
q
, which we explained before. So, by continuing
this process, we get a contradiction unless
hf
s
1
(
w
1
)
···
f
s
i
1
−
1
(
w
i
1
−
1
) =
f
t
1
(
v
1
)
···
f
t
j
1
−
1
(
v
j
1
−
1
)
.
(2.5)
Note that
s
1
, . . . , s
i
1
−
1
≤
N
, which implies
f
s
1
(
w
1
) =
···
=
f
s
i
1
−
1
(
w
i
1
−
1
) =
e.
Similarly,
t
1
, . . . , t
j
1
−
1
≤
N
implies that
f
t
1
(
v
1
) =
···
=
f
t
j
1
−
1
(
v
j
1
−
1
) =
e.
So, from (
2.5
) we get
h
=
e
, which is a contradiction.
13
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