AN ERGODIC THEOREM FOR THE QUASI-REGULAR
REPRESENTATION OF THE FREE GROUP

arXiv:1601.00668v1 [] 4 Jan 2016

ADRIEN BOYER AND ANTOINE PINOCHET LOBOS
Abstract. In [BM11], an ergodic theorem à la Birkhoff-von Neumann for the action of the
fundamental group of a compact negatively curved manifold on the boundary of its universal
cover is proved. A quick corollary is the irreducibility of the associated unitary representation.
These results are generalized [Boy15] to the context of convex cocompact groups of isometries
of a CAT(-1) space, using Theorem 4.1.1 of [Rob03], with the hypothesis of non arithmeticity
of the spectrum. We prove all the analog results in the case of the free group Fr of rank r even
if Fr is not the fundamental group of a closed manifold, and may have an arithmetic spectrum.

1. Introduction
In this paper, we consider the action of the free group Fr on its boundary B, a probability
space associated to the Cayley graph of Fr relative to its canonical generating set. This action
is known to be ergodic (see for example [FTP82] and [FTP83]), but since the measure is not
preserved, no theorem on the convergence of means of the corresponding unitary operators had
been proved. Note that a close result is proved in [FTP83, Lemma 4, Item (i)].
We formulate such a convergence theorem in Theorem 1.2. We prove it following the ideas of
[BM11] and [Boy15] replacing [Rob03, Theorem 4.1.1] by Theorem 1.1.
1.1. Geometric setting and notation. We will denote Fr = ha1 , ..., ar i the free group on r
±1
generators, for r ≥ 2. For an element γ ∈ Fr , there is a unique reduced word in {a±1
1 , ..., ar }
which represents it. This word is denoted γ1 · · · γk for some integer k which is called the length of
γ and is denoted by |γ|. The set of all elements of length k is denoted Sn and is called the sphere
of radius k. If u ∈ Fr and k ≥ |u|, let us denote P ru (k) := {γ ∈ Fr | |γ| = k, u is a prefix of γ}.
±1
Let X be the Cayley graph of Fr with respect to the set of generators {a±1
1 , ..., ar }, which is
a 2r-regular tree. We endow it with the (natural) distance, denoted by d, which gives length 1
to every edge ; for this distance, the natural action of Fr on X is isometric and freely transitive
on the vertices ; the space X is uniquely geodesic, the geodesics between vertices being finite
sequences of successive edges. We denote by [x, y] the unique geodesic joining x to y.
We fix, once and for all, a vertex x0 in X. For x ∈ X, the vertex of X which is the closest to x
in [x0 , x], is denoted by ⌊x⌋ ; because the action is free, we can identify ⌊x⌋ with the element γ
that brings x0 on it, and this identification is an isometry.
The Cayley tree and its boundary. As for any other CAT(−1) space, we can construct a boundary of X and endow it with a distance and a measure. For a general construction, see [Bou95].
The construction we provide here is elementary.
±1
Let us denote by B the set of all right-infinite reduced words on the alphabet {a±1
1 , ..., ar }.
This set is called the boundary of X.
We will consider the set X := X ∪ B.
For u = u1 · · · ul ∈ Fr \ {e}, we define the sets
Xu := {x ∈ X | u is a prefix of ⌊x⌋}
Bu := {ξ ∈ B | u is a prefix of ξ}
2010 Mathematics Subject Classification. Primary 37; Secondary 43, 47.
Key words and phrases. boundary representations, ergodic theorems, irreducibility, equidistribution, free
groups.
Weizmann Institute of Science, aadrien.boyer@gmail.com.
Universit d’Aix-Marseille, CNRS UMR7373, a.p.lobos@outlook.com.
1

AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP

2

Cu := Xu ∪ Bu
We can now define a natural topology on X by choosing as a basis of neighborhoods
(1) for x ∈ X, the set of all neighborhoods of x in X
(2) for ξ ∈ B, the set {Cu | u is a prefix of ξ}
For this topology, X is a compact space in which the subset X is open and dense. The
induced topology on X is the one given by the distance. Every isometry of X continuously
extend to a homeomorphism of X.
Distance and measure on the boundary. For ξ1 and ξ2 in B, we define the Gromov product
of ξ1 and ξ2 with respect to x0 by
(ξ1 |ξ2 )x0 := sup {k ∈ N | ξ1 and ξ2 have a common prefix of length k}
and

dx0 (ξ1 , ξ2 ) := e−(ξ1 |ξ2 )x0 .
Then d defines an ultrametric distance on B which induces the same topology ; precisely, if
ξ = u1 u2 u3 · · · , then the ball centered in ξ of radius e−k is just Bu1 ...uk .
On B, there is at most one Borel regular probability measure which is invariant under the
isometries of X which fix x0 ; indeed, such a measure µx0 must satisfy
1
µx0 (Bu ) =
2r(2r − 1)|u|−1

and it is straightforward to check that the ln(2r − 1)-dimensional Hausdorff measure verifies
this property.
If ξ = u1 · · · un · · · ∈ B, and x, y ∈ X, then (d(x, u1 · · · un ) − d(y, u1 · · · un ))n∈N is stationary.
We denote this limit βξ (x, y). The function βξ is called the Busemann function at ξ.
Let us denote, for ξ ∈ B and γ ∈ Fr the function
P (γ, ξ) := (2r − 1)βξ (x0 ,γx0 )
The measure µx0 is, in addition, quasi-invariant under the action of Fr . Precisely, the RadonNikodym derivative is given for γ ∈ Γ and for a.e. ξ ∈ B by
dγ∗ µx0
(ξ) = P (γ, ξ),
dµx0
where γ∗ µx0 (A) = µx0 (γ −1 A) for any Borel subset A ⊂ B.
The quasi-regular representation. Denote the unitary representation, called the quasi-regular
representation of Fr on the boundary of X by
π : Fr → U (L2 (B))
γ 7→ π(γ)
defined as


1
π(γ)g (ξ) := P (γ, ξ) 2 g(γ −1 ξ)
for γ ∈ Fr and for g ∈ L2 (B). We define the Harish-Chandra function
Z
1
P (γ, ξ) 2 dµx0 (ξ),
(1.1)
Ξ(γ) := hπ(γ)1B , 1B i =
B

where 1B denotes the characteristic function on the boundary.
For f ∈ C(X), we define the operators
1 X
π(γ)g
(1.2)
Mn (f ) : g ∈ L2 (B) 7→
f (γx0 )
∈ L2 (B).
|Sn |
Ξ(γ)
γ∈Sn

We also define the operator
(1.3)

M (f ) := m(f|B )P1B

where m(f|B ) is the multiplication operator by f|B on L2 (B), and P1B is the orthogonal projection on the subspace of constant functions.

AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP

3

Results. The analog of Roblin’s equidistribution theorem for the free group is the following.
Theorem 1.1. We have, in C(X × X)∗ , the weak-∗ convergence
1 X
Dγx0 ⊗ Dγ −1 x0 ⇀ µx0 ⊗ µx0
|Sn |
γ∈Sn

where Dx denotes the Dirac measure on a point x.
Remark 1. It is then straightforward to deduce the weak-∗ convergence
X
Dγx0 ⊗ Dγ −1 x0 ⇀ µx0 ⊗ µx0
kmΓ ke−δn
|γ|≤n

mΓ denoting the Bowen-Margulis-Sullivan measure on the geodesic flow of SX/Γ (where SX is
the “unit tangent bundle”) and δ denoting ln(2r − 1), the Hausdorff measure of B.
(1) Notice that in our case, the spectrum is Z so the geodesic flow is not topologically mixing,
according to [Dal99] or directly by [CT01, Ex 1.3].
(2) Notice also that our multiplicative term is different of that of [Rob03, Theorem 4.1.1],
which shows that the hypothesis of non-arithmeticity of the spectrum cannot be removed.
We use the above theorem to prove the following convergence of operators.
Theorem 1.2. We have, for all f in C(X), the weak operator convergence
Mn (f ) −→ M (f ).
n→+∞

In other words, we have, for all f in C(X) and for all g, h in L2 (B), the convergence
1 X
hπ(γ)g, hi
f (γx0 )
−→ hM (f )g, hi.
n→+∞
|Sn |
Ξ(γ)
γ∈Sn

We deduce the irreducibility of π, and give an alternative proof of this well known result (see
[FTP82, Theorem 5]).
Corollary 1.3. The representation π is irreducible.
Proof. Applying Theorem 1.2 to f = 1X shows that the orthogonal projection onto the space of
constant functions is in the von Neumann algebra associated with π. Then applying Theorem
1.2 to g = 1B shows that the vector 1B is cyclic. Then, the classical argument of [Gar14,
Lemma 6.1] concludes the proof.

Remark 2. For α ∈ R∗+ , let us denote by Wα the wedge of two circles, one of length 1 and
the other of length α. Let p : Tα ։ Wα the universal cover, with Tα endowed with the distance making p a local isometry. Then F2 ≃ π1 (Wα ) acts freely properly discontinously and
cocompactly on the 4-regular tree Tα (which is a CAT(-1) space) by isometries. For α ∈ R \ Q,
the analog of Theorem 1.2 for the quasi-regular representation πα of F2 on L2 (∂Tα , µα ) for a
Patterson-Sullivan measure associated to a Bourdon distance is known to hold ([Boy15]) because
[Rob03, Theorem 4.1.1] is true in this setting. Now if α1 and α2 are such that α1 6= α±1
2 , then
the representations πα are not unitarily equivalent ([Gar14, Theorem 7.5]). For α ∈ Q∗+ \ {1}, it
would be interesting to formulate and prove an equidistribution result like Theorem 1.1 in order
to prove Theorem 1.2 for πα .
2. Proofs
2.1. Proof of the equidistribution theorem. For the proof of Theorem 1.1, let us denote


Z


X
1
f (γx0 , γ −1 x0 ) →
E := f : C(X × X) |
f d(µx0 ⊗ µx0 )


|Sn |
X×X
γ∈Sn

The subspace E is clearly closed in C(X × X) ; it remains only to show that it contains a
dense subspace of it.

AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP

4

Let us define a modified version of certain characteristic functions : for u ∈ Fr we define

 max{1 − dX (x, Cu ), 0} if x ∈ X
0
if x ∈ B \ Bu
χu (x) :=

1
if x ∈ Bu

It is easy to check that he function χu is a continuous function which coincides with χCu on
Fr x0 and B.
The proof of the following lemma is straightforward.
X
Lemma 2.1. Let u ∈ Fr and k ≥ |u|, then χu −
χγ has compact support included in X.
γ∈P ru (k)

Proposition 2.2. The set χ := {χu | u ∈ Fr \ {e}} separates points of B, and the product
of two such functions of χ is either in χ, the sum of a function in χ and of a function with
compact support contained in X, or zero.
Proof. It is clear that χ separates points. It follows from Lemma 2.1 that χu χv = χv if u is a
proper prefix of v, that χ2u − χu has compact support in X, and that χu χv = 0 if none of u and
v is a proper prefix of the other.

Proposition 2.3. The subspace E contains all functions of the form χu ⊗ χv .
Proof. We make the useful observation that
|Snu,v |
1 X
(χu ⊗ χv )(γx0 , γ −1 x0 ) =
|Sn |
|Sn |
γ∈Sn

Snu,v

where
is the set of reduced words of length n with u as a prefix and v −1 as a suffix. We
easily see that this set is in bijection with the set of all reduced words of length n − (|u| + |v|)
that do not begin by the inverse of the last letter of u, and that do not end by the inverse of
±1
the first letter of v −1 . So we have to compute, for s, t ∈ {a±1
1 , ..., ar } and m ∈ N, the cardinal
of the set Sm (s, t) of reduced words of length m that do not start by s and do not finish by t.
Now we have
Sm = Sm (s, t) ∪ {x | |x| = m and starts by s} ∪ {x | |x| = m and ends by t}.
Note that the intersection of the two last sets is the set of words both starting by s and
ending by t, which is in bijection with Sm−2 (s−1 , t−1 ).
We have then the recurrence relation :
|Sm (s, t)| = 2r(2r − 1)m−1 − 2(2r − 1)m−1 + |Sm−2 (s−1 , t−1 )|
m−3 + |S
= 2(r − 1)(2r − 1)m−1 + 2(r − 1)(2r
m−4 (s, t)|
 − 1)
.
2
2(r
−
1)
(2r
−
1)
+
1
+
|S
(s,
t)|
= (2r − 1)m
m−4
(2r − 1)3
We set C :=

2(r−1)((2r−1)2 +1)
,
(2r−1)3

n = 4k + j with 0 ≤ j ≤ 3 and we obtain

s,t
s,t
|
|S4k+j
| = C(2r − 1)4k+j + |S4(k−1)+j
s,t
4k+j
4(k−1)+j
|
= C(2r − 1)
+ C(2r − 1)
+ |S4(k−2)+j

= C

k
X
(2r − 1)4i+j + |Sjs,t |
i=1

= C(2r − 1)4+j

= (2r − 1)1+j
Now we can compute

(2r − 1)4k − 1
+ |Sj (s, t)|
(2r − 1)4 − 1

(2r − 1)4k − 1
+ |Sj (s, t)|
2r

AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP
u,v
|S4k+j
|

|S4k+j |

−1
)
S4k+j−(|u|+|v|)(u|u| , v|v|

=

|S4k+j |
(2r − 1)1+j

=
=

5

(2r − 1)4k−(|u|+|v|) − 1
−1
)
+ Sj (u|u| , v|v|
2r
2r(2r − 1)4k+j−1

1
1
+ o(1)
|u|−1
2r(2r − 1)
2r(2r − 1)|v|−1

= µx0 (Bu )µx0 (Bv ) + o(1)
when k → ∞, and this proves the claim.



Corollary 2.4. The subspace E is dense in C(X × X).
Proof. Let us consider E ′ , the subspace generated by the constant functions, the functions which
can be written as f ⊗ g where f, g are continuous functions on X and such that one of them
has compact support included in X, and the functions of the form χu ⊗ χv . By Proposition
2.2, it is a subalgebra of C(X × X) containing the constants and separating points, so by the
Stone-Weierstraß theorem, E ′ is dense in C(X × X). Now, by Proposition 2.3, we have that
E ′ ⊆ E, so E is dense as well.

2.2. Proof of the ergodic theorem. The proof of Theorem 1.2 consists in two steps:
Step 1: Prove that the sequence Mn is bounded in L(C(X), B(L2 (B))).
Step 2: Prove that the sequence converges on a dense subset.
2.2.1. Boundedness. In the following 1X denotes the characteristic function of X. Define


Fn := Mn (1X ) 1B .

We denote by Ξ(n) the common value of Ξ on elements of length n.
1
P
(P (γ, ξ)) 2 is constant equal to |Sn | × Ξ(n).
Corollary 2.5. The function ξ 7→
γ∈Sn

Proof. This function is constant on orbits of the action of the group of automorphisms of X
fixing x0 . Since it is transitive on B, the function is constant. By integrating, we find
Z X
X
1
1
2
(P (γ, ξ))
(P (γ, ξ)) 2 dµx0 (ξ)
=
B γ∈S
γ∈Sn
n
XZ
1
=
(P (γ, ξ)) 2 dµx0 (ξ)

B
γ∈S
Xn
Ξ(n)
=
γ∈Sn

= |Sn |Ξ(n),
Lemma 2.6. The function Fn is constant, equal to 1B .
Proof. Because Ξ depends only on the length, we have that
1

Fn (ξ) :=
=

1 X (P (γ, ξ)) 2
|Sn |
Ξ(γ)
γ∈Sn
X
1
1
(P (γ, ξ)) 2
|Sn |Ξ(n)
γ∈Sn

=

1,

and the proof is done.



AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP

6

It is easy to see that Mn (f ) induces continuous linear transformations of L1 and L∞ , which
we also denote by Mn (f ).
Proposition 2.7. The operator Mn (1X ), as an element of L(L∞ , L∞ ), has norm 1; as an
element of B(L2 (B)), it is self-adjoint.
Proof. Let h ∈ L∞ (B). Since Mn (1X ) is positive, we have that




Mn (1X ) h ∞ ≤
Mn (1X ) 1B ∞ khk∞
= kFn k∞ khk∞
= khk∞
so that kMn (1X )kL(L∞ ,L∞ ) ≤ 1.
The self-adjointness follows from the fact that π(γ)∗ = π(γ −1 ) and that the set of summation
is symmetric.

Let us briefly recall one useful corollary of Riesz-Thorin’s theorem :
Let (Z, µ) be a probability space.
Proposition 2.8. Let T be a continuous operator of L1 (Z) to itself such that the restriction T2
to L2 (Z) (resp. T∞ to L∞ (Z)) induces a continuous operator of L2 (Z) to itself (resp. L∞ (Z)
to itself ).
Suppose also that T2 is self-adjoint, and assume that kT∞ kL(L∞ (Z),L∞ (Z)) ≤ 1.
Then kT2 kL(L2 (Z),L2 (Z)) ≤ 1.
Proof. Consider the adjoint operator T ∗ of (L1 )∗ = L∞ to itself. We have that
kT ∗ kL(L∞ ,L∞ ) = kT kL(L1 (Z),L1 (Z)) .
Now because T2 is self-adjoint, it is easy to see that T ∗ = T∞ . This implies
1 ≥ kT ∗ kL(L∞ ,L∞ ) = kT kL(L1 (Z),L1 (Z)) .
Hence the Riesz-Thorin’s theorem gives us the claim.

Proposition 2.9. The sequence (Mn )n∈N is bounded in L(C(X), B(L2 (B))).
Proof. Because Mn (f ) is positive in f , we have, for every positive g ∈ L2 (B), the inequality
−kf k∞ [Mn (1X )]g ≤ [Mn (f )]g ≤ kf k∞ [Mn (1X )]g
from which we deduce, for every g ∈ L2 (B)
k[Mn (f )]gkL2

≤ kf k∞ k[Mn (1X )]gkL2
≤ kf k∞ kMn (1X )kB(L2 ) kgkL2

which allows us to conclude that
kMn (f )kB(L2 ) ≤ kMn (1X )kB(L2 ) kf k∞ .
This proves that kMn kL(C(X),B(L2 )) ≤ kMn (1X )kB(L2 ) .
Now, it follows from Proposition 2.7 and Proposition 2.8 that the sequence (Mn (1X ))n∈N is
bounded by 1 in B(L2 ), so we are done.

2.2.2. Estimates for the Harish-Chandra function. The values of the Harish-Chandra are known
(see for example [FTP82, Theorem 2, Item (iii)]). We provide here the simple computations we
need.
We will calculate the value of
Z
1
P (γ, ξ) 2 dµx0 (ξ).
hπ(γ)1B , 1Bu i =
Bu

AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP

7

Lemma 2.10. Let γ = s1 · · · sn ∈ Fr . Let l ∈ {1, ..., |γ|}, and u = s1 · · · sl−1 tl tl+1 · · · tl+k 1, with
tl 6= sl and k ≥ 0, be a reduced word. Then
1
hπ(γ)1B , 1Bu i =
|γ|
2r(2r − 1) 2 +k
and
2r − 1
hπ(γ)1B , 1Bγ i =
|γ|
2r(2r − 1) 2
Proof. The function ξ 7→ βξ (x0 , γx0 ) is constant on Bu equal to 2(l − 1) − |γ|.
So hπ(γ)1B , 1Bu i is the integral of a constant function:


Z
1
log(2r−1) (l−1)− |γ|
2
P (γ, ξ) 2 dµx0 (ξ) = µx0 (Bu ) e
Bu

1

=

2r(2r − 1)

|γ|
+k
2

·

The value of hπ(γ)1B , 1Bγ i is computed in the same way.



Lemma 2.11. (The Harish-Chandra function)
Let γ = s1 · · · sn in Sn written as a reduced word. We have that


|γ|
r−1
|γ| (2r − 1)− 2 .
Ξ(γ) = 1 +
r
Proof. We decompose B into the following partition:

B=

G

Bu1

u1 6=s1

|γ|
G

⊔


l=2

G

u=s1 ···sl−1 tl
tl 6∈{sl ,(sl−1 )−1 }





Bu  ⊔ Bγ


and Lemma 2.10 provides us the value of the integral on the subsets forming this partition.
A simple calculation yields the announced formula.

The proof of the following lemma is then obvious :
Lemma 2.12. If γ, w ∈ Fr are such that w is not a prefix of γ, then there is a constant Cw not
depending on γ such that
Cw
hπ(γ)1B , 1Bw i
≤
.
Ξ(γ)
|γ|
2.2.3. Analysis of matrix coefficients. The goal of this section is to compute the limit of the
matrix coefficients hMn (χu )1Bv , 1Bw i.
Lemma 2.13. Let u, w ∈ Fr such that none of them is a prefix of the other (i.e. Bu ∩ Bw = ∅).
Then
lim hMn (χu )1B , 1Bw i = 0
n→∞

Proof. Using Lemma 2.12, we get
hMn (χu )1B , 1Bw i =
=

hπ(γ)1B , 1Bw i
1 X
χu (γx0 )
|Sn |
Ξ(γ)
γ∈Sn
X
hπ(γ)1B , 1B i
1

|Sn |
Ξ(γ)
γ∈Cu ∩Sn
X
Cw
1
≤
|Sn |
|γ|
 γ∈C
 u ∩Sn
1
= O
n
1For l = 1, s · · · s
1
l−1 is e by convention.

w

AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP

8


Lemma 2.14. Let u, v ∈ Fr . Then
lim suphMn (χu )1Bv , 1B i ≤ µx0 (Bu )µx0 (Bv )
n→∞

Proof.

hMn (χu )1Bv , 1B i = hMn (χu )∗ 1B , 1Bv i
hπ(γ)1B , 1Bv i
1 X
χu (γ −1 x0 )
=
|Sn |
Ξ(γ)
γ∈Sn
1 X
≤
χu (γ −1 x0 )χv (γx0 )
|Sn |
γ∈Sn
hπ(γ)1B , 1Bv i
1 X
χu (γ −1 x0 )
+
|Sn |
Ξ(γ)
γ∈Sn
γ6∈Cv

1 X
χu (γ −1 x0 )χv (γx0 )
|Sn |
γ∈Sn

1
+O
n
Hence, by taking the lim sup and using Theorem I, we obtain the desired inequality.
=



Proposition 2.15. For all u, v, w ∈ Fr , we have
lim hMn (χu )1Bv , 1Bw i = µx0 (Bu ∩ Bw )µx0 (Bv )

n→∞

Proof. We first show the inequality
lim suphMn (χu )1Bv , 1Bw i ≤ µx0 (Bu ∩ Bw )µx0 (Bv ).
n→∞

If none of u and w is a prefix of the other, we have nothing to do according to Lemma 2.13.
Let us assume that u is a prefix of w (the other case can be treated analogously). We have, by
Lemma 2.14, that
µx0 (Bw )µx0 (Bv ) ≥ lim suphMn (χw )1Bv , 1B i
n→∞

≥ lim suphMn (χw )1Bv , 1Bw i
n→∞

≥ lim suphMn (χw )1Bv , 1Bw i +
n→∞

X

lim suphMn (χγ )1Bv , 1Bw i

γ∈P ru (|w|)\{w}

n→∞

= lim suphMn (χu )1Bv , 1Bw i
n→∞

We now compute the expected limit. Let us define
Su,v,w := {(u′ , v ′ , w′ ) ∈ Fr | |u| = |u′ |, |v| = |v ′ |, |w| = |w′ |}.
Then
1 = lim inf hMn (1X )1B , 1B i
n→∞

≤ lim inf hMn (χu )1Bv , 1Bw i +
n→∞

X

lim suphMn (χu′ )1Bv′ , 1Bw′ i

X

lim suphMn (χu′ )1Bv′ , 1Bw′ i

(u′ ,v′ ,w ′ )∈Su,v,w \{u,v,w}

≤ lim suphMn (χu )1Bv , 1Bw i +
n→∞

(u′ ,v′ ,w ′ )∈Su,v,w \{u,v,w}

X

≤ µx0 (Bu ∩ Bw )µx0 (Bv ) +

(u′ ,v′ ,w ′ )∈S

n→∞

n→∞

µx0 (Bu′ ∩ Bw′ )µx0 (Bv′ )

u,v,w \{u,v,w}

= 1
This proves that all the inequalities above are in fact equalities, and moreover proves that
the inequalities
lim inf hMn (χu )1Bv , 1Bw i ≤ lim suphMn (χu )1Bv , 1Bw i ≤ µx0 (Bu ∩ Bw )µx0 (Bv )
n→∞

n→∞

AN ERGODIC THEOREM FOR THE QUASI-REGULAR REPRESENTATION OF THE FREE GROUP

are in fact equalities.

9



Proof of Theorem 1.2. Because of the boundedness of the sequence (Mn )n∈N proved in Proposition 2.9, it is enough to prove the convergence for all (f, h1 , h2 ) in a dense subset of C(X) ×
L2 × L2 , which is what Proposition 2.15 asserts.

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