Special algebraic subvarieties in the case of
PSL(2, C)
Bertrand Deroin
August 2011
1
Introduction
The mini-course will be concerned with the study of holomorphic families of subgroups
of PSL(2, C), from a dynamical as well as an algebraic point of view. This later group
has been chosen for several reasons. First, most of the methods that will be used are
based on complex analysis/potential theory, so that we need a complex Lie group. The
second reason is that we want to be as elementary as possible, but in a setting where
interesting phenomenom occurs. Thus the choice of PSL(2, C), namely the first Lie
group which contains free subgroups, is very natural. However, many of the covered
results have generalisations to other Lie groups.
Consider a finitely generated marked subgroup of PSL(2, C), namely this is a finitely
generated group G together with a system of generators g1 , . . . , gr . The set
Hom(G, PSL(2, C))
consisting of all the representations of G in PSL(2, C) can be identified with a complex
algebraic submanifold Λ(G) ⊂ PSL(2, C)r , see section 2, which will be referred to as a
”special” submanifold, following a terminology proposed by Breuillard.
The special subvarieties are contained in the so-called bifurcation locus, whose
exterior is the set of structurally stable representations. Sullivan proved that the
stability locus corresponds to the set of representations where the isomorphism type
of the image of the representation does not change by small perturbations. In other
words, he proved that the union of the special subvarieties is dense in the bifurcation
locus. Most of the topic will be to answer the following question: how do the special
subvarieties fill in the bifurcation locus?
We first review some results of Glutsyuk and Kaloshin-Rodnianski concerning quantitative estimates on the speed at which a given representation can be approximated
by representations with relations of controlled length.
We then prove that given a special subvariety Λ(G), certain special subvarieties of
Λ(G) equidistribute in the whole bifurcation locus of Λ(G). Namely, if we consider a
word in the generators g1 , . . . , gr taken at random, and that we consider the quotient of
G obtained by adding as a relation the square of this word (or any power p > 1) then
the corresponding special subvarieties become more and more dense in the bifurcation
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locus, and equidistribute themselves towards a certain current in Λ(G). This is a result
that we obtained in collaboration with Dujardin.
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2
3
Algebraic instability
The automorphism group of the Riemann sphere P1 is the group of Möbius maps
preserving orientation, and acting on P1 by z 7→ az+b
. We think of an automorphism
cz+d
1
of P as an invertible 2 × 2-matrix modulo multiplication by a constant. Thus the
automorphism group of P1 identifies with PSL(2, C).
Elliptic, parabolic and loxodromic elements of PSL(2, C) are elements conjugated
to respectively, a rotation z 7→ eiθ z with θ ∈ R, the translation z 7→ z + 1, or a map
of the form z 7→ λz with |λ| =
6 0, 1. These types have different dynamical behaviour:
existence of diffuse invariant measure for the first, only one fixed point which attracts
every compact set in the future and in the past for the second, attractive/expanding
dynamics for the third. Observe that the set of loxodromic elements is a dense open
subset of PSL(2, C) (this is a crucial difference with PSL(2, R)!).
Let G be a finitely generated marked group, i.e. we have the additional data of
a system of generators of G. A representation of G in PSL(2, C) is a map ρ : G →
PSL(2, C) which is such that ρ(gg 0 ) = ρ(g)◦ρ(g 0 ) for every g, g 0 ∈ G. We denote Λ(G) =
Hom(G, PSL(2, C)) the set of representations of G in PSL(2, C). This is a complex
algebraic submanifold of PSL(2, C)r where r is the number of the finite generating
set of G. This is due to the fact that a r-tuple (Ar )r∈R defines a representation of G
if and only if the relations in G are satisfied by this tuple. Since these relations are
polynomial in the Ar ’s, and that an intersection of complex algebraic submanifolds is
an algebraic submanifold by the Hibert basis theorem, the claim follows.
When G is the non abelian free group Fr on r generators, an element of Λ(G) is
just the data of the images of those generators, namely Λ(G) ' PSL(2, C)r .
We will be mostly interested in the non elementary representations. An elementary representation is a representation which preserves a probability measure on the
Riemann sphere. The reader can check as an exercise that a non elementary representation is either conjugated to a subgroup of SU (2), or to a subgroup of the affine
groups consisting of transformations of the form z 7→ az + b, or to a subgroup of the
group generated by the inversion z 7→ −1/z and the linear maps z 7→ λz. Hence, the
set of elementary representations is a real algebraic submanifold of Λ(G).
2.1
Algebraic versus structural stability
Let {ρλ }λ∈Λ be a holomorphic family of representations of G in PSL(2, C), that is a
holomorphic map from a complex manifld Λ to Λ(G).
A parameter λ0 ∈ Λ is algebraically stable if for λ sufficiently close to λ0 , the kernels
of ρλ are constant for λ in a neighborhood of λ0 . Hence, the isomorphism types of the
image of ρλ does not change.
A parameter λ0 is called structurally stable if for every λ in a neighborhood of λ0 ,
the representation ρλ is topologically conjugated to ρλ0 . We define the stability locus
as the set of structurally stable representations. This is a closed subset of Λ(G). The
bifurcation locus is the complement of the stability locus. The following result is due
to Sullivan [16].
Theorem 2.1 (Equivalence between algebraic and structural stability). Let λ0 be a
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parameter corresponding to a non elementary representation (i.e. ρλ0 (G) does not
preserve a probability measure on P1 ). Then, λ0 is algebraically stable if and only if it
is structurally stable.
Proof. We only review here the main ideas of the proof, and refer to the original
article [16, Theorem 2] for details.
It is clear that structural stability implies algebraic stability. Hence we need to
prove that if λ0 is algebraically stable, and that ρλ0 is non elementary, then λ0 is
structurally stable.
Consider a neighborhood U of λ0 in Λ which is homeo to a ball and such that for
every λ ∈ U , the kernel of ρλ is the same as the kernel of ρλ0 . Then, observe that
for every g ∈ G, the types (elliptic, parabolic, loxodromic) of ρλ0 (g) and ρλ (g) are the
same (exercise! hint: use the fact that if for some g ∈ G the matrices ρλ (g) have not
a constant type in a neighborhood of λ0 , then in any neighborhood of λ0 there exist a
parameter λ such that ρλ (g) is a finite order elliptic element).
Then, if ρλ0 (g) is loxodromic, it has two fixed points, and these fixed points can
be followed analytically for λ ∈ U since ρλ (g) is always loxodromic, and since U is a
ball. The graphs of these fixed points in U × P1 determines a family of curves which
are transverse to the vertical fibration of U × P1 .
Observe that these curves are disjoint. This can be deduced by the following facts.
If two loxodromic elements have a common fixed point, then they generate a solvable
group. However, if two loxodromic elements do not have a common fixed point, then
the group generated by these elements is not solvable, since some powers of these
elements generate a Schottky group, which is free on 2 generators.
Hence, if F Lλ denotes the set of loxodromic fixed points of elements of ρλ (G), then
we have a map HM : U × F Lλ0 → P1 such that
1) HM (λ0 , ·) = id|F Lλ0
2) for every λ ∈ U , the map HM (λ, ·) is a bijection between F Lλ0 and F Lλ
3) for every p ∈ F Lλ0 , the map HM (·, p) is holomorphic.
Such a map is called a holomorphic motion. By construction, it is equivariant in the
following sense:
∀λ ∈ U, p ∈ F Lλ0 , g ∈ G
HM (λ, ρλ0 (g)p) = ρλ (g)HM (λ, p)
(1)
The proof is concluded by using the famous λ-lemma of Mañe/Sad/Sullivan, which
states that a holomorphic motion is indeed continuous. Our holomorphic motion can
thus be extended continuously to a holomorphic motion of the set ρλ0 (G), which is the
limit set of the representation ρλ0 .
To prove the structural stability on the whole sphere, we then have to extend our
topological conjugacy on the limit set to a topological conjugacy on the whole sphere.
Observe that this is not necessary for most representation in the bifurcation locus since
most of them have full limit set. The proof in general uses Ahlors finiteness theorem.
We refer to [16, p. 252].
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Corollary 2.2. Given any finitely generated group G, the set of special subvarieties of
Λ(G) is dense in the intersection of Λ(G) with the bifurcation locus.
We end the section by reviewing the classification of the stability locus obtained
by Sullivan in [16]. A discrete subgroup G ⊂ PSL(2, C) is called convex cocompact if
there exists a G-invariant convex subset C of the hyperbolic 3-space whose quotient
by G is compact.
The typical example of a convex cocompact group is a Schottky group. A Schottky
group is a group generated by Möbius maps A1 , . . . , Ar such that there exist disjoint
closed discs Ui , Vi in P1 such that Ai (Uic ) = Int(Vi ) for i = 1, . . . , r. A Schottky group is
free, in the sense that for every non trivial word ω ∈ Fr , the relation ω(A1 , . . . , Ar ) = I
holds. We give the proof of this fact, which relies on a so-called ”Ping-Pong” argument.
Recall that a word ω is non trivial if it does not contain successively a letter
S and its
1
inverse. Let us prove that ω(A) 6= I for such a word ω. Let p ∈ P \ (Ui ∪ Vi ).
We will prove that ω(A)(p) belongs to one of the Ui or Vi . To this end, define the
attracting and repelling balls of A = A±
i to be att(Ai ) = Vi and rep(Ai ) = Ui , and
−1
−1
att(Ai ) = Ui , rep(Ai ) = Vi . Observe that if A is one of the maps A±
i , and if a
point q ∈ P1 \ exp(A), then A(q) belongs to att(A). The first element A we apply
will then map p to the attracting basin of A. Then we apply a letter B different from
A. Hence the expanding basin of B is disjoint from the attracting basin of A, and so
BA(p) belongs to att(B). By induction, we prove with this argument that the point
p is mapped by a reduced word of the form X . . . BA to the basin of attraction of X.
Hence to a point different from p. We let as an exercise for the reader the fact that a
Schottky group is a discrete subgroup of PSL(2, C).
Other examples of convex cocompact groups are given by quasi-Fuchsian subgroups.
We will not describe them now. Let us state the following fundamental result of
Sullivan.
Theorem 2.3 (Stability locus). A stable finitely generated subgroup G ⊂ PSL(2, C) is
either rigid or convex cocompact.
In particular, if G is the free group, then the set stability locus consists of Schottky
representations.
2.2
Quantitative estimates
In this section, we are interested in measuring quantitatively the algebraic instability.
To this end, let us introduce a distance on Λ(G), for any marked finitely generated
subgroup G of PSL(2, C), as follows. Given two representations ρ1 , ρ2 ∈ Λ(G), let
||ρ1 − ρ2 || be the maximum of the numbers ||ρ1 (g) − ρ2 (g)|| where g is an element of
the generating set. Then, given a (non structurally stable) representation ρ ∈ Λ(G),
and a positive integer l, define
cl (ρ) = inf ||ρ0 − ρ||,
where ρ0 ∈ Λ(G) is a representation whose kernel contains an element which is not
contained in the kernel of ρ and which can be expressed as a word in the generators
and their inverses of length bounded by n. For a non stable representation ρ, the
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sequence cn (ρ) tends to 0 when n tends to infinity, but we are interested in precise
estimates for the behaviour of the sequence.
Let us mention the following result of Glutsyuk. It is also valid for more general Lie
groups, and indeed it gives the proof of the algebraic instability in higher dimensions.
See [9].
Theorem 2.4 (Quantitative instability). Let κ = ln 1.5/ ln 9, and let ρ ∈ Λ(Fr ) be a
representation with non discrete image. There is a constant c > 0 such that, for every
positive integer l, cn ≤ exp(cnκ ).
Here ||ρ0 − ρ|| means the maximum of the norms ||ρ(g) − ρ0 (g)||, g being a generator
of Fr .
question: are analogous estimates valid for a representation ρ in the boundary of
the stability locus?
Let us observe that a generic element of the component of Λ(G) containing the
representation id is faithful. Here generic means appart from a countable union of
algebraic subvarieties of positive codimension. Indeed, any non trivial element g ∈ G
defines an algebraic sub-manifold {ρ ∈ Λ(G) | ρ(g) = I} which is not the whole
component, hence necessarily a subvariety of positive codimension.
It is tempting to think of faithful representations as the ”irrational” ones, and the
non faithful as the ”rational” ones. This analogy is supported by the following example:
1
2
consider the family of representations ρα : Z → R which send the generators
0
0
and
of Z2 to 1 and α respectively. Then the faithful representations correspond
1
exactly to the α irrationals, and the non faithful ones to the rationals. Recall that
a real number α ∈ R is called diophantine if it is not well-approximable by rational
numbers. More precisely, if there exist constants C, ε > 0 such that for every prime
integers p, q with q > 0, we have |α − pq | ≥ Cq −2−ε . It is well-known that almost
every Lebesgue number is Diophantine (exercise!). This analogy suggests that most
representations are badly approximated. In this direction, we have the following result.
Theorem 2.5 (Bad approximation in Λ(G)). For Lebesgue-a.e. ρ ∈ Λ(G), there is a
constant α > 0 such that if n is large enough, cn ≥ exp(−αn2 ).
This result is certainly not satisfactory. One would like to have an exponential
bound, namely the existence of a constant α > 0 such that for n large,
cn ≥ e−αn ,
(2)
but this seems to be a difficult problem.
The way one can prove theorem 2.5 is by proving that for a Lebesgue generic
representation, the identity is badly approximated by words in the generators. Namely,
it is related to the following problem:
Given a Lebesgue generic ρ ∈ Λ(G), bound from below the quantity
dn = inf{||(g) − I|| where g ∈ G is such that length(g) ≤ n}
(3)
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Indeed, an element ρ ∈ Λ(G) being given, suppose that ρ0 is δ-close to A and such that
ρ0 (g) = I for a word g of length ≤ n. Then
dn ≤ ||ρ(g) − I|| ≤ ||ρ(g) − ρ0 (g)|| ≤ C n ||ρ0 − ρ||
for some constant C > 1 depending only on δ. This proves that
cn ≥ dn C −n .
(4)
Thus any bound from below of dn will give a bound from below of cn . We mention
here that Gamburd, Jacobson and Sarnak asked the following question in the case of
representations taking values in SU(2): is it true that for a.e. A, there exists α s.t. (2)
holds for n large enough? See [8].
By (4), theorem 2.5 follows from the following result.
Theorem 2.6 (Bad approximation by words). For Lebesgue a.e. ρ ∈ Λ(G), there
exists a constant α > 0 such that for n large enough dn ≥ exp(−αn2 ).
This theorem has been proved by Kaloshin and Rodnianski in [12] in the case of
SU(2). The proof that we give here follows the ideas from [4].
Proof. We will provide a proof in the case where there exists a convex cocompact
representation ρC ∈ Λ(G). This gives the result for G a free group or a surface group.
This is instructive that the result, which typically concerns dense representations, can
be proved by using a discrete one.
The proof is a quite straightforward consequence of (non trivial) estimates of the
volume of sublevel sets of psh functions. The precise result that we use is the following
estimates due to Hörmander.
Lemma 2.7 (Volume of sublevel sets of psh functions). Let B be a complex manifold
equipped with a volume form v, K be a compact subset of B, and p be a point of B.
Let a < b be some numbers, and ψ be a psh function defined on B, which is such that
ψ(p) > a and ψ < b on B.
Then:
(5)
Z
exp(−cψ)dv ≤ d,
K
where c, d > 0 are constants depending only on B, K, p, a, b.
The proof of this lemma uses the compactness (in L1loc ) of the set of psh functions
verifying (5) and [11, Theorem 4.4.5, p. 97].
We apply this lemma to the family of functions
u(ρ, g) :=
1
log ||ρ(g) − I||,
length(g)
which are psh functions, if for instance ||.|| is chosen to be the L2 -norm. We equip
Λ(G) with a volume v, and consider any compact subset K of Λ(G). We then choose
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a connected relatively compact open subset of Λ(G) which contains K and the convex
cocompact representation p = ρC . We have seen as an exercise that
u(p, g) > a,
where a is a constant which is independant of g. Because the norms of the image of the
generators are bounded on B (since B is relatively compact), there exists a constant
b > 0 such that
u(·, g) < b
on B. By applying lemma 2.7 to the functions ψ = u(·, g), we get
Z
exp(−cu(·, g))dv ≤ d.
K
The Markov inequality then shows that for any s > 0, we have
v({ρ ∈ K s.t. exp(−cu(ρ, g)) ≥ s}) ≤ d/s.
(6)
Let us introduce a parameter κ > 0 and let n = length(g) and s = exp(−κn). The
inequality (6) rewrites as
κ
v({ρ ∈ K s.t. ||ρ(g) − I|| ≤ − n2 }) ≤ d exp(−κn).
c
(7)
We choose κ such that the number of words of length n is bounded by exp(κn) exp(−n).
Then (7) gives
κ
v({ρ ∈ K s.t. there exists g of length n s.t ||ρ(g) − I|| ≤ − n2 }) ≤ d exp(−n).
c
This estimates together with Borel-Cantelli lemma shows the theorem with α = κ/c in
the case of PSL(2, C).
exercise: prove that the exponential bound dn ≥ exp(−αn) is not satisfied in a
Gδ -dense set of points in the bifurcation locus.
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9
Equidistribution of some special subvarieties
Let G be a finitely generated marked subgroup of PSL(2, C), and {ρλ }λ∈Λ be a holomorphic family of representations from G to PSL(2, C), where Λ is a connected complex
manifold. We will assume that
1) for every parameter λ ∈ Λ the representation ρλ is non elementary
2) the conjugacy class of ρλ is not constant.
Hence, there exists an element g ∈ G such that the function λ 7→ tr2 (ρλ (g)) is a non
constant holomorphic function.
For any number t ∈ C, let us introduce the following complex subvarieties of Λ:
Λg,t := {λ ∈ Λ such that tr2 (ρλ (g)) = t},
(8)
We will be only interested in those varieties corresponding to elements g such that
λ 7→ tr2 (ρλ (g)) is not constant, and we will assume that Λg,t is empty if this is not
the case (anyway, for every parameter t appart from a countable number, there is no
ambiguity with this convention). We would like to understand quantitatively how these
manifolds distribute themselves in the bifurcation locus (observe that when t ∈ [0, 4],
Λg,t is contained in the bifurcation locus) when g is a word of long length in the
generators of G.
We will use a random walk on the group G to produce a long word in the generators
of G. Namely, let µ be a probability measure on G (here µ is assumed to have finite
support) whose support generates G as a semi-group. For instance, µ could be chosen
to be the uniform measure on the union of the set of generators and their inverses. We
consider the product measure µN on GN and for a.e. g ∈ GN , we define
ln (g) := gn . . . g1 .
(9)
This is a left random product of n elements of G chosen independently with proba µ.
We will prove that a.s. the codimension 1 submanifolds Λln (g),t equidistributes towards
a current Tbif whose support is the whole bifurcation locus.
We obtained the following result in collaboration with Dujardin. See [4].
Theorem 3.1 (Equidistribution towards the bifurcation current). There exists a current Tbif on Λ, such that for every complex number t ∈ C, and µN -a.e. every g ∈ GN ,
the following convergence holds
1
Λl (g),t −→n→∞ Tbif
2n n
(10)
in the sense of currents. Moreover, the support of Tbif is the intersection of bifurcation
locus with Λ.
We deduce immediately from theorem 3.1 the following
Corollary 3.2. For every t ∈ C, every compact set K contained in the bifurcation
locus, and every ε > 0, there exists an element g ∈ G such that Λg,t is ε-dense in K.
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Observe that if t = 4 cos2 (2πq/p) with p, q relatively prime numbers, and p > 1,
and Λ = Λ(G), then Λg,t consists of irreducible components of the special variety
Λ(G0 ), where G0 is the smallest quotient of the group G containing g p in its kernel.
Thus, theorem 3.1 gives a result concerning equidistribution for some families of special
subvarieties.
3.1
A review on random matrix products
The theory of random matrices has two aspects, which are dependant to each other
and are often studied at the same time: one is to understand the matrices ρ(ln (g)) or
ρ(rn (g)) when n tends to infinity, the other is to understand the Markov process on the
Riemann sphere defined by the transition probabilities px = µ ∗ δx for x ∈ P1 (namely
how behave the sequences ρ(ln (g)(x) where x ∈ P1 ).
First recall the random ergodic theorem in our
R context. The markov process can be
studied by the convolution operator P (f )(x) = f (ρ(g)(x))dµ(g) acting on the space
of continuous functions. Such an operator acts dually on the space of measures by the
formula P ∗ (m) = µ ∗ m, by preserving the compact convex set of probability measures.
Hence by Kakutani’s theorem there exists an invariant measure ν. Such a measure is
called stationary. The random ergodic theorem states that if ν is extremal (in the set
of stationary measures), then for ν-a.e. x ∈ P1 and for µN -a.e. g), then the sequence
ρ(ln (g))(x) equidistributes itself with respect to ν.
A first fundamental result says that a.s. the norm of the matrices ρ(ln (g)) or
ρ(rn (g)) tends to infinity exponentially when n tends to infinity, as soon as the representation is non elementary. This result is due to Furstenberg. See [7]. Furstenberg
introduces the quantity
Z
1
log ||ρ(g)||dµn (g).
(11)
χ(ρ) := lim
n→∞ n
In fact, the subadditive theorem applied to the shift and the Bernouilli measure implies
that there exists a number χ(ρ) such that for µN -a.e. g, we have
1
log ||ρ(ln (g))||.
n
(12)
1
log ||ρ(rn (g))||.
n→∞ n
(13)
χ(ρ) = lim
n→∞
Indeed, one can also prove that
χ(ρ) = lim
This number is called the Lyapunov exponent. Furstenberg’s theorem can be restated
as
Theorem 3.3 (Positivity of Lyapunov exponent). If ρ is non elementary, then χ(ρ)
is positive.
Moreover, if z0 ∈ P1 is fixed, then for µN a.e. g,
lim
n→∞
1
log ||ρ(ln (g))Z0 || = χ(ρ),
n
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where Z0 denotes any lift of z0 ∈ P1 to C2 .
Furthermore, we have the formula
Z Z
||ρ(g)(Z)||
log
χ(ρ) =
dµ(g)dν(z)
||Z||
P1 G
(14)
(again Z is any lift of z).
Let us try to understand more in details the behaviour of the matrices ρ(ln (g))
and ρ(rn (g)) in terms of their so-called KAK-decomposition. Let us think about this
decomposition in geometric terms. For a matrix M ∈ SL(2, C) with a large norm,
one can find two balls att(M ) and rep(M ) in P1 of radius of the order of magnitude
of 1/||M ||, such that M (P1 \ rep(M )) = att(M ). Because of the positivity of the
Lyapunov exponent, a.s. the attracting and repelling balls of ρ(rn (g)) or ρ(ln (g))
shrink exponentially. The following heuristic shows that the attracting ball of ρ(rn (g))
and the repelling ball of ρ(ln (g)) tends respectively to some points θ(g) and θ̌(g) on
P1 : if we multiply a matrix of large norm M on the right by a matrix A of bounded
norm, we have
rep(AM ) ∼ rep(M )
and
att(M A) ∼ att(M ).
(15)
The functions θ, θ̌ : GN → P1 are called boundary maps. This terminology will be
justified later.
To understand the behaviour of the repelling balls of ρ(rn (g)) and ρ(ln (g)), one has
to use another heuristic: with M and A as before, we have
att(AM ) ∼ A · att(M )
and
rep(M A) ∼ A−1 · rep(M ).
(16)
Hence, one deduces that the attrating ball of ρ(ln (g)) distributes itself towards the
stationary measure ν, and the repelling ball of ρ(rn (g)) distributes itself towards the
stationary measure induced by the inverse process. In fact, we have the more precise
result
Proposition 3.4 (Behaviour of n-th step random matrix products). A.s. the repelling
ball of ρ(ln (g)) tends to a point θ̌(g) and the attracting ball equidistributes itself with
respect to the stationary measure ν. Moreover, for any ε > 0, the probability that the
attracting ball and the repelling ball of ln (g) are exp(−εn)-distant decreases exponentially.
This result can be deduced from the previous discussion and another fundamental
result concerning the convolution operator induced by the previously defined Markov
process on the sphere. A theorem of Le Page says that if m is a probability measure
on the sphere, then the measures µn ∗ m converges to a probability measure ν on the
sphere, and the convergence is exponential if tested on Hölder functions. See [14]. More
precisely
Theorem 3.5 (Exponential convergence of the convolution operator). If ρ is non
elementary, then there exists constants α, β, C, > 0 such that for every Hölder function
f : P1 → R of exponent α, and every integer n ≥ 0
Z
n
|P (f ) − f dν| ≤ C exp(−βn)||f ||α .
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As an exercise, the reader can prove the estimate 3.4 by using the exponential
convergence of P . This estimates was obtained by Aoun recently to obtain the following
nice result. See [1].
Theorem 3.6 (Random subgroups are Schottky groups). Let r ≥ 1 be an integer.
Then for (µN )r a.e. r-tuple (g1 , . . . , gr ), the group generated by ρλ (gn1 ), . . . , ρλ (gnr ) is a
Schottky group if n is large enough.
We will not give the proof of this result, but instead let it as an exercise assuming
proposition 3.4.
3.2
Equidistribution theorem
Here we prove the equidistribution statement in theorem 3.1, and postpone the study
of the support of Tbif for later. The proof relies on potential theory and on an estimate
due to Guivarc’h on the trace of a random matrix products. We only sketch it.
Let us recall the Poincaré-Lelong formula. Let D be a divisor, defined by an equation {f = 0}, where f is a holomorphic function. Then we have
[D] = ddc log |f |,
the last equality has to be understood in the sense of currents. Namely, for every
smooth (d − 1, d − 1)-form ω on Λ with compact support, we have
Z
Z
ω=
log |f |ddc ω.
D
Λ
The function log |f | is then called a potential of D. Then to prove that a sequence of
codimension 1 submanifolds Dn converges to a current, it suffices to prove that their
potential log |fn | converge in L1loc to a psh function which is locally integrable.
The potentials of Λg,t are the functions defined on Λ by
u(λ, g, t) := log |tr2 (ρλ (g)) − t|.
1
We thus have to prove that 2n
u(·, ln (g), t) converges in L1loc . The convergence at a
given parameter is an immediate consequence of the following convergence result of
Guivarc’h:
1
lim log |tr(ρλ (ln (g)))| = χ(λ).
(17)
n→∞ n
See [10].
Let us explain why Guivarc’h estimates (17) is true. Indeed, one has to prove that
the trace of a large random product of matrices is approximately its norm. This is not
always the case since for instance a parabolic map can have arbitrarily large norm, but
its trace is always 2. However, if a matrix of large norm in SL(2, C) is loxodromic and
its two fixed points are sufficiently separated, then its trace is approximately its norm:
more precisely we have the following estimates for a matrix A which is not parabolic
nor the identity:
p
|tr2 A − 4|
||A|| ∼ max(1,
)
δ
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where δ is the distance between the two fixed points of A. Here ∼ means equality up
to a bounded multiplicative error.
To deduce the convergence in L1 , we use the Hartogs lemma, an a priori bounds on
the L1 -norm of u(·, ln (g), t), and the fact that the Lyapunov exponent is continuous at
a non elementary parameter. We refer to [4] for more details.
3.3
The Poisson boundary and the double ergodicity theorem
First recall the definition of the Poisson boundary of the pair (G, µ). This is a measurable space P = P (G, µ) on which G acts by measurable maps, and it is equipped
with a measure ν such that µ ∗ ν = ν.
For its construction, we follow the presentation of Kaimanovich. Let us consider
the path space of the Markov process on G, consisting of the all random sequences
r ∈ GN . An initial measure m on G determines a Markov measure m, the image of
m × µN by the map
R(g) = (rn (g)).
Consider the smallest σ-algebra whose measurable sets are the measurable sets of the
path space which do not distinguish between two sequences that have the same tail:
(rn ) ∼ (rn0 ) if and only if rn = rn0 is n is large enough.
If the initial measure is the counting measure on G, then the shift (rn ) 7→ (rn+1 )
preserves the Markov measure m, and this is a measurable map with respect to the tail
algebra. The space of ergodic components of the shift is a measurable space called the
Poisson boundary, which is a quotient of the path space. The action of the group G on
the path space induces an action of G on the Poisson boundary by measurable maps.
The image of the Markov measure δe is a measure ν on P , which verifies µ ∗ ν = ν.
A result of Furstenberg states that if ρ is non elementary, then there is a unique
ρ-equivariant map θ from the Poisson boundary P to P1 . We borrow notations from
subsection 3.1.
Theorem 3.7 (Boundary map). The map
θ(g) = lim att(ρ(rn (g))),
n→∞
is defined for µN a.e. g, and induces a ρ-equivariant map θ : P → P1 .
remark: This map is indeed an isomorphism if ρλ is discrete and faithful, by a
theorem of Ledrappier. But in the case where ρλ is dense, it can be a strict quotient.
Applications to rigidity.
We conclude by the double ergodicity theorem of Kaimanovich:
Theorem 3.8 (Double ergodicity). The action of G on P (G, µ) × P (G, µ̌) is doubly
ergodic.
Proof. Let us sketch the proof of double ergodicity. Let us consider a G-invariant
function i : P (G, µ) × P (G, µ̌) → R, and prove that i is constant ν × ν̌ a.e. To a
given bi-infinite sequence g = (gn )n∈ , we associate two sequences g+ = (gn )n≥1 and
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Xavier’s birthday
−1
g− = (g−n
)n≥0 , hence two points r = r(g) and ř = r(ǧ) in the respective Poisson
boundaries P (G, µ) and P (G, µ̌). The map g 7→ (r, ř) sends the measure µZ on GZ
to a measure in the measure class of P (G, µ) × P (G, µ̌). Now, if σ(gn ) = (gn+1 ) is
the bilateral shift acting on GZ , we have the immediate formulas r(σg) = g1−1 r(g) and
ř(σg) = g1−1 ř(g). Hence we deduce that the function i(r, ř) on GZ is invariant under
the bilateral shift, hence constant µZ -a.e. by ergodicity. We conclude that i is almost
everywhere constant on P (G, µ) × P (G, µ̌).
3.4
The support theorem
Here we prove the second part of theorem 3.1. It is useful to introduce the notion of
a holomorphic variation of the Poisson boundary. We say that the Poisson boundary
varies holomorphically on an open subset U ⊂ Λ if the maps θλ can be glued together
to a map θ : U × P → P1 such that θ(λ, ·) = θλ a.e., and θ is holomorphic in the first
variable.
Proposition 3.9. If U is an open set where the bifurcation current vanishes, then
there exists on every relatively compact set of U a holomorphic variation of the Poisson
boundary.
Proof. We will be interested in the asymptotic behaviour of the graphs of the functions
f (g) : λ ∈ U 7→ ρλ (g)(z) ∈ P1 , where g is a typical random product of length n. Here
z is a point of the sphere that will be kept fixed. Our interest in those functions comes
from the fact that for a fixed parameter λ, f (rn (g))(λ) tends to θλ (g) when n tends
to infinity. The idea is that if the bifurcation current does not vanish, then typically
the graph of f (rn (g)) has a volume of the order of magnitude of n, but if not then its
volume is bounded in average independantly of n.
Suppose for simplicity that Λ has dimension 1. Let us observe that, if f = [f0 :
f1 ] : U → P1 is a holomorphic function on U , then the volume of its graph is given by
Z
ddc log(|f0 |2 + |f1 |2 ).
vol(graph(f )) = vol(U ) +
U
This is due to the fact that the Fubini-Study metric on P1 is given by ddc log(|z0 |2 +
|z1 |2 ), where z = [z0 , z1 ] is the affine coordinate. If we apply this to the functions f (g),
we get
Z
vol(graph(f (g))) = vol(U ) +
ddc log ||ρλ (g)(Z)||2 ,
U
z0
where Z =
.
z1
R
Now, by introducing the function f (z) = log ||ρ(g)(Z)||
dµ(g), one sees that
||Z||
Z
||ρλ (g)(Z)|| n
log
dµ (g) = f (z) + P (f )(z) + . . . + P n−1 (f )(z),
||Z||
so that by combining theorems 3.5 and 3.3, one obtain the estimates
Z
1
log ||ρλ (g)(Z)||dµn (g) = χ(λ) + O(1/n)
n
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when n tends to infinity, and the O(1/n) being uniform in every relatively compact set
of U . Thus, if χ is harmonic on U , then
Z
Z
n
vol(graph(f (g)))dµ (g) = vol(U ) +
ddc ϕn ,
U
where
Z
ϕn =
log ||ρλ (g)(Z)||2 − 2χ(λ).
is bounded on every relatively compact subset of U . Hence,
Z
vol(graph(f (g)|U 0 ))dµn (g)
is bounded on every relatively compact subset U 0 ⊂ U .
Then, the game consists in using Bishop theorem in order to extract from sequences
f (rn (g)) subsequences that converges to a holomorphic function interpolating λ 7→
θλ (g).
Proposition 3.10. A holomorphic variation of the Poisson boundary consisting of non
elementary representations is a holomorphic motion.
Proof. We give the proof for a symmetric measure µ. Observe that we can suppose
that U is of complex dimension 1. We will prove that the graphs of the functions
θ(., g), for g) in the Poisson boundary, do not intersect, or are equal. This will provide
the desired holomorphic motion.
For any compact set K ⊂ U , let us introduce the function iK : P (G, µ)2 → N
which assigns to a couple (g, g0 ) ∈ P (G, µ)2 the number of intersection points of the
graphs of the functions θ(., g) and θ(., g0 ) (this number can a priori be infinite). These
functions are invariant by the diagonal action of G, hence by the double ergodicity
theorem of Kaimanovich, they are constant almost every where to a constant iK ∈ N
(we let as an exercise the fact that this integer is indeed finite). The function K 7→ iK
can be extended to a measure with integer values. Such a measure is necessarily a
linear combination of Dirac masses supported on a finite set of points of U . Appart
from this finite set the holomorphic variation is a holomorphic motion. The reader can
check as an exercise that the holomorphic motion extends at these points.
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Bertrand Deroin
Université Paris-Sud & CNRS, Lab. de Mathématiques, Bât 425
91405 Orsay Cedex, France
[email protected]