Parametrizing Hitchin components

0.36 MB·English

by  Francis Bonahon

PARAMETRIZING HITCHIN COMPONENTS
2 FRANCISBONAHONANDGUILLAUMEDREYER
1
0
Abstract. We construct a geometric, real analytic parametrization of the
2
Hitchincomponent Hitn(S)ofthePSLn(R)–character varietyRPSLn(R)(S)of
p a closed surface S. The approach is explicit and constructive. In essence,
e ourparametrizationisanextensionofThurston’sshearingcoordinatesforthe
S Teichmu¨ller space of aclosedsurface, combined withFock-Goncharov’s coor-
dinates for the moduli space of positive framed local systems of a punctured
6
surface. More precisely, given a maximal geodesic lamination λ ⊂ S with
1
finitely many leaves, our coordinates are of two types, and consist of shear
invariants associatedwitheachleafofλ,andoftriangleinvariantsassociated
T] with each component of the complement S −λ. Besides, we compute and
describevariousidentitiesandrelationsbetween thesetwoinvariants.
G
.
h
t For a closed, connected, oriented surface S of genus g > 1, this article is con-
a
m cerned with the space of homomorphisms ρ: π1(S) → PSLn(R) from the funda-
mental group π (S) to the Lie group PSL (R) (equal to the special linear group
[ 1 n
SL (R) if n is odd, and to SL (R)/±Id if n is even), and more precisely with a
n n
1 preferred component Hit (S) of the character variety
n
v
6 RPSLn(R)(S)={homomorphisms ρ: π1(S)→PSLn(R)}//PSLn(R),
2
5 where the group PSL (R) acts on homomorphisms π (S)→PSL (R) by conjuga-
n 1 n
.3 tion. The precise definition of the character variety RPSLn(R)(S) requires that the
9 quotient be taken in the sense of geometric invariant theory [MFK]; however, for
0 the component Hit (S) that we are interested in, this quotient construction coin-
n
2 cides with the usual topological quotient [Hi]. Also, note that the consideration
1 of homomorphisms π (S) → PSL (R) is essentially equivalent, by arguments in-
: 1 n
v volvingthecohomologygroupsH1(S;R∗)andH1(S;Z ),totheanalysisofgeneral
2
i representations π (S)→GL (R).
X 1 n
r In the case where n = 2, the character variety RPSL2(R)(S) has 4g−3 compo-
a nents [Go ]. Two of these components correspond to all injective homomorphisms
1
ρ: π (S) → PSL (R) whose image ρ π (S) is discrete in PSL (R). Identifying
1 2 1 2
PSL2(R) with the orientation-preserv(cid:0)ing iso(cid:1)metry group of the hyperbolic plane
H2, the orientation of S then singles out one of these two components, where the
naturalequivalence relationS →H2/ρ π (S) has degree+1. This preferredcom-
1
ponent of RPSL2(R)(S) is the Teichmu¨ll(cid:0)er com(cid:1)ponent T(S). By the Uniformization
Theorem,theTeichmu¨llercomponentT(S)isdiffeomorphictothespaceofcomplex
structures on S, and consequently plays a fundamental rˆole in complex analysis as
well as in 2–dimensional hyperbolic geometry. In particular, a classical result is
that it is diffeomorphic to R6(g−1).
Date:September 18,2012.
ThisresearchwaspartiallysupportedbythegrantsDMS-0604866andDMS-1105402fromthe
NationalScienceFoundation.
1
2 FRANCISBONAHONANDGUILLAUMEDREYER
In the general case, there is a preferred homomorphism PSL (R) → PSL (R)
2 n
coming from the unique n–dimensional representation of SL (R) (or, equivalently,
2
from the natural action of SL2(R) on the vector space R[X,Y]n−1 ∼= Rn of homo-
geneouspolynomialsofdegreen−1intwovariables). Thisprovidesanaturalmap
RPSL2(R)(S) → RPSLn(R)(S). The Hitchin component Hitn(S) is the component of
RPSLn(R)(S)thatcontainsthe imageofthe Teichmu¨llercomponentofRPSL2(R)(S).
A Hitchin representation is a homomorphism ρ: π (S) → PSL (R) representing
1 n
an element of the Hitchin component. The terminology is motivated by the fol-
lowing fundamental result of N. Hitchin [Hi], who was the first to single out this
component.
Theorem 1 (Hitchin). When n >3, the character variety has 3 or 6 components
according to whether n is odd or even, and the Hitchin component Hit (S) is dif-
n
feomorphic to R2(g−1)(n2−1).
Hitchin’sproofoftheseresultsisbasedonthetheoryofHiggsbundles,andrelies
on techniques of geometric analysis. Hitchin notes in [Hi] that these methods do
notprovideanygeometricinformationonindividualHitchinrepresentations. Afew
yearsafterHitchin’swork,S.ChoiandW.Goldman[ChGo]showedthat,inthecase
where n = 3, the Hitchin component Hit (S) parametrizes the deformation space
n
of real convex projective structures on S. In particular, using this point of view,
Goldman [Go ] independently provided an explicit parametrization of the Hitchin
2
component Hit (S) by R16(g−1), via an extenion of the classical Fenchel-Nielsen
3
coordinates for the Teichmu¨ller space T(S).
A decade later, F. Labourie [La] showed, among other properties, that Hitchin
representationsareinjective andhavediscrete imagein PSL (R). He achievedthis
n
by establishing a very powerful Anosov property for Hitchin representations. This
Anosov property associates to each Hitchin representation ρ: π (S) → PSL (R) a
1 n
certainflag curve valuedinthe spaceFlag(Rn)ofallflagsinRn,whichisinvariant
under the imageρ π (S) ⊂PSL (R). The same invariantflag curvewassimilarly
1 n
providedbyindepe(cid:0)ndent(cid:1)workofV.FockandA.Goncharov[FoG],whoinaddition
established a certain positivity condition for this flag curve. This approach also
proves the faithfulness and the discreteness of Hitchin representations. The point
of view of Fock and Goncharov is algebaic geometric and relies on G. Lusztig’s
notion of positivity [Lu1, Lu2]; in particular, it is very different from Labourie’s.
Themainachievementofthecurrentpaperistoprovideanewparametrizationof
theHitchincomponentHit (S)byR2(g−1)(n2−1),whichismuchmorecloselyrelated
n
tothe geometryofHitchinrepresentationsthanHitchin’soriginalparametrization.
It relies on the methods developed by Labourie and Fock-Goncharov,and is some-
whatreminiscent of the classicalFenchel-Nielsencoordinates (see for instance [Hu,
§7.6]) for the Teichmu¨ller space T(S). When n = 2, this parametrization coin-
cides with the parametrization of the Teichmu¨ller space via the shear coordinates
associatedwithamaximalgeodesiclaminationλthatweredevelopedin[Th ,Bo ].
2 1
We begin with some topological data on the surface S, consisting of a maximal
geodesic lamination λ with finitely many leaves. For instance, in the spirit of the
Fenchel-Nielsen coordinates, such a geodesic lamination can be obtained from a
decomposition of S into pairs of pants, and by cutting each pair of pants along
three bi-infinite curves spiraling around the boundary components to obtain three
infinite triangles; the geodesic lamination λ then consists of the 3(g −1) closed
PARAMETRIZING HITCHIN COMPONENTS 3
(a) (b)
Figure 1. A finite geodesic lamination coming from a pair of
pants decomposition
curves of the pair of pants decomposition, together with the 6(g−1) spiraling bi-
infinite curves; see Figure 1(a) for an example, while Figure 1(b) illustrates how
to split a pair of pants along three spiraling curves to obtain two infinite triangles.
In general,for an arbitraryauxiliarymetric of negative curvature onS, a maximal
geodesiclaminationwithfinitelymanyleavesconsistsofsdisjointclosedgeodesics,
with16s63(g−1),andof6(g−1)disjointbi-infinitegeodesicswhoseendsspiral
around these closed geodesics and which split the surface S into 4(g−1) infinite
triangles.
Given a Hitchin representation ρ: π (S)→PSL (R), the Anosov structure dis-
1 n
covered by Labourie and the positivity property introduced by Fock-Goncharov
enableustoreadacertainnumberofinvariantsofρ. Theseinclude 1(n−1)(n−2)
2
real numbers (called triangle invariants) associated with each of the 4(g−1) tri-
angles of S −λ, and n−1 real numbers (called shear invariants) associated with
each of the 6(g−1)+s leaves of the geodesic lamination λ.
The triangle invariants, and the shear invariants associated with the infinite
leaves, were introduced by Fock and Goncharov in their parametrization [FoG] of
the so-calledmoduli space of positive framed localsystems of a surface S, where S
isrequiredtohaveatleastonepuncture. Thismodulispaceisthenaturalextension
of the Hitchin component to punctured surfaces; see [BAG] for the Higgs bundle
point of view on this space. The construction of the shear invariants associated
with closed leaves is very similar to that of infinite leaves.
A major difference with the punctured-surface case of Fock and Goncharov lies
in the fact that, when the surface S is closed, the triangle and shear invariants are
notindependentofeachother. Indeed,theysatisfyn−1linearequalitiesandn−1
linear inequalities for each of the s > 1 closed leaves of λ. It turns out that these
equalities and inequalities are the only relations satisfied by these invariants, and
that they can be used to parametrize Hit (S).
n
Theorem 2. The above triangle and shear invariants provide a real-analytic pa-
rametrization of theHitchin component Hit (S)by theinterior of aconvex polytope
n
of dimension 2(g−1)(n2−1).
In the special case where n=3, and where λ is a maximal lamination obtained
by adding spiraling bi-infinite curves to a pair of pants decomposition of S as in
Figure 1, our parametrization of Hit (S) is similar in spirit to the one developed
n
by Goldman [Go ], but different in its details.
2
When n = 2, there are no triangle invariants and, as indicated earlier, the
parametrizationofTheorem2coincideswiththeparametrizationoftheTeichmu¨ller
space T(S) by shear coordinates [Th , Bo ]. In the general case, because S −
2 1
λ consists of 4(g −1) triangles, the triangle invariants define a map Hit (S) →
n
4 FRANCISBONAHONANDGUILLAUMEDREYER
R2(g−1)(n−1)(n−2). It turns out that there are global linear relations between these
triangle invariants:
Proposition 3. The image of the map Hit (S) → R2(g−1)(n−1)(n−2) defined by
n
triangle invariants is contained in a linear subspace of codimension ⌊1(n−1)⌋ in
2
R2(g−1)(n−1)(n−2) (where ⌊x⌋ denotes the largest integer 6x).
The existence of constraints for the triangle invariants were somewhat unex-
pected to us. They can be explained by a more conceptual approach that uses the
lengthfunctionsof[Dr],combinedwithahomologicalargument;see[BoD]. Infact,
the abstract proof of [BoD] preceded the explicit computational argument that we
give in the current article.
1. Generic triples and quadruples of flags
The construction of our invariants of Hitchin representations heavily relies on
finite collections of flags in Rn.
1.1. Flags. A flag in Rn is a family F of nested linear subspaces F(0) ⊂ F(1) ⊂
···⊂F(n−1) ⊂F(n) of Rn where each F(a) has dimension a.
Apairofflags(E,F)isgeneric ifeverysubspaceE(a) ofE istransversetoevery
subspace F(b) of F. This is equivalent to the property that E(a)∩F(n−a) = 0 for
every a.
Similarly, a triple of flags (E,F,G) is generic if each triple of subspaces E(a),
F(b), G(c), respectively in E, F, G, meets transversely. Again, this is equivalent to
the property that E(a)∩F(b)∩G(c) =0 for every a, b, c with a+b+c=n.
1.2. Triple ratios ofgenericflag triples. Elementarylinearalgebrashowsthat,
′ ′
for any two generic flag pairs (E,F) and (E ,F ), there is a linear isomorphism
Rn →Rn sending E to E′ and F to F′. However, the same is not true for generic
flagtriples. Indeed,thereisawholemodulispaceofgenericflagtriplesmodulothe
action of PSL (R), and this moduli space can be parametrized by invariants that
n
we now describe. These invariants are expressed in terms of the exterior algebra
Λ(Rn) of Rn.
Let (E,F,G) be a generic flag triple. For each a, b, c between 0 and n, the
spaces Λa E(a) , Λb F(b) and Λc G(c) are all isomorphic to R. Choose non-zero
elements e(cid:0)(a) ∈(cid:1)Λa (cid:0)E(a) (cid:1), f(b) ∈(cid:0)Λb F(cid:1)(b) and g(c) ∈ Λc G(c) . We will use the
sameletterstodeno(cid:0)tethe(cid:1)irimagese(a(cid:0)) ∈Λ(cid:1)a(Rn),f(b) ∈Λb(cid:0)(Rn)(cid:1)andg(c) ∈Λc(Rn).
Given integers a, b, c > 1 with a+b+c = n, we then define the (a,b,c)–triple
ratio of the generic flag triple (E,F,G) as the number
e(a+1)∧f(b)∧g(c−1)
T (E,F,G)=
abc e(a−1)∧f(b)∧g(c+1)
e(a)∧f(b−1)∧g(c+1) e(a−1)∧f(b+1)∧g(c)
e(a)∧f(b+1)∧g(c−1) e(a+1)∧f(b−1)∧g(c)
where each of the six wedge products are elements of Λn(Rn) ∼= R. The fact that
the flag triple (E,F,G) is generic guarantees that these wedge products are non-
zero, so that the three ratios make sense. Also, because all the spaces Λa′ E(a′) ,
Λb′ F(b′) andΛc′ G(c′) involvedintheexpressionareisomorphictoR,th(cid:0)istrip(cid:1)le
rati(cid:0)o is i(cid:1)ndepende(cid:0)nt of(cid:1)the choice of the non-zero elements e(a′) ∈ Λa′ E(a′) ,
(cid:0) (cid:1)
PARAMETRIZING HITCHIN COMPONENTS 5
f(b′) ∈ Λb′ F(b′) and g(c′) ∈ Λc′ G(c′) ; indeed, each of these elements appears
twice in the(cid:0)expr(cid:1)ession, once in a n(cid:0)umer(cid:1)ator and once in a denominator.
The natural action of the linear group GL (R) on the flag variety Flag(Rn)
n
descends to an action of the projective linear group PGL (R), quotient of GL (R)
n n
by its center R∗Id consisting of all non-zero scalar multiples of the identity. Note
that the projective special linear group PSL (R) is equal to PGL (R) if n is odd,
n n
and is an index 2 subgroup of PGL (R) otherwise.
n
Proposition 4. Two generic flag triples (E,F,G) and (E′,F′,G′) are equivalent
under the action of PGL (R) if and only if T (E,F,G) = T (E′,F′,G′) for
n abc abc
every a, b, c>1 with a+b+c=n.
Inaddition, for anysetofnon-zeronumberst ∈R∗,thereexistsagenericflag
abc
triple(E,F,G)suchthatT (E,F,G)=t foreverya,b,c>1witha+b+c=n.
abc abc
Proof. See [FoG, §9]. (cid:3)
Note the elementary property of triple ratios under permutation of the flags.
Lemma 5.
T (E,F,G)=T (F,G,E)=T (F,E,G)−1. (cid:3)
abc bca bac
1.3. Quadruple ratios of generic flag triples. In addition to triple ratios, a
similar type of invariants of generic flag triples will play an important rˆole in our
analysis of Hitchin representations.
For an integer a with 16a6n−1, the a–th quadruple ratio of the generic flag
triple (E,F,G) is the number
e(a−1)∧f(n−a)∧g(1) e(a)∧f(1)∧g(n−a−1)
Q (E,F,G)=
a e(a)∧f(n−a−1)∧g(1) e(a−1)∧f(1)∧g(n−a)
e(a+1)∧f(n−a−1) e(a)∧g(n−a)
e(a+1)∧g(n−a−1) e(a)∧f(n−a)
where,asbefore,weconsiderarbitrarynon-zeroelementse(a′) ∈Λa′ E(a′) ,f(b′) ∈
Λb′ F(b′) andg(c′) ∈Λc′ G(c′) ,andwheretheratiosarecomputedi(cid:0)nΛn(R(cid:1)n)∼=R.
As(cid:0)with t(cid:1)riple ratios, the(cid:0)numb(cid:1)er Qa(E,F,G)∈R∗ is well-defined, independent of
choices,andinvariantundertheactionofPGL (R)onthesetofgenericflagtriples.
n
Note that Q (E,G,F)) = Q (E,F,G)−1, but that this quadruple ratio usually
a a
does not behave well under the other permutations of the flags E, F and G, as E
plays a special rˆole in Q (E,F,G).
a
By Proposition 4, a generic flag triple (E,F,G) is completely determined by its
triple ratios modulo the action of the linear group GL (R). It is therefore natural
n
to expect that the quadruple ratio can be expressed in terms of the triple ratios of
(E,F,G). This is indeed the case,and the correspondingexpressionis particularly
simple.
Lemma 6. For a=1, 2, ..., n−1,
Q (E,F,G)= T (E,F,G)
a abc
b+cY=n−a
where the product is over all integers b, c > 1 with b+c = n−a. In particular,
Qn−1(E,F,G)=1 and Qn−2(E,F,G)=T(n−2)11(E,F,G).
Proof. When computing the right-hand side of the equation, most terms e(a′) ∧
f(b′)∧g(c′) cancel out and we are left with the eight terms of Q (E,F,G). (cid:3)
a
6 FRANCISBONAHONANDGUILLAUMEDREYER
1.4. Double ratios of generic flag quadruples. We now consider quadruples
(E,F,G,G′) of flags E, F, G, G′ ∈ Flag(Rn). Such a flag quadruple is generic if
each quadruple of subspaces E(a), F(b), G(c), G′(d) meets transversely. As usual,
we can restrict attention to the cases where a+b+c+d=n.
For16a6n−1,thea–thdoubleratio ofthegenericflagquadruple(E,F,G,G′)
is
e(a)∧f(n−a−1)∧g(1) e(a−1)∧f(n−a)∧g′(1)
′
D (E,F,G,G)=−
a e(a)∧f(n−a−1)∧g′(1) e(a−1)∧f(n−a)∧g(1)
where we choose arbitrary non-zero elements e(a′) ∈ Λa′(E(a′)), f(b′) ∈ Λb′(F(b′)),
g(1) ∈ Λ1(G(1)) and g′(1) ∈ Λ1(G′(1)). As usual, D (E,F,G,G′) is independent
a
of these choices. The minus sign is motivated by the notion of positivity that is
describedin§1.5andplaysaveryimportantrˆoleinthisarticle(seeProposition10).
Lemma 7.
D (E,F,G′,G)=D (E,F,G,G′)−1
a a
and Da(F,E,G,G′)=Dn−a(E,F,G,G′)−1. (cid:3)
1.5. Positivity. A flag triple (E,F,G) is positive if it is generic and if all its
triple ratios T (E,F,G) are positive. By Proposition 4, positive flag triples form
abc
a component in the space of all generic flag triples. Lemma 5 also shows that
positivity of the triple (E,F,G) is preservedunder all permutations of the flags E,
F and G∈Flag(Rn).
′
Agenericflagquadruple(E,F,G,G)ispositive ifitisgeneric,ifthe twotriples
′ ′
(E,F,G) and (E,F,G) are positive, and if all double ratios D (E,F,G,G) are
a
positive.
See [FoG, Lu2, Lu1] for a more conceptual and general definition of positivity,
valid for k–tuples of (i.e. partial) flags.
2. Invariants of Hitchin representations
WenowdefineseveralinvariantsofHitchinrepresentationsρ: π (S)→PSL (R).
1 n
These invariants require that we are givena certaintopologicalinformationon the
surface.
2.1. The topological data. Sincewearegoingtousetheterminologyofgeodesic
laminations, it is convenient to endow the surface S with a riemannian metric of
negative curvature. However, it is well-known that geodesic laminations can also
be defined in a metric independent way, and in particular are purely topological
objects. See for instance [Th , PeH, Bo ].
1 2
Let λ be a maximal geodesic lamination with finitely many leaves. Namely λ
is the union of finitely many disjoint simple closed geodesics c , c , ..., c and of
1 2 s
finitelydisjointmanybi-infinitegeodesicsg ,g ,...,g inthecomplementofthec ,
1 2 t i
insucha waythat eachendofag spiralsalongsomec ,andthatthe complement
j i
S−λ consists of finitely many infinite triangles T , T , ..., T .
1 2 u
An Euler characteristic argument shows that, if g is the genus of the surface S,
then the number u of components of S−λ is equal to 4(g−1), while the number
t of infinite leaves of λ is equal to 6(g−1). The number s of closed leaves of λ can
be any integer between 1 and 3(g−1).
For instance, λ can be obtained from a family of disjoint simple closed curves
c1, c2,...,c3g−3 decomposingS intopairsofpants,andthenbydecomposingeach
PARAMETRIZING HITCHIN COMPONENTS 7
pair of pants into 2 infinite triangles along 3 infinite geodesicsspiraling aroundthe
boundary. Figure 1 describes one such example associated with a pair of pants
decomposition, and Figure 2 shows another example with only one closed leaf.
Figure 2. A finite geodesic lamination with exactly one closed leaf
Weneedmoredata,inadditiontothefinite-leavedmaximalgeodesiclamination
λ. One is the choice of an orientation on each leaf of λ. This choice is completely
free and arbitrary. In particular, we are not making any assumption of continuity
regardingtheseorientations,orofarelationshipbetweentheseorientationsandthe
directions in which infinite leaves spiral around closed leaves.
Finally, for each closed leaf c , we choose an arc k that is transverse to λ, cuts
i i
c in exactly one point, and meets no other closed leaf c .
i j
For the reader who is familiar with the case where n = 2, we can indicate that
thechoiceoforientationsfortheleavesofλisirrelevantinthatcase. Regardingthe
need for the transverse arcs k , it corresponds to a well-known technical difficulty
i
in the definition of the Fenchel-Nielsen coordinates: the twist parameters are rela-
tively easy to define modulo the length parameters, but require more cumbersome
topological information to be well-defined as real numbers.
2.2. The flag curve of a Hitchin representation. Oncewearegiventhistopo-
logicaldata,the keytoolforthe constructionofourinvariantsis the Anosovstruc-
ture for Hitchin representations discovered by F. Labourie [La]. Another good
reference for Labourie’s work is [Gui].
We begin with what is actually a corollary of these results.
Proposition 8 (Labourie). Let ρ: π (S)→PSL (R) be a Hitchin representation.
1 n
Then, for every non-trivial γ ∈π (S), the element ρ(γ)∈PSL (R) has real eigen-
1 n
values and their absolute values are distinct. (cid:3)
When n is even, the eigenvalues of ρ(γ) ∈ PSL (R) = SL (R)/{±Id} are only
n n
defined up to sign. However, we can be a little more specific.
Lemma 9. Let ρ: π (S)→PSL (R) be a Hitchin representation. Then, for every
1 n
non-trivial γ ∈ π (S), the element ρ(γ) ∈ PSL (R) admits a lift ρ(γ)′ ∈ SL (R)
1 n n
whose eigenvalues are distinct and all positive.
Proof. Let the non-trivial element γ ∈π (S) be fixed.
1
Theproperty“alleigenvaluesofaliftρ(γ)′ ∈SL (R)ofρ(γ)∈PSL (R)havethe
n n
same sign” is open and closed in the space of Hitchin representations, since these
eigenvalues are real and non-zero. This property holds in the special case where ρ
is the composition of a Teichmu¨ller representation ρ : π (S) →PSL (R) with the
2 1 2
naturalembeddingPSL (R)→PSL (R);indeed,ifρ (γ)hasaliftρ (γ)′ ∈SL (R)
2 n 2 2 2
with eigenvalues a and a−1, then ρ(γ) has a lift ρ(γ)′ ∈ SL (R) with eigenvalues
n
8 FRANCISBONAHONANDGUILLAUMEDREYER
an−2k+1 as k ranges over all integers with 1 6 k 6 n. Therefore, the property
holds for every Hitchin representation by connectedness of the space of Hitchin
representations.
This proves that the eigenvalues of any lift ρ(γ)′ ∈ SL (R) of ρ(γ) ∈ PSL (R)
n n
have the same sign. If these eigenvalues are all negative, note that n is even since
ρ(γ)′hasdeterminant+1. Then−ρ(γ)′ ∈SL (R)isanotherliftofρ(γ)∈PSL (R),
n n
whose eigenvalues are all positive (and distinct by Proposition 8). (cid:3)
IfρisaHitchinrepresentationandifγ ∈π (S)isnon-trivial,letρ(γ)′ ∈SL (R)
1 n
be the lift of ρ(γ)∈PSL (R) given by Lemma 9. Let
n
mρ(γ)>mρ(γ)>···>mρ(γ)>0
1 2 n
′
be the eigenvalues of ρ(γ), indexed in decreasing order. Since these eigenvalues
′
are distinct, ρ(γ) is diagonalizable. Let L be the (1–dimensional) eigenspace
a
corresponding to the eigenvalue mρ(γ).
a
This associates to ρ(γ) two preferred flags E, F ∈ Flag(Rn) defined by the
property that
a n
E(a) = L and F(a) = L .
b b
Mb=1 b=nM−a+1
By definition, E ∈ Flag(Rn) is the stable flag of ρ(γ) ∈ PSL (R), and F is its
n
unstable flag.
Let S be the universal covering of the surface S, and let ∂∞S be its circle at
infinity.eRecall that every non-trivial γ ∈ π1(S) fixes two pointseof ∂∞S, one of
them attracting and the other one repelling.
e
Proposition10(Labourie,Fock-Goncharov). GivenaHitchinrepresentationρ: π (S)→
1
PSLn(R), there exists a unique continuous map Fρ: ∂∞S →Flag(Rn) such that:
(1) ifx∈∂∞S istheattractingfixedpointofγ ∈π1(eS),thenFρ(x)∈Flag(Rn)
is the stable flag of ρ(γ)∈PSL (R);
e n
(2) F is equivariant with respect to the Hitchin homomorphism ρ: π (S) →
ρ 1
PSL (R), in the sense that F (γx) = ρ(γ)(x) for every γ ∈ π (S) and
n ρ 1
every x∈∂∞S;
(3) for any two deistinct points x, y ∈ ∂∞S, the flag pair Fρ(x),Fρ(y) is
generic; (cid:0) (cid:1)
e
(4) foranythreedistinctpointsx,y,z ∈∂∞S,theflagtriple Fρ(x),Fρ(y),Fρ(z)
is positive; (cid:0) (cid:1)
′ e
(5) for any four distinct points x, y, z, z occurring in this order around the
circle at infinity ∂∞S, the flag quadruple Fρ(x),Fρ(y),Fρ(z),Fρ(z′) is
positive. (cid:0) (cid:1) (cid:3)
e
By definition, this curve Fρ: ∂∞S → Flag(Rn) is the flag curve of the Hitchin
representationρ: π (S)→PSL (R).
1 n e
The first three properties of Proposition 10 are immediate consequences of the
Anosov structure of [La]. The positivity properties of the last two conditions of
Proposition 10 were proved by Fock and Goncharov [FoG]; see also the hypercon-
vexity property of [La, Gui].
PARAMETRIZING HITCHIN COMPONENTS 9
2.3. Invariants of triangles. Given a finite maximalgeodesic lamination λ as in
§2.1 and a Hitchin representation ρ: π (S) → PSL (R), the first set of invariants
1 n
of ρ is associated with the components of the complement S−λ. Recall that each
of these components is an ideal triangle.
Consider such a triangle T , and select one of its vertices v . (Such a vertex is
j j
of course not an actual point of the surface S; we let the reader devise a formal
definition for a vertex of the ideal triangle T ⊂ S.) Lift T to to an ideal triangle
j j
Tj in the universalcoveringS, and let vj ∈∂∞S be the vertex of Tj corresponding
′ ′′
teo the vertexvj ofTj. Labelethe verticeesofTj aes vj, vj andvj ∈∂e∞S in clockwise
order around T . We can then consider the flag triple F (v ),F (v′),F (v′′) ,
j e e e eρ j ρe j ρ j
which is positive by Proposition 10. (cid:0) (cid:1)
We can theneconsider the (positive) triple ratios of this posietive flagetriple,eand
their logarithms
τρ (T ,v )=logT F (v ),F (v′),F (v′′)
abc j j abc ρ j ρ j ρ j
(cid:0) (cid:1)
definedfor everya, b, c>1 witha+b+c=n. By ρ–equivarianceofthe flagcurve
e e e
F , these triple ratio logarithms depend only on the triangle T and on the vertex
ρ j
v of T , and not on the choice of the lift T .
j j j
Lemma 5 indicates how the invariant τρ (T ,v ) ∈ R changes if we choose a
eabc i j
different vertex of the triangle T .
j
Lemma 11. If v , v′ and v′′ are the vertices of T , indexed clockwise around T ,
j j j j j
then
τρ (T ,v )=τρ (T ,v′)=τρ (T ,v′′). (cid:3)
abc i j bca i j cab i j
2.4. Shear invariants of infinite leaves. Let g be an infinite leaf of λ.
j
Lift g to a leaf g of the preimage λ of λ in the universal covering S. This leaf
j j
′
is isolated in λ, and is adjacent to two components T and T of the complement
e e ′ e
S−λ. Choosethe notationso that T andT are respectivelyto the left andto the
e e e
right of g for the orientation of g coming from the orientation of g .
e e j j e e j
Let x and y ∈ ∂∞S be the positive and negative end points of gj, respectively,
e e
′ ′
of gj. Let z, z ∈ ∂∞Se be the third vertices of T and T , respectiveely, namely the
verticesofthesetrianglesthatareneitherxnory. See Figure3. Considerthe flags
E =e F (x), F = F (ye), G = F (z) and G′ = Fe(z′) asseociated with these vertices
ρ ρ ρ ρ
by the flag curve Fρ: ∂∞S →Flag(Rn).
For 1 6 a 6 n−1, we can now consider the double ratio D (E,F,G,G′) as
e a
in §1.4. This double ratio is positive by Proposition 10. The a–th shear invariant
of the Hitchin homomorphism ρ along the oriented leaf g is then defined as the
j
logarithm
σρ(g )=logD (E,F,G,G′).
a j a
This invariant σ (g ) is clearly independent of the choice of the lift g of the leaf
a j j
g to S.
j
By Lemma 7, reversing the orientation of g replaces σρ(g ) by σρe (g ).
e j a j n−a j
2.5. Shear invariants of closed leaves. The shear invariants of a closed leaf c
i
are defined in very much the same way as for infinite leaves, except that we need
to use the transverse arc k that is part of the topological data to single out two
i
′
triangles T and T that are located on either side of c .
i
10 FRANCISBONAHONANDGUILLAUMEDREYER
z
T
x
g
j
y e
′
eT
e
′
z
Figure 3. The construction of shear invariants of infinite leaves
Moreprecisely,letc beacomponentofthepre-imageofc intheuniversalcover
i i
S, and orient it by the orientation of c . Lift the arc k to an arc k that meets c
i i i i
e ′
in one point. Let T and T be the two triangle components of S−λ that contain
e ′ e e
the end points of k , in such a way that T and T are respectively to the left and
ei e e e
to the right of c for the orientation of c lifting the orientation of c .
i e i e e i
Let x and y ∈ ∂∞S be the positive and negative end points of ci, respectively.
e e
Among the vertices oef T, let z ∈ ∂∞S be the one that is farthesteaway from ci;
namely, z is adjacent to the two components of S − T that do not contain c .
′ e ′ e ei
Similarly, let z be the vertex of T that is farthest away from c . See Figures 4(a)
and (b) for two of the four possible configurations,eaccoerding tio the directionseof
the spiraling of infinite leaves around c . e
i
z z
T T
e e
y x y x
k k
i i
′ ′
T T
e e
e e
′ ′
z z
(a) (b)
Figure 4. The construction of shear invariants of closed leaves
Finally, let E = F (x), F = F (y), G = F (z) and G′ = F (z′) be the flags
ρ ρ ρ ρ
associated to these vertices by the flag curve Fρ: ∂∞S →Flag(Rn).
Withthisdata,weagainconsiderfor16a6n−1thedoubleratioD (E,F,G,G′)>
e a
0asin§1.4andProposition10. Thea–th shear invariant ofthe Hitchinhomomor-
phism ρ along the oriented closed leaf c is defined as the logarithm
i
σρ(c )=logD (E,F,G,G′).
a i a
By ρ–equivariance of the flag curve F , this shear invariant σ (c ) is clearly inde-
ρ a i
pendent of the choice of the component c of the preimage of c , and of the lift k
i i i
of the arc k .
i e e