Triangle invariants

Let %:π1(S)!PSLn(R) be a Hitchin homomorphism. We will use a maximal geodesic laminationλto construct invariants of the corresponding character%∈Hitn(S).

3.1. The flag curve

The key to the definition of these invariants is the following construction of Labourie [27].

LetT¹S and T¹Sebe the unit tangent bundles of the surface S and of its universal coverS, respectively. For convenience, lift the homomorphisme %:π1(S)!PSLn(R) to a homomorphism%⁰:π1(S)!SLn(R). The fact that such a lift exists is classical whenn=2, and therefore also holds when%:π1(S)!PSLn(R) comes from a discrete representation π1(S)!PSL2(R); the existence of the lift in the general case follows by connectedness of the Hitchin component Hitn(S), and by homotopy invariance of the obstruction to lift.

We can then consider the twisted product

T¹S×%⁰Rⁿ=T¹Se×Rⁿ π₁(S) ,

where the fundamental groupπ₁(S) acts onT¹Seby its usual action on the universal cover S, and acts one Rⁿ by%⁰. The natural projection T¹S×_%⁰Rⁿ!T¹S presentsT¹S×_%⁰Rⁿ as a vector bundle overT¹S with fiberRⁿ.

Endow the surfaceS with an arbitrary metric of negative curvature. This defines a circle at infinity∂_∞Sefor the universal coverS, and a geodesic flow on the unit tangente bundleT¹S. It is well known (see for instance [20], [11] and [19]) that these objects are actually independent of the choice of the negatively curved metric, at least if we do not care about the actual parametrization of the geodesic flow (which is the case here).

The geodesic flow (g_t)_t∈RofT¹S has a natural flat lift to a flow (G_t)_t∈Ron the total space T¹S×_%⁰Rⁿ. The flatness property here just means that the flow (G_t)_t∈R is the projection of the flow (Ge_t)_t∈RonT¹S×e Rⁿ that acts by the geodesic flow (˜g_t)_t∈RofT¹Se on the first factor, and by the identity Id_Rn on the second factor.

Endow each fiber of the vector bundleT¹S×_%⁰Rⁿ!T¹Swith a normk · kdepending continuously on the corresponding point ofT¹S.

Theorem 3.1. (Labourie [27, Theorem 4.1]) If %:π1(S)!PSLn(R) is a Hitchin homomorphism,the vector bundle T¹S×%⁰Rⁿ!T¹S admits a unique decomposition as a direct sum L1⊕L2⊕...⊕Ln of nline sub-bundles La!T¹S such that

(1) each line bundle La is invariant under the lift (Gt)t∈Rof the geodesic flow; (2) for every a>b,there exist constantsAab, Bab>0such that,for every va∈La and vb∈Lb in the same fiber of T¹S×%⁰Rⁿ and for every t>0,

kGt(vb)k kvbk 6Aab

kGt(va)k kvak e^−Babt.

The second property is clearly independent of the choice of the norm k · k. It is referred to as theAnosov propertyof the Hitchin homomorphism%. This relative property does not say anything about whether the flow (G_t)_t∈Rn expands or contracts the fibers of any individual sub-bundleLa, but states that, whena<b, the flow (Gt)_t∈Rn contracts the fibers ofLb much more than those ofLa. Writing this in a more intrinsic way, this means that (Gt)_t∈Rⁿ induces on the line bundle Hom(La, Lb) a flow that is uniformly contracting whena>b.

Lift the sub-bundles La of T¹S×%⁰Rⁿ=(T¹Se×Rⁿ)/π1(S) to sub-bundles ˜La of T¹Se×Rⁿ. Because the line sub-bundles La are invariant under the lift (Gt)_t∈R of the geodesic flow, the fiber of ˜Laover ˜x∈Seis of the form{˜x}×L˜a(g) for some line ˜La(g)⊂Rⁿ depending only on the orbitg of ˜xfor the geodesic flow of T¹S.e

The line ˜La(g)⊂Rⁿ depends on the orbitg of the geodesic flow of T¹Se or, equiv-alently, on the corresponding oriented geodesic g of S. The Anosov property has thee following relatively easy consequence. Define a flagE(g)∈Flag(Rⁿ) by the property that E(g)^(a)= ˜L1(g)⊕L˜2(g)⊕...⊕L˜a(g); thenE(g) depends only on the positive endpoint ofg.

More precisely, we have the following result.

Proposition 3.2. (Labourie [27, Theorem 4.1]) For a Hitchin homomorphism

%:π1(S)−!PSLn(R), there exists a unique map F%:∂_∞Se!Flag(Rⁿ)such that

(1) F% is H¨older continuous for the visual metric of ∂_∞Se and for an arbitrary riemannian metric on Flag(Rⁿ);

(2) for every oriented geodesic g of Se with positive endpoint x˜+∈∂_∞S,e the image F%(˜x+)is equal to the flag E(g)defined above.

In addition, F% is %-equivariant in the sense that F%(γx)=%(γ)(˜ F%(˜x)) for every x∈∂˜ _∞Se and γ∈π₁(S).

By definition, this mapF%:∂_∞Se!Flag(Rⁿ) is the flag curve of the Hitchin homo-morphism %:π1(S)!PSLn(R). It is independent of the choice of the lift %⁰:π1(S)! SLn(R) of %:π1(S)!PSLn(R), and of the negatively curved metric onS used to define the geodesic flow of the unit tangent bundleT¹S.

Note that the line decompositionRⁿ=Ln

a=1L˜a(g), associated by Theorem3.1with an oriented geodesicg ofS, is easily recovered from the flag curvee F%:∂∞Se!Flag(Rⁿ) by the property that ˜La(g)=F%(˜x+)^(a)∩F%(˜x−)^(n−a+1)for the positive and negative end-points ˜x+, ˜x−∈∂∞Seof g. The main content of Proposition 3.2is that, for two oriented geodesicsgandg⁰ofSethat share the same positive endpoint ˜x+∈∂∞S, the correspondinge line decompositionsRⁿ=Ln

a=1L˜a(g) andRⁿ=Ln

a=1L˜a(g⁰) may be different but define the same linear subspacesF%(˜x+)^(a)=La

b=1L˜_b(g)=La

b=1L˜_b(g⁰) for everya=1,2, ..., n.

See also [21] for a nice exposition of Theorem3.1and Proposition3.2.

The flag curveF% has the following important positivity property.

Theorem3.3. (Fock–Goncharov [18, Theorem 1.15]) For every finite set of distinct points x1, x2, ..., xk∈∂_∞Se occurring in this order on the circle at infinity ∂_∞S,e the flag k-tuple (F%(x1),F%(x2), ...,F%(xk))is positive in the sense of §1.5.

3.2. Triangle invariants of Hitchin characters

We now define a first set of invariants for the Hitchin character represented by a homo-morphism%:π1(S)!PSLn(R).

The complement of the maximal geodesic lamination λ consists of finitely many infinite trianglesT1, T2, ..., Tm, each with three spikes.

Consider such a triangle componentT ofS\λ, and select one of its spikess. LiftT to an ideal triangleTein the universal coverS, and let ˜e sbe the spike ofTecorresponding tos. The spike ˜suniquely determines a point of the circle at infinity∂∞S, which we wille also denote by ˜s.

Label the spikes of T as s, s⁰ and s⁰⁰ in counterclockwise order around T, and let ˜s,˜s⁰,s˜⁰⁰∈∂∞Se be the corresponding points of the circle at infinity. The flag triple (F%(˜s),F%(˜s⁰),F%(˜s⁰⁰)), associated with ˜s,s˜⁰,s˜⁰⁰∈∂∞Seby the flag curve

F%:∂∞Se−!Flag(Rⁿ),

is positive by Theorem3.3. We can therefore consider the logarithms τ_abc^% (s) = logTabc(F%(˜s),F%(˜s⁰),F%(˜s⁰⁰))

of its triple ratios, defined for everya,b,c>1 witha+b+c=n. By%-equivariance of the flag curve F%, these triple ratio logarithms depend only on the triangle T and on the spikesofT, and not on the choice of the liftTe.

The following statement indicates how the invariantτ_abc^% (s)∈Rchanges if we choose a different vertex of the triangleT.

Lemma3.4. If s,s⁰and s⁰⁰are the three spikes of the component T of S\λ,indexed counterclockwise around T,then

τ_abc^% (s) =τ_bca^% (s⁰) =τ_cab^% (s⁰⁰).

Proof. This is an immediate consequence of Lemma1.1.

By invariance of triple ratios under the action of PGL_n(R) on Flag(Rⁿ), it is imme-diate that the triangle invariantsτ_abc^% (s) depend only on the character%∈Hit_n(S), and not on the homomorphism%:π1(S)!PSLn(R) representing it.

Because of Lemma 3.4, we can think of the invariant τ_abc^% (s) as mainly associated with the triangle componentT ofS\λthat has the spikesas a vertex, since choosing a different vertex ofT only affects the order in which the indicesa,bandcare considered.

For this reason, we will refer to the τ_abc^% (s) as the triangle invariants of the Hitchin character%∈Hitn(S).

Remark 3.5. The companion article [9] uses a clockwise labeling convention for the vertices of a triangle. As a consequence, the triangle invariants of [9] are the opposite of those introduced here.