Condiciones de Seguridad e Higiene

Atrain track on a surface Σ is an embedded 1–complex τ _⊂Σ such that:

i The edges (called branches) are smooth arcs and the tangent vectors at the

endpoints are well–defined.

ii At any vertex v (calledswitch) the incident edges are mutually tangent and are

divided into “incoming” and “outgoing” branches atv(according to their inward

pointing tangent vectors at the switch).

iii Each closed curve component of τ has a unique bivalent switch, and all other

switches are at least trivalent.

iv The complementary regions of the train track have negative Euler characteristic (that is, they are different from disks with 0, 1 or 2 cusps at the boundary and different from annuli and once-punctured disks with no cusps at the boundary, where a cusp is a non-smooth point).

A train track is calledgeneric if all switches are at most trivalent. In the case of a

trivalent vertex, there is one incoming branch and two outgoing ones. A train track is calledmaximal if each complementary region is either a trigon or a once-punctured

monogon.

Denote B = B(τ) the set of branches of τ. A function w: B −→ R>0 (resp. w: _{B −→}R) is atransverse measure (resp. weighting) forτ if it satisfies theswitch condition, that is for all switchesv, we wantP

iw(ei) =Pjw(Ej) where the ei are

the incoming branches atv andEj are the outgoing ones.

A train track is calledrecurrent if it admits a transverse measure which is positive

on every branch. A train track τ is called transversely recurrent if every branch

b _{∈ B}(τ) is intersected by an embedded simple closed curve c = c(b) _⊂ Σ which

intersectsτ transversely and is such that Σ₋τ₋cdoes not contain an embedded

bv(n) v

cv(n)

incoming branch

outgoing branches

Figure 1.8: A switchv of a train track with one black incoming edge and two red

weighted outgoing edges.

c cuts a branch of τ transversally and a cusp is formed by two mutually tangent

branches ofτ which meet at a switch, either both incoming or both outgoing.) A

recurrent and transversely recurrent train track is calledbirecurrent.

A geodesic lamination (or a train track)λiscarried by a train trackτ if there is

a mapF: Σ_−→Σ of classC1 which is isotopic to the identity and which mapsλto

τ in such a way that the restriction of its differentialdF to every tangent line ofλis

non–singular. A generic transversely recurrent train track which carries a complete geodesic lamination is calledcomplete, where we define a geodesic lamination to be complete if there is no geodesic lamination that strictly contains it.

Given a generic birecurrent train trackτ _⊂Σ, we define_V(τ) to be the collection

of all (not necessary nonzero) transverse measures supported onτ and let_W(τ) be

the vector space of all weightings, that is assignments of (not necessary non-negative) real numbers, one to each branch of τ, which satisfy the switch conditions. By

splitting, we can arrangeτ to be generic. Since Σ is oriented, we can distinguish the

right and left hand outgoing branches. Ifn,n0 ∈ W(τ) are weightings onτ, then we

denote bybv(n), cv(n) the weights of the left hand and right hand outgoing branches

atv respectively; see Figure 1.8. Thurston’s product ΩTh: W(τ)× W(τ)−→ R is

defined as ΩTh(n,n0) = 1 2 X v bv(n)cv(n0)−bv(n0)cv(n) . (1.1)

In Theorem 3.1.4 of Penner and Harer [38] it is proved that, if the train track

τ ⊂Σ is complete, then the interior int (V(τ)) ofV(τ) can be thought of as a chart

on the PIL manifold ML(Σ) of measured laminations (with compact support), as underlined by Penner and Harer in Section 3.1 of [38]. See Section 1.3.1 for the definition of ML(Σ). (PIL is short forpiecewise–integral–linear, see [38, Section 3.1]

for the definition.) In addition, in this case, we can identifyW(τ) with the tangent

space to ML(Σ) at a point in int (_V(τ)). The Thuston product ΩTh defined above

We now add a short remark which is not used in the following sections, but which could be useful for some readers. Anorientationon a train trackτ is a specification

of orientation for each branch of τ such that incoming branches points towards

outgoing ones, see page 31 of Penner and Harer for a more precise definition. It is interesting to note that, ifτ is oriented, then there is a natural maphτ: W(τ)−→

H1(Σ;R), see Section 3.2 [38], which is related to Thurston’s product by the following

result. For a generalisation of this result to the case of an arbitrary (not necessarily orientable) train trackτ ∈Σ, see Section 3.2 [38].

Proposition 1.3.1(Lemma 3.2.1 and 3.2.2 [38]). For any train trackτ,ΩTh(·,·)is

a skew-symmetric bilinear pairing on_W(τ). In addition, ifτ is connected, oriented and recurrent, then for any n,n0 ∈ W(τ), ΩTh(n,n0) is the homology intersection

number of the classes hτ(n) and hτ(n0).

As before, we add now a remark not important for the future results, but that we find really interesting and which could help in understanding the meaning of ΩTh. We need to recall the last Proposition of Section 3.2 of [38] and some notation

from Bonahon’s work (see his survey paper [6] for a general introduction to the argument and for other further references). After rigorously defining the tangent space TηML(Σ) with η ∈ ML(Σ), Bonahon proves in [4] that we can interpret

any tangent vectorv∈TηML(Σ) as a geodesic lamination with a transverse H¨older

distribution. Theorem 11 of Bonahon [5] shows that the space of H¨older distributions on a track τ coincides with the space_W(τ), the vector space of all assignments of

not necessary non-negative real numbers, one to each branch of τ, which satisfy

the switch conditions. As a corollary of this result, one can prove that the space of H¨older distributions on a trackτ is finite dimensional. In particular, ifτ is maximal,

then the real dimension of this space is dim_R(W(τ)) = 2ξ(Σg,b) = 6g−6 + 2b.

Bonahon also characterised which geodesic laminations with transverse distributions correspond to tangent vectors to ML(Σ). Notice that, if the laminationη_∈ML(Σ)

is carried by the trackτ, we can locally identify TηML(Σ) with W(τ).

Theorem 1.3.2 (Theorem 3.2.4 [38]). For any surface Σ, Thurston’s product is a skew-symmetric, nondegenerate, bilinear pairing on the tangent space to the PIL manifoldML(Σ).