A LENGTH COMPARISON THEOREM FOR GEODESIC CURRENTS JENYA SAPIR

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A LENGTH COMPARISON THEOREM FOR GEODESIC

CURRENTS

JENYA SAPIR

Abstract. We work with the space C(S) of geodesic currents on a closed sur-

face Sof negative Euler characteristic. By prior work of the author with Se-

bastian Hensel, each ﬁlling geodesic current µhas a unique length-minimizing

metric Xin Teichm¨uller space. In this paper, we show that, on so-called

thick components of X, the geometries of µand Xare comparable, up to a

scalar depending only on µand the topology of S. We also characterize thick

components of the projection using only the length function of µ.

1. Introduction

Let Sbe a closed, oriented, ﬁnite type surface with negative Euler characteristic.

The space of geodesic currents, C(S), contains many of the structures one might

wish to study on S. For example, it contains the set of closed curves up to homotopy,

as well as an embedded copy of Teichm¨uller space, T(S). These sets are united by

an intersection pairing i(·,·) on C(S). If µ, ν ∈ C(S) represent two closed curves,

then i(µ, ν) is just their geometric intersection number. And if µrepresents a

metric, while νrepresents a closed curve, then i(µ, ν) is the length of the geodesic

representative of νin the metric µ.

We let Cf ill(S) denote the set of ﬁlling currents, that is, those currents that have

positive intersection with all other geodesic currents. Then all currents representing

metrics are examples of ﬁlling currents. We extend the notion of length function

from currents representing metrics to all of Cfill(S). If µ∈ Cfill(S), we deﬁne its

length function `µ:C(S)→Rso that

`µ(ν) := i(µ, ν)

for all ν∈ C(S).

In [HS21], Hensel and the author show that there is a continuous projection

π:Cfill(S)→ T (S)

that minimizes the length of µin the sense that `π(µ)(µ)< `X(µ) for all X6=

π(µ)∈ T (S). We call π(µ) the length minimizer of µ. The goal of this paper is

to compare the length function of a ﬁlling current µto the length function of its

length minimizer π(µ).

To state our result precisely, let cbbe the Bers constant. Then any two curves

of length less than cbwith respect to π(µ) are disjoint. Cut π(µ) along all such

curves. Then the connected components of the result are the thick components

of π(µ). We describe how to identify these components in Theorems 1.2 and 1.3

below.

In [BIPP19], they show that if µ∈ Cfill(S), then its systolic length sys(µ) =

infα⊂Si(µ, α) is positive, where the inﬁmum is taken over all simple closed curves

arXiv:2210.00925v2 [math.GT] 18 Apr 2023

2 JENYA SAPIR

on S. For any subsurface Yof S, they also deﬁne the Y-systolic length to be

sysY(µ) = inf

α⊂Yi(µ, α)

where the inﬁmum is taken over all essential, non-peripheral simple closed curves

in Y. Then we show the following:

Then we show that the geometries of µand π(µ) are comparable on thick com-

ponents:

Theorem 1.1. Let µ∈ Cfill(S), and let π(µ)be its length minimizer. Let Ybe a

thick component of π(µ). Then for all essential, non-peripheral simple closed curves

αin Y,

`µ(α)

`π(µ)(α)sysY(µ)

where sysY(µ)is the Y-systolic length of µ, and the constants depend only on the

Euler characteristic χ(S).

1.1. Notation. Given quantities A, B, we use say A≺Bwith constants depending

only on Cif there is a constant cdepending only on Cso that A≤cB. Likewise,

we say ABif A≥c0Bfor some c0>0 depending only on C, and ABif

A≺Band B≺A.

1.2. Identifying thick components of π(µ).We also characterize when curves

in π(µ) are short, and when subsurfaces are thick, in terms of the length function

of µ.

First, it turns out that a simple closed curve αis short in π(µ) if all simple closed

curves βcrossing αare relatively long with respect to µ.

Theorem 1.2. For every  > 0, there are constants N1, N2so that for any µ∈

P Cfill(S), any simple closed curve α, and any simple closed curve βwith i(α, β)≥

(1) If `π(µ)(α)< , then

i(µ, β)> N1i(µ, α)

(2) If

i(µ, β)> N2i(µ, α)

then `π(µ)(α)< .

Note that this theorem is a coarse biconditional. The constant N1grows coarsely

like the width of the collar about α, while N2comes from the constant in Theorem

1.1, and is more mysterious.

Once we know which curves are short, the following theorem gives a more prac-

tical characterization of thick components of π(µ).

Theorem 1.3. Let Y⊂π(µ)be a subsurface so that `π(µ)(β)< cbfor each bound-

ary component βof Y, where cbis the Bers constant. Then for each essential

simple closed curve αin Y,

`π(µ)(α)1

if and only if there exists a marking Γof Yso that

i(µ, γ)sysY(µ)

for all γ∈Γ, where all constants depend only on S.

A LENGTH COMPARISON THEOREM FOR GEODESIC CURRENTS 3

1.3. Outline and idea of proof. The paper is organized as follows.

(1) In Section 3, we prove Proposition 3.1. This is a preliminary version of

Theorem 1.1. It shows that, if µ∈ Cfill(S) and Yis a thick component of

π(µ), then

sysY(µ)2

i(µ, Γ) ≺i(µ, α)

i(π(µ), α)≺i(µ, Γ)

where Γ is a shortest marking for Ywith respect to π(µ) (see Section 3 for a

deﬁnition). We spend the rest of the paper to show that i(µ, Γ) sysY(µ).

This result is completely independent from the rest of the paper. In fact,

it relies only on analogues of results in [Raf07], which we prove for geodesic

currents rather than ﬂat structures.

(2) In Sections 4 - 7 we prove Theorem 7.3, giving a mixed collar lemma for

geodesic currents and their length minimizers. Roughly, this theorem says

the following. Let µ∈ Cfill(S), and let αand βbe two simple closed curves

with i(α, β)≥1. If αis short and βis long in π(µ), then µmust intersect

βmuch more than α. An idea of the proof is given in Section 4.

(3) In Sections 8 and 9, we prove a few more mixed collar theorems that follow

from Theorem 7.3.

(4) In Section 10, we prove Proposition 10.1, which says that the hyperbolically

short marking Γ of a thick subsurface Yis actually µ-short:

i(µ, Γ) sysY(µ)

The theorem then follows directly from Propositions 3.1 and 10.1.

(5) In Section 11, we prove Theorems 1.2 and 1.3 that show how to identify

short curves and thick subsurfaces of π(µ).

(6) Section 12 is an appendix proving a few identities about the collars of

geodesics in hyperbolic surfaces.

1.4. Acknowledgements. The author would like to thank D´ıdac Mart´ınez-Granado,

Giuseppe Martone and Sebastian Hensel for many helpful discussions.

2. Background and connection to related results

2.1. Geodesic currents. Geodesic currents on Swere ﬁrst deﬁned by Bonahon in

[Bon85] as follows. Fix a complete hyperbolic metric Xfor S. Identify its universal

cover with H2. Let Gbe the set of all unparameterized, unoriented geodesics in

H2. Each geodesic in H2is uniquely determined by its endpoints on the circle S1

at inﬁnity. Thus Gcan be identiﬁed with S1×S1\∆/∼, where ∆ is the diagonal

in S1×S1, and ∼is the relation so that (a, b)∼(b, a). Since π1(S) acts on H2

by isometries, it also acts on G. A geodesic current is a π1(S)-invariant, Borel

measure on G. We let C(S) denote the space of currents and endow it with the

weak* topology. It turns out that C(S) is independent of the choice of metric X.

In fact, the space of currents deﬁned using any other metric Yis H¨older equivalent

to C(S) [Bon85].

The set of closed geodesics on Xembeds into the space of currents as follows.

Given any closed geodesic γon X, we can lift it to a subset ˜γ∈ G. Then the

current corresponding to γwill be the Dirac measure on ˜γ. Abusing notation, we

will still let γrefer to this current. There is a natural action of R+on C(S) by

scaling. It turns out that the R+orbit of the set of closed geodesics is dense in C(S)

[Bon85]. Moreover, Bonahon also shows that the geometric intersection function

4 JENYA SAPIR

i(·,·) extends continuously from pairs of closed geodesics to a bilinear, symmetric

intersection form on pairs of currents.

By work of Bonahon, Teichm¨uller space also embeds in C(S). Moreover, suppose

γis a closed geodesic and Y∈ T (S). We abuse notation slightly, and let γand Y

still denote the corresponding currents. Then,

i(γ, Y ) = `Y(γ)

In fact, many spaces of metrics on Sembed into C(S): spaces of singular ﬂat

metrics [DLR10], metrics of variable negative curvature [Ota90], and others. The

embeddings are all characterized by the fact that intersecting a metric with a closed

curve γgives the length of the geodesic representative of γwith respect to that

metric.

2.2. Relationship to the thick-thin decomposition for quadratic diﬀeren-

tials. This theorem is inspired by an analogous result of Raﬁ in [Raf07]. Raﬁ shows

the following. Each holomorphic quadratic diﬀerential on Sinduces a singular ﬂat

metric q. By [DLR10], this set of singular ﬂat metrics can be viewed as ﬁlling

currents in the sense that if qis a ﬂat structure and γis a curve, then i(q, γ) is

the length of the q-geodesic representative of γin q. Moreover, qhas a unique

hyperbolic metric Xin its conformal class. So, we get a projection from singular

ﬂat structures (which we will think of as a subset of Cfill(S)) to T(S).

Given a hyperbolic metric Xconformally equivalent to a ﬂat structure q, Raﬁ

considers the thick components of X, deﬁned the same way as above. He shows

that if Yis a thick component of X, and αis any essential, non-peripheral simple

closed curve in Y, then, in the language of this paper,

`q(α)

`X(α)sysY(q)

where the constants depend only on χ(S) [Raf05, Theorem 1.3]. Thus, the state-

ment of the theorem of Raﬁ is analogous to ours, although the projections to Te-

ichm¨uller space are diﬀerent. It would be interesting to see how diﬀerent the two

projections are.

Question 1. Given a singular ﬂat metric qcoming from a holomorphic quadratic

diﬀerential, let Xbe the hyperbolic metric in its conformal class, and let π(q)be its

length minimizing metric. Are Xand π(q)at a uniformly bounded distance, with

respect to some natural metric on T(S)?

The method of proof of Theorem 1.1 is also inspired by the proof in [Raf07].

Raﬁ’s proof relies crucially on a mixed collar lemma for quadratic diﬀerentials,

which he proves in [Raf05]. His theorem says that if Xis the hyperbolic metric

in the conformal class of a holomorphic quadratic diﬀerential q, then for any two

intersecting simple closed curves αand β,

`q(α)≥d·`q(β)

where d=d(`X(β)) depends only on the length of βwith respect to X, and on

χ(S) [Raf05, Theorem 1.3].

The ﬁrst half of this paper involves proving an analogous result (Theorem 7.3).

If αand βare intersecting simple closed curves, and π(µ) is the length minimizer

A LENGTH COMPARISON THEOREM FOR GEODESIC CURRENTS 5

of some µ∈ Cfill(S), we show that

`µ(α)≥D

i(α, β)`µ(β)

where D=D(`π(µ)(β)) depends only on the length of βwith respect to π(µ). The

factor of i(α, β) in the denominator is due to the diﬀerences in the methods of proof.

We use a careful geometric argument to show how the length of µwill change if

we pinch a metric Xalong α. If µintersects αmuch less than β, then pinching α

increases the length of µcoming from crossing α, while decreasing the contribution

to length coming from intersecting β. Balancing these two eﬀects gives our mixed

collar lemma.

We can prove a version of the length comparison theorem (Proposition 3.1) using

techniques similar to those of [Raf07, Theorem 1]. We need this result to prove the

main theorem, however its statement is rather unsatisfying on its own. To get the

full version of Theorem 1.1, we have to work speciﬁcally with length minimizing

metrics, and the techniques are rather diﬀerent.

2.3. Connection to Higher Teichm¨uller Spaces. Giuseppe Martone recently

made us aware of a similar length comparison theorem in Higher Teichm¨uller theory.

Take the space of representations ρ:π1(S)→P SL(3,R). Consider its Hitchin

component TH, which is a connected component of discrete, faithful, orientation-

preserving representations. There is a map from THto Cfill(S). This map is

natural in the sense that, if ρ∈ Cfill(S) represents an element of TH, then for

any closed curve γ,i(ρ, γ) gives the so-called Hilbert length of γwith respect to ρ

[BCLS18, MZ19].

Labourie [Lab07] and Loftin [Lof01] showed independently that there is a map-

ping class group-invariant projection from THto Teichm¨uller space. This is part

of a much larger, quite active research program: see [Wie19, Conjecture 14] for an

overview of the broader context.

As noted in [DM20, Lemma 5.1], work of Tholozan [Tho17, Theorem 3.9, Corol-

lary 3.10] implies a length comparison result for all but a bounded set of represen-

tations ρ∈ THwith projection X∈ T (S). For all (not necessarily simple) closed

curves γon S,

`ρ(γ)

`X(γ)1

h(ρ)

where h(ρ) is the topological entropy of the Hilbert length of ρ, and the length

functions are the ones deﬁned above for the associated currents. In fact, it follows

from [MZ19, Theorem 1.4, Corollary 1.5] that sys(ρ)≺1/h(ρ)≺sysY(ρ) for a thick

subsurface Yof π(ρ). The upper bound is not stated this way in their paper, but it

can be deduced using Theorem 7.3 that the function Kρin [MZ19] is the Y-systole

of ρon a certain subsurface Y. Thus, we get the inequality

sys(ρ)≺`ρ(γ)

`X(γ)≺sysY(ρ)

where the lower bound is the systole of ρon all of S, and the upper bound is the

Y-systole for a certain Y⊂S.

It should be noted that the constants in this inequality depend on the projection

Xof ρ. It would be interesting to see if one could state this theorem with constants

that depend only on the topology of S, and simultaneously tighten the two bounds

to get a statement similar to Theorem 1.1. Moreover, it would again be interesting

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ALENGTHCOMPARISONTHEOREMFORGEODESICCURRENTSJENYASAPIRAbstract.WeworkwiththespaceC(S)ofgeodesiccurrentsonaclosedsur-faceSofnegativeEulercharacteristic.BypriorworkoftheauthorwithSe-bastianHensel,eachllinggeodesiccurrenthasauniquelength-minimizingmetricXinTeichmullerspace.Inthispaper,weshowthat,onso...

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