AUTOMORPHISMS OF SOME VARIANTS OF FINE GRAPHS FRÉDÉRIC LE ROUX AND MAXIME WOLFF Abstract. Recently Bowden Hensel and Webb deﬁned the ﬁne curve graph for

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AUTOMORPHISMS OF SOME VARIANTS OF FINE GRAPHS

FRÉDÉRIC LE ROUX AND MAXIME WOLFF

Abstract. Recently Bowden, Hensel and Webb deﬁned the ﬁne curve graph for

surfaces, extending the notion of curve graphs for the study of homeomorphism

or diﬀeomorphism groups of surfaces. Later Long, Margalit, Pham, Verberne

and Yao proved that for a closed surface of genus g⩾2, the automorphism group

of the ﬁne graph is naturally isomorphic to the homeomorphism group of the

surface. We extend this result to the torus case g=1; in fact our method works

for more general surfaces, compact or not, orientable or not. We also discuss the

case of a smooth version of the ﬁne graph.

1. Introduction

1.1. Context and results. For a connected, compact surface Σgof genus g⩾1,

Bowden, Hensel and Webb [2] recently introduced the ﬁne curve graph C†(Σ), as the

graph whose vertices are all the essential closed curves on Σ, with an edge between

two vertices aand bwhenever a∩b=∅, if g⩾2, and whenever ∣a∩b∣⩽1if g=1.

They proved that for every g⩾1, the graph C†(Σ)is hyperbolic, and derived a

construction of an inﬁnite dimensional family of quasi-morphisms on Homeo0(Σ),

thereby answering long standing questions of Burago, Ivanov and Polterovich.

The ancestor of the ﬁne graph is the usual curve complex of a surface Σ,i.e., the

complex whose vertices are the isotopy classes of essential curves, with an edge (or

a simplex, more generally) between some vertices if and only if they have disjoint

representatives. Since its introduction by Harvey [5], the curve complex of a surface

has been an extremely useful tool for the study of the mapping class group Mod(Σ)

of that surface, as it acts on it naturally. In particular, the fact that this complex is

hyperbolic, discovered by Masur and Minsky, has greatly improved the understand-

ing of the mapping class groups (see [12, 11]). The result of Bowden, Hensel and

Webb, promoting the hyperbolicity of the curve complex to that of the ﬁne curve

graph, opens the door both to the study of what classical properties of usual curve

complexes have counterparts in the ﬁne curve graph, and to the use of this graph

to derive properties of homeomorphism groups. A ﬁrst step in this direction was

taken by Bowden, Hensel, Mann, Militon and Webb [1], who explored the metric

properties of the action of Homeo(Σ)on this hyperbolic graph.

A classical theorem by Ivanov [7] states that, when Σis a closed surface of genus

g⩾2, the natural map Mod(Σ)→Aut(C(Σ)) is an isomorphism. Recently Long,

Margalit, Pham, Verberne and Yao [10] proved the following natural counterpart

of Ivanov’s theorem for ﬁne graphs: provided Σis a compact orientable surface of

genus g⩾2, the natural map

Homeo(Σ)Ð→Aut(C†(Σ))

is an isomorphism. They also suggested that this map (with the appropriate version

of C†) may also be an isomorphism when g=1, and conjectured that the automor-

phism group of the ﬁne curve graph of smooth curves, should be nothing more than

Diﬀ(Σ).

arXiv:2210.05460v1 [math.GT] 11 Oct 2022

2 FRÉDÉRIC LE ROUX AND MAXIME WOLFF

In this article, we address both these questions. Our motivation originates from

the case of the torus: excited by [1], we wanted to understand more closely the

relation between the rotation set of homeomorphisms isotopic to the identity and

the metric properties of their actions on the ﬁne graph. This subject will be treated

in another article, joint with Passeggi and Sambarino [9]. The methods developed

in the present article are valid not only for the torus but for a large class of surfaces.

We work on nonspherical surfaces (i.e., surfaces not embeddable in the 2-sphere, or

equivalently, containing at least one nonseparating simple closed curve), orientable

or not, compact or not. We consider the graph NC†

⋔(Σ), whose vertices are the

nonseparating simple closed curves, and with an edge between two vertices aand

bwhenever they are either disjoint, or have exactly one, topologically transverse

intersection point (see the beginning of Section 2 for more detail). Our ﬁrst result

answers a problem raised in [10].

Theorem 1.1. Let Σbe a connected, nonspherical surface, without boundary. Then

the natural map Homeo(Σ)→Aut(NC†

⋔(Σ)) is an isomorphism.

Our second result concerns the smooth version of ﬁne graphs. We consider the

graph NC†∞

⋔(Σ)whose vertices are the smooth nonseparating curves in Σ, with an

edge between aand bif they are disjoint or have one, transverse intersection point,

in the diﬀerentiable sense (in particular, NC†∞

⋔(Σ)is not the subgraph of NC†

⋔(Σ)

induced by the vertices corresponding to smooth curves: it has fewer edges). The

following result partially conﬁrms the conjecture of [10]; here we restrict to the case

of orientable surfaces for simplicity.

Theorem 1.2. Let Σbe a connected, orientable, nonspherical surface, without

boundary. Then all the automorphisms of NC†∞

⋔(Σ)are realized by homeomor-

phisms of Σ.

In other words, if we denote by Homeo∞⋔(Σ)the subgroup of Homeo(Σ)preserv-

ing the collection of smooth curves and preserving transversality, then the natural

map

Homeo∞⋔(Σ)Ð→Aut(NC†∞

⋔(Σ))

is an isomorphism. We were surprised to realize however that Homeo∞⋔(Σ)is strictly

larger than Diﬀ(Σ).

Proposition 1.3. Every surface Σadmits a homeomorphism fsuch that fand f−1

preserve the set of smooth curves, and preserve transversality, but such that neither

fnor f−1is diﬀerentiable. In particular, the natural map

Diﬀ(Σ)Ð→Aut(NC†∞

⋔(Σ))

is not surjective.

1.2. Idea of the proof of Theorem 1.1. The main step in this proof is the

following.

Proposition 1.4. If {a, b}, or {a, b, c}is a 2-clique or a 3-clique of NC†

⋔(Σ)then,

from the graph structure of NC†

⋔(Σ), we can tell the type of the clique.

If {a1,...,an}is an n-clique in the graph NC†

⋔(Σ), the homeomorphism type of

the subset ∪n

j=1ajof Σwill be called the type of the n-clique. We will explore this

only for 2and 3-cliques. A 2-clique {a, b},i.e., an edge of the graph NC†

⋔(Σ), may

have two distinct types: the intersection a∩bmay be empty or not. For a 3-clique

AUTOMORPHISMS OF SOME VARIANTS OF FINE GRAPHS 3

{a, b, c}, up to permuting the curves a,band c, the cardinals of the intersections

a∩b,a∩cand b∩c, respectively, may be (1,1,1), or (1,1,0), or (1,0,0), or (0,0,0).

This determines the type of the 3-clique, except in the case (1,1,1), where the

intersection points a∩b,a∩cand b∩cmay be pairwise distinct, in which case we

will speak of a 3-clique of type necklace, or these intersection points may be equal,

in which case we will speak of a 3-clique of type bouquet, see Figure 1.

Figure 1. A bouquet (left) and a necklace (right) of three circles.

The main bulk of the proof of Proposition 1.4 consists in distinguishing the 3-

cliques of type necklace from any other 3-clique of NC†

⋔(Σ). Here, the key is that

among all the 3-cliques, the cliques {a, b, c}of type necklace are exactly those such

that the union a∪b∪ccontains nonseparating simple closed curves other than a,b

and c. In terms of the graph structure, this leads to the following property, denoted

by N(a, b, c), which turns out to characterize these cliques:

There exists a ﬁnite set Fof at most 8vertices of NC†

⋔(Σ), all distinct from a,b

and c, such that every vertex dconnected to a,band cin this graph, is connected

to at least one element of F.

From this, we will easily characterize all the conﬁgurations of 2-cliques and 3-

cliques in terms of similar statements in the ﬁrst order logic of the graph NC†

⋔(Σ).

Now, let T(NC†

⋔(Σ))denote the set of edges {a, b}of NC†

⋔(Σ)satisfying ∣a∩b∣=1.

Then we have a map

Point∶T(NC†

⋔(Σ))→Σ,

which to each edge {a, b}of NC†

⋔(Σ), associates the intersection point a∩b. The

next step in the the proof of Theorem 1.1 now consists in characterizing the equality

Point(a, b)=Point(c, d)in terms of the structure of the graph. This characterization

shows that every automorphism of NC†

⋔(Σ)is realized by some bijection of Σ; then

we prove that such a bijection is necessarily a homeomorphism (see Proposition 3.1).

In order to characterize the equality Point(a, b)=Point(c, d), we introduce on

T(NC†

⋔(Σ)) the relation ygenerated, essentially (see section 3.2 for details), by

(a, b)y(b, c)if (a, b, c)is a 3-clique of type bouquet. Obviously, if (a, b)y(c, d)

then Point(a, b)=Point(c, d). Interestingly, the converse is false, but we can still

use this idea in order to characterize the points of Σin terms of the graph structure

of NC†

⋔(Σ).

This subtlety between the relation yand the equality of points is related to

the non smoothness of the curves involved, and more precisely, to the fact that a

curve may spiral inﬁnitely with respect to another curve in a neighborhood of a

common point. We think that this phenomenon is of independent interest and we

investigate it in Section 4. In particular, we can easily state, in terms of the graph

structure of NC†

⋔(Σ), an obstruction for a homeomorphism to be conjugate to a

C1-diﬀeomorphism, see Section 4.6.

4 FRÉDÉRIC LE ROUX AND MAXIME WOLFF

1.3. Ideas of the proof of Theorem 1.2. In the smooth case, the adaptation of

our proof of Theorem 1.1 fails from the start: indeed, the closed curves contained

in the union a∪b∪cof a necklace, and distinct from a,band c, are not smooth.

This suggests the idea to use sequences of curves (at the expense of losing the

characterizations of conﬁgurations in terms of ﬁrst order logic).

This time it is easiest to ﬁrst characterize disjointness of curves (see Lemma 5.7),

and then recover the diﬀerent types of 3-cliques. Then the strategy follows the C0

case.

Once we start to work with sequences, it is natural to say that a sequence (fn)of

curves not escaping to inﬁnity converges to ain some weak sense, if for every vertex

dsuch that {a, d}is an edge of the graph, {fn, d}is also an edge for all nlarge

enough. As it turns out, this property implies convergence in C0-sense to a, and

is implied by convergence in C1-sense. But it is not equivalent to the convergence

in C1-sense, and it is precisely this default of C1-convergence that enables us to

distinguish between disjoint or transverse pairs of curves.

Interestingly, this simple criterion for disjointness has no counterpart in the C0-

setting. Indeed, in that setting, no sequence of curves converges in this weak sense:

given a curve a, and a sequence (fn)of curves with, say, some accumulation point in

a, we can build a curve dintersecting aonce transversally (topologically), but oscil-

lating so much that it itersects every fnseveral times. From this perspective, none

of our approaches in the C0-setting and in the C∞-setting are directly adaptable to

the other.

1.4. Further comments. We can imagine many variants of ﬁne graphs. For ex-

ample, in the arXiv version of [2], for the case of the torus they worked with the

graph NC†

⋔(Σ)on which we are working here, whereas in the published version,

they changed to a ﬁne graph in which two curves aand bare still related by an edge

when they have one intersection, not necessarily transverse.

More generally, in the spirit of Ivanov’s metaconjecture, we expect that the group

of automorphisms should not change from any reasonable variant to another. And

indeed, using the ideas of [10, Section 2] and those presented here, we can navigate

between various versions of ﬁne graphs, and recover, from elementary properties of

one version, the conﬁgurations deﬁning the edges in another version, thus proving

that their automorphism groups are naturally isomorphic. From this perspective,

it seems satisfying to recover the group of homeomorphisms of the surface as the

automorphism group of any reasonable variant of the ﬁne graph. In this vein,

we should mention that the results of [10, Section 2] directly yield a natural map

Aut(C†(Σ)) →Aut(NC†

⋔(Σ)), and from there, our proof of Theorem 1.1 may be

used as an alternative proof of their main result.

All reasonable variants of the ﬁne graphs should be quasi-isometric, and a unifying

theorem (yet out of reach today, as it seems to us) would certainly be a counterpart

of the theorem by Raﬁ and Schleimer [13], which states that every quasi-isometry

of the usual curve graph is bounded distance from an isometry.

1.5. Organization of the article. Section 2 is devoted to the recognition of the

3-cliques in the C0-setting, and of some other conﬁgurations regarding the nonori-

entable case. We encourage the reader to skip, at ﬁrst reading, everything that

concerns the nonorientable case: these points shoud be easily identiﬁed, and this

halves the length of the proof. In Section 3 we prove Theorem 1.1. In Section 4

we characterize, from the topological viewpoint, the relation yintroduced above in

AUTOMORPHISMS OF SOME VARIANTS OF FINE GRAPHS 5

terms of the graph structure, and deduce our obstruction to diﬀerentiability. Finally

in Section 5 we prove Theorem 1.2 and Proposition 1.3.

Acknowledgments. We thank Kathryn Mann for encouraging discussions, and

Dan Margalit for his extensive feedback on a preliminary version of this manuscript.

2. Recognizing configurations of curves

2.1. Standard facts and notation. We will use, often without mention, the fol-

lowing easy or standard facts for curves on surfaces.

The ﬁrst is the classiﬁcation of connected, topological surfaces with boundary

(not necessarily compact). In particular, every topological surface admits a smooth

structure. Given a closed curve ain a surface Σ, we can apply this classiﬁcation

to Σ∖aand understand all possible conﬁgurations of simple curves; this is the

so-called change of coordinates principle in the vocabulary of the book of Farb and

Margalit [3].

In particular, every closed curve ahas a neighborhood homeomorphic to an an-

nulus or a Möbius strip in which ais the “central curve”. Very often in this article,

we will consider the curves a′obtained by deforming ain such a neighborhood, so

that a′is disjoint from ain the ﬁrst case, or intersects it once, transversely, in the

second, as in Figure 2. We will say that a′is obtained by pushing aaside.

The change of coordinates principle also applies to ﬁnite graphs embedded in Σ:

there is a homeomorphism of Σthat sends any given graph to a smooth graph,

such that all edges connected to a given vertex leave it in distinct directions. In

the simple case when the graph is the union of two or three simple closed curves

that pairwise intersect at most once, this observation justiﬁes the description of the

possible conﬁgurations of cliques in the introduction. This also enables, provided

two curves aand bintersect at a single point (or more generally at a ﬁnite number of

points), to speak of a transverse (also called essential), or to the contrary inessential,

intersection point, as we did in the introduction.

Here are two other useful facts.

Fact 2.1. A simple closed curve ain a surface, is nonseparating if and only if

there exists a closed curve b, such that a∩bis a single point and this intersection is

transverse.

Fact 2.2. Let p, q be two distinct points, and x, x′, x′′ three simple arcs, each with

end-points pand q, such that

x∩x′=x′∩x′′ =x∩x′′ ={p, q}.

If two of the three curves x∪x′, x′∪x′′, x′′ ∪xare separating, then the third one is

also separating.

Proof. Denote y=x∖{p, q}, the arc xwithout its ends, and similarly, deﬁne y′

and y′′. Suppose x∪x′and x∪x′′ are separating. Denote by Σ1,Σ2, resp. Σ3,Σ4,

the components of Σ∖(x∪x′), resp. Σ∖(x∪x′′), where Σ2contains y′′ and Σ4

contains y′. By looking at neighborhoods of pand q(see Figure 2, left), we see that

Σ′=Σ2∩Σ4is non empty, and that the arc ybounds Σ1on one side, and Σ3on the

other, so Σ′′ =Σ1∪y∪Σ3is a surface. Now, Σ∖(x′∪x′′)=Σ′∪Σ′′, and Σ′and Σ′′

are disjoint by construction. 

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AUTOMORPHISMSOFSOMEVARIANTSOFFINEGRAPHSFRÉDÉRICLEROUXANDMAXIMEWOLFFAbstract.RecentlyBowden,HenselandWebbdenedthenecurvegraphforsurfaces,extendingthenotionofcurvegraphsforthestudyofhomeomorphismordieomorphismgroupsofsurfaces.LaterLong,Margalit,Pham,VerberneandYaoprovedthatforaclosedsurfaceofgenusg...

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