Finite rigid sets of the non-separating curve complex_2

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FINITE RIGID SETS OF THE NON-SEPARATING CURVE

COMPLEX

RODRIGO DE POOL

Abstract. We prove that the non-separating curve complex of every

surface of ﬁnite type and genus at least three admits an exhaustion by

ﬁnite rigid sets.

1. Introduction

Let Sbe a connected, orientable, ﬁnite-type surface. The curve complex

is the simplicial complex C(S) whose k-simplices are sets of to k+ 1 distinct

isotopy classes of essential simple curves on Sthat are pairwise disjoint.

The extended mapping class group, denoted Mod±(S), acts naturally on

the set of curves up to isotopy on S. This action preserves disjointness of

curves, and therefore extends to an action on the complex C(S). Via this

action, the curve complex works as a combinatorial model to study proper-

ties of Mod±(S). For instance, a celebrated theorem of Ivanov in [12] asserts

that for suﬃciently complex surfaces the group Mod±(S) is isomorphic to

the group of simplicial automorphisms of the curve complex, a result com-

monly known as simplicial rigidity. In turn, this result is a key ingredient

in establishing the isomorphism Aut(Mod±(S)) ∼

=Mod±(S).

The curve complex, and its applications to the mapping class group, has

motivated the study of similar complexes associated to surfaces. For exam-

ple, simplicial rigidity has been established for: the arc complex [11]; the

non-separating curve complex [8]; the separating curve complex [3], [13];

the Hatcher-Thurston complex [10]; and the pants graph [15] (see [17] for a

survey on complexes associated to surfaces).

Another notion of rigidity which has been of recent interest is that of

ﬁnite rigidity: the simplicial complex C(S) is said to be ﬁnitely rigid if there

exists a ﬁnite subcomplex Xsuch that any locally injective simplicial map

φ:X→ C(S)

is induced by a unique mapping class, that is, there exists a unique h∈

Mod±(S) such that the simplicial action h:C(S)→ C(S) satisﬁes h|X=φ.

Such Xis called a ﬁnite rigid set of C(S) with trivial pointwise stabilizer.

The ﬁnite rigidity of the curve complex was proven by Aramayona and

Leininger in [1], thus answering a question by Lars Louder. Furthermore,

they constructed in [2] an exhaustion of C(S) by ﬁnite rigid sets with trivial

pointwise stabilizers, thus recovering Ivanov’s result [12] on the simplicial

rigidity of C(S).

Following the result of Aramayona and Leininger, ﬁnite rigidity has been

proven for other complexes: Shinkle proved it for the arc complex [18] and

the ﬂip graph [19]; Hern´andez, Leininger and Maungchang proved a slightly

arXiv:2210.05317v2 [math.GT] 29 Nov 2023

2 RODRIGO DE POOL

diﬀerent notion for the pants graph [5], [16]; and Huang and Tshishiku

proved a weaker notion for the separating curve complex [6].

The main goal of this article is to prove the ﬁnite rigidity of the non-

separating curve complex,N(S), which is the subcomplex of C(S) spanned

by the non-separating curves. To prove the ﬁnite rigidity of N(S), one

would like to restrict the ﬁnite rigid set of C(S) in [1] to N(S), however it is

not clear why this restriction would yield a ﬁnite rigid set in N(S) as their

proof uses separating curves in a fundamental way. Below, we construct a

diﬀerent subcomplex of N(S) and prove its rigidity.

Our main result is compiled in the next theorem.

Theorem 1.1. Let Sbe a connected, orientable, ﬁnite-type surface of genus

g≥3. There exists a ﬁnite simplicial complex X⊂ N (S)such that any

locally injective simplicial map

φ:X→ N (S)

is induced by a unique h∈Mod±(S).

Our second result produces an exhaustion of the non-separating curve

complex by ﬁnite rigid sets:

Theorem 1.2. Let Sbe an orientable ﬁnite-type surface of genus g≥3.

There exist subcomplexes X1⊂X2⊂ · · · ⊂ N (S)such that

∞

[

i=1

Xi=N(S)

and each Xiis a ﬁnite rigid set with trivial pointwise stabilizer.

From Theorem 1.2 we can recover the simplicial rigidity of N(S) (see [9,

Theorem 1.1]):

Corollary 1.3. Let Sbe a connected, orientable, ﬁnite-type surface of genus

g≥3. Any locally injective simplicial map φ:N(S)→ N (S)is induced by

a unique h∈Mod±(S). In particular, this yields an isomorphism between

Mod±(S)and the group of simplicial automorphisms of N(S).

Plan of the paper. In Section 2we introduce some basic deﬁnitions that

will be required. In Section 3we introduce the notion of ﬁnite rigid sets.

Sections 4and 5deal with the proofs of Theorems 1.1 and 1.2 for closed

surfaces. Lastly, sections 6and 7present the proofs of theorems 1.1 and 1.2

for punctured surfaces.

Acknowledgements. The author would like to thank his supervisor Javier

Aramayona for suggesting the problem and for the guidance provided.

The author acknowledges ﬁnancial support from grant CEX2019-000904-

S funded by MCIN/AEI/ 10.13039/501100011033.

2. Preliminaries

Let Sbe connected, orientable surface without boundary. We will further

assume that Shas ﬁnite type, i.e, π1(S) is ﬁnitely generated. As such,

Sis homeomorphic to Sg,n, the result of removing npoints from a genus g

FINITE RIGID SETS OF THE NON-SEPARATING CURVE COMPLEX 3

surface. We refer to the removed points as punctures. If Shas no punctures,

we will say Sis closed. Otherwise, we will refer to Sas a punctured surface.

Before fervently jumping into the proofs, we warn the reader that the clas-

siﬁcation of surfaces, the change of coordinates principle and the Alexander

method will be frequently used in proofs, sometimes without mention. For

these and other fundamental results on mapping class groups, we refer the

reader to [4].

2.1. Curves. By a curve cin Swe will mean the isotopy class of an unori-

ented simple closed curve that does not bound a disk or a punctured disk.

Throughout the article, we will make no distinction between curves and their

representatives. We will say cis non separating if any representative γof c

has connected complement in S.

The intersection number i(a, b) between two curves aand bis the mini-

mum intersection number between representatives of aand b. If i(a, b) = 0,

we will say the curves aand bare disjoint. Given two representatives α∈a

and β∈b, we say they are in minimal position if i(a, b) = |α∩β|. A fact

that will be often used is that for any set of curves we may pick a single

representative for each curve, such that the representatives are pairwise in

minimal position (see [4, Chapter 1.2]).

Given a set of curves {c1, . . . , ck}, consider representatives γi∈cipairwise

in minimal position. We will denote a regular neighborhood of Sγiby

N(Sγi). The set of curves in the boundary of N(Sγi) will be denoted

∂(c1, . . . , ck).

We emphasize that implicit in the deﬁnition of b∈∂(c1, . . . , ck), is that bis

an isotopy class of a simple closed curve which does not bound a disk or a

punctured disk.

2.2. Non-separating curve complex. The non-separating curve complex

N(S) is the simplicial complex whose k-simplices are sets of k+ 1 isotopy

classes of pairwise disjoint curves.

Note that we can endow N(S) with a metric by declaring each k-simplex

to have the standard euclidean metric and considering the resulting path

metric on N(S).

2.2.1. Pants decompositions. The dimension of N(Sg,n) is 3g−3+nand the

vertex set of a top-dimensional simplex in N(Sg,n) is called a non-separating

pants decomposition of Sg,n. If P={c1, . . . , ck}is a pants decomposition

of S, then S\Sciis a union of pairs of pants (i.e, a union of subsurfaces

homeomorphic to S0,3).

Let P={c1, . . . , ck}be a pants decomposition of the surface S. Two

curves ci, cj∈Pare said to be adjacent rel to P if there exists a curve

ck∈Psuch that ci, cj, ckbound a pair of pants in S; see Figure 1a for an

example.

We record the following observation for future use.

Remark 2.1. Let Pbe a non-separating pants decomposition of Sg,n, where

g≥3:

•If n≤1, then every curve in Pis adjacent rel to Pto at least three

other curves.

4 RODRIGO DE POOL

•If n > 1, then every curve is adjacent to at least two other curves.

Consider A⊂P, where Pis a pants decomposition of S. We say that a

set of curves ˜

Asubstitutes Ain Pif

(˜

A∪P)\A

is a pants decomposition. In words, we say ˜

Asubstitutes Ain Pif both sets

have no curves in common and we can replace the curves in Aby the curves

in ˜

Aand still get a pants decomposition.

3. Finite rigid sets

For a simplicial subcomplex X⊂ N (S), a map φ:X→ N (S) is said to

be a locally injective simplicial map if φis simplicial and injective on the

star of each vertex. A ﬁrst elementary observation is the following:

Lemma 3.1. Let φ:X→ N (S)be a locally injective simplicial map. If

P⊂Xis a pants decomposition, then φ(P)is a pants decomposition.

Proof. Take a vertex p∈P. Since φis injective in the star of pand Pis

a simplex, then φis injective on P. Thus, φ(P) is a maximal-dimensional

simplex, i.e, φ(P) is a pants decomposition. □

As mentioned in the introduction, the main goal of this article is to con-

struct a ﬁnite subcomplex X⊂ N (S) with the following properties:

Deﬁnition 3.2 (Finite rigid set).Aﬁnite rigid set Xof N(S)is a ﬁnite

subcomplex such that any locally injective simplicial map φ:X→ N (S)is

induced by a mapping class, i.e, there exists h∈Mod±(S)with h|X=φ.

In addition, if his unique, we say Xhas trivial pointwise stabilizer.

Observe that a subcomplex X⊂ N (S) has trivial pointwise stabilizer if

and only if the inclusion X → N (S) is induced uniquely by the identity

1∈Mod±(S), hence the name.

Remark 3.3. By the change of coordinates principle (see [4, Chapter 1.3]),

every vertex {v}⊂N(S)is a ﬁnite rigid set. However, the stabilizer of {v}

is not trivial.

Remark 3.4. If Xis a ﬁnite rigid set and X⊂Y, then Ymay not be a

ﬁnite rigid set. For example:

Consider two disjoint curves v1, v2such that S\Sviis connected, and two

disjoint curves v′

1, v′

2such that S\Sv′

iis disconnected. Now, take X={v1},

Y={v1, v2}and the locally injective simplicial map φ(vi) = v′

i. Clearly, φ

is not induced by a mapping class and so Yis not a ﬁnite rigid set of N(S).

Note that Xis a ﬁnite rigid set of N(S)by the remark above.

Following Aramayona and Leininger in [1], we will say that a subcomplex

X⊂ N (S)detects the intersection of two curves a, b ∈X, if every locally

injective simplicial map φ:X→ N (S) satisﬁes

i(a, b)̸= 0 ⇔i(φ(a), φ(b)) ̸= 0.

4. Finite rigid sets for closed surfaces

In this section we construct ﬁnite rigid sets for closed surfaces and prove

their rigidity. This will establish Theorem 1.1 for closed surfaces.

FINITE RIGID SETS OF THE NON-SEPARATING CURVE COMPLEX 5

4.1. Constructing the ﬁnite rigid set. Let Sbe a closed surface of genus

g≥3. We will start by deﬁning the curves in the ﬁnite rigid set. The reader

should keep ﬁgures 1a,1b,1c,1d,1f and 1e in mind throughout the section.

Fix a set {p1, c1, . . . , pg, cg, pg+1}of non-separating curves such that

i(ci, pi) = i(ci, pi+1) = 1 and the rest of the curves are pairwise disjoint (see

ﬁgures 1a and 1b). Such a set of curves is unique up to homeomorphism.

Let cg+1 be a curve such that i(p1, cg+1) = i(pg+1, cg+1) = 1 and is disjoint

from every other curve in the set above. We deﬁne

C={c1, . . . , cg+1}.

Notice that S\Spihas two connected components (S\Spi)+and (S\

Spi)−, we will call (S\Spi)+the top component and (S\Spi)−the bottom

component. In the same fashion, S\Scihas two connected components

(S\Sci)+and (S\Sci)−, we will call (S\Sci)+the front component and

(S\Sci)−the back component.

For each k= 2, . . . , g−1, the set ∂(p1, c1, . . . , pk, ck) consists of two curves:

one of them in (S\Spi)+and the other one in (S\Spi)−. We will call

kthe curve of ∂(p1, c1, . . . , pk, ck) contained in (S\Spi)+and by p−

kthe

curve of ∂(p1, c1, . . . , pk, ck) in (S\Spi)−. We set

P={p1...,pg+1}∪{p+

2, p−

2, . . . , p+

g−1, p−

g−1}.

Notice that Pis a pants decomposition (see Figure 1a).

For each k= 2, . . . , g −1, the set ∂(pk−1, ck, pk) has two curves, one in

(S\Spi)+and the other one in (S\Spi)−. We will denote by ukthe curve

in (S\Spi)+and by dkthe curve in (S\Spi)−(see Figure 1c). We set

U={u2, . . . , ug−1}

and

D={d2, . . . , dg−1}.

Given k∈ {2, . . . , g −1}the set ∂(pk, ck, p+

k) contains two curves and

only one of them is also a curve in P. We will denote by lkthe curve in

∂(pk, ck, p+

k) not already in P(see Figure 1d). We set

L={l2, . . . , lg−2}.

Analogously, let R={r2, . . . , rg−2}be the set of curves where rkis the

unique curve in ∂(pk+1+, ck+1, pk+2) that is not in P(see Figure 1d).

The set ∂(p2, c2, . . . pg−1, cg−1, pg) has two curves, one curve in each

component of S\Spi. Let bbe the curve contained in the bottom component

(S\Spi)−. Then, the set ∂(c1, b,cg) has exactly two curves, one curve

contained in (S\Sci)+and the other in (S\Sci)−. Denote by nd the

curve in (S\Sci)+(see Figure 1e).

Lastly, consider the torus T1that contains p1and is bounded by the curves

2, p−

2. Let nl be the unique curve contained in T1\(nd ∪c1) distinct from

c1and p+

2. In the same way, p+

g−1, p−

g−1bound a torus Tgsuch that pg⊂Tg,

let nr be the unique curve in Tg\(nd ∪cg) distinct from cgand p+

g−1(see

Figure 1f). We set

N={nl, nr, nd}.

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FINITERIGIDSETSOFTHENON-SEPARATINGCURVECOMPLEXRODRIGODEPOOLAbstract.Weprovethatthenon-separatingcurvecomplexofeverysurfaceoffinitetypeandgenusatleastthreeadmitsanexhaustionbyfiniterigidsets.1.IntroductionLetSbeaconnected,orientable,finite-typesurface.ThecurvecomplexisthesimplicialcomplexC(S)whosek-s...

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