uniform hyperbolicity for curve graphs of nonorientable surfacesmok7/slide/kuno_s.pdf ·...

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Uniform hyperbolicity for curve graphs of

nonorientable surfaces

Erika Kuno

Osaka university

December 26, 2014

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 1 / 29

Contents

...1 Motivation

...2 Preliminaries

...3 Unicorn paths

Definition of unicorn pathsUnicorn paths are paths in arc graphs

...4 Curve graphs are hyperbolic

Main theoremConstruction of a retractionOutline of proof of main theorem

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 2 / 29

Motivation

.Theorem 1.1 (Hensel-Przytycki-Webb 2013)..

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S: an orientable surfaceC(S): the curve graph of SIf C(S) is connected, then it is 17-(Gromov) hyperbolic.

.Theorem 1.2 (Bestvina-Fujiwara 2007, Masur-Schleimer2013)..

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N : a nonorientable surfaceC(N): the curve graph of NEach C(N) is Gromov hyperbolic.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 3 / 29

Motivation

.Theorem 1.1 (Hensel-Przytycki-Webb 2013)..

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S: an orientable surfaceC(S): the curve graph of SIf C(S) is connected, then it is 17-(Gromov) hyperbolic.

.Theorem 1.2 (Bestvina-Fujiwara 2007, Masur-Schleimer2013)..

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N : a nonorientable surfaceC(N): the curve graph of NEach C(N) is Gromov hyperbolic.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 3 / 29

Motivation

.Question 1.3..

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Is C(N) uniformly hyperbolic?How large is the hyperbolicity constant?

We try applying Hensel-Przytycki-Webb’s argument

to the case of nonorientable surfaces!

[Reference]S. Hensel, P. Przytycki, and R. C. H. Webb,Slim unicorns and uniform hyperbolicity for arc graphs and curvegraphs,to appear in J. of Europ. Math. Soc.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 4 / 29

Motivation

.Question 1.3..

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Is C(N) uniformly hyperbolic?How large is the hyperbolicity constant?

We try applying Hensel-Przytycki-Webb’s argument

to the case of nonorientable surfaces!

[Reference]S. Hensel, P. Przytycki, and R. C. H. Webb,Slim unicorns and uniform hyperbolicity for arc graphs and curvegraphs,to appear in J. of Europ. Math. Soc.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 4 / 29

Motivation

.Question 1.3..

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Is C(N) uniformly hyperbolic?How large is the hyperbolicity constant?

We try applying Hensel-Przytycki-Webb’s argument

to the case of nonorientable surfaces!

[Reference]S. Hensel, P. Przytycki, and R. C. H. Webb,Slim unicorns and uniform hyperbolicity for arc graphs and curvegraphs,to appear in J. of Europ. Math. Soc.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 4 / 29

Nonorientable surfaces

N = Ng,n: the nonorientable surface　　　　　 of genus g with n boundary components

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 5 / 29

Arc curve graphs, arc graphs, and curve graphs

arcs on N :properly embedded and essential , i.e. is not homotopicinto ∂N

curves on N :properly embedded and essential , i.e. does not bound adisk or a Mobius band, and is not homotopic into ∂N

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 6 / 29

Arc curve graphs, arc graphs, and curve graphs

.Definition 2.1..

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Arc curve graph AC(N)

The vertex set AC(0)(N) = { homotopy classes of arcs and curves onN }(a, b): edge in AC(N), (a, b ∈ AC(0)(N))⇔ the corresponding arcs or curves can be realized disjointly.Arc graph A(N)

⇔ the subgraph of AC(N) consists of homotopy classes of arcs on N

Curve graph C(N)

⇔ the subgraph of AC(N) consists of homotopy classes of curves onN .

We consider the graphs as geodesic spaces.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 7 / 29

Gromov hyperbolic graphs

A graph is hyperbolic⇔ there exists a constant k ≥ 0 (we call this constant hyperbolicityconstant) such that for all geodesic triangles T = abd (a, b, d:vertices) in the space, there exists a vertex p such that distancesbetween edges of the triangle and p are at most k (p is calledk-center of T ).

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 8 / 29

Gromov hyperbolic graphs

.Definition 2.2..

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Uniformly hyperbolic⇔ the hyperbolicity constant does not depend on topological typesof sufaces.

The important point of Main theorem

C(N) is 17-hyperbolic

independently of the topological types of N !

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 9 / 29

Gromov hyperbolic graphs

.Definition 2.2..

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Uniformly hyperbolic⇔ the hyperbolicity constant does not depend on topological typesof sufaces.

The important point of Main theorem

C(N) is 17-hyperbolic

independently of the topological types of N !

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 9 / 29

Definition of unicorn paths

αα′a: the subarc of an arc a whose endpoints are α and α′

.Definition 3.1 (Unicorn arc)..

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a, b: arcs on N which are in minimal positionα, β: one of the endpoints of a and bπ ∈ a ∩ ba′ ⊂ a: the arc whose endpoints are α and πb′ ⊂ b: the arc whose endpoints are β and πWhen a′ ∪ b′ is an embedded arc in N ,we say that a′ ∪ b′ is a unicorn arcobtained from aα, bβ and π.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 10 / 29

Definition of unicorn paths

Example

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 11 / 29

Definition of unicorn paths

.Definition 3.2 (Order)..

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a′ ∪ b′, a′′ ∪ b′′: two unicorn arcs obtained from aα, bβ

a′ ∪ b′ ≤ a′′ ∪ b′′ ⇔ a′′ ⊆ a′ and b′ ⊆ b′′

≤ is total order.

(c1, c2, ..., cn−1): ordered set of unicorn arcs obtained from aα, bβ

.Definition 3.3 (Unicorn path)..

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P(aα, bβ) := (a = c0, c1, ..., cn−1, cn = b)is called the unicorn path between aα and bβ.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 12 / 29

Definition of unicorn paths

.Definition 3.2 (Order)..

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a′ ∪ b′, a′′ ∪ b′′: two unicorn arcs obtained from aα, bβ

a′ ∪ b′ ≤ a′′ ∪ b′′ ⇔ a′′ ⊆ a′ and b′ ⊆ b′′

≤ is total order.

(c1, c2, ..., cn−1): ordered set of unicorn arcs obtained from aα, bβ

.Definition 3.3 (Unicorn path)..

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P(aα, bβ) := (a = c0, c1, ..., cn−1, cn = b)is called the unicorn path between aα and bβ.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 12 / 29

Definition of unicorn paths

.Definition 3.2 (Order)..

......

a′ ∪ b′, a′′ ∪ b′′: two unicorn arcs obtained from aα, bβ

a′ ∪ b′ ≤ a′′ ∪ b′′ ⇔ a′′ ⊆ a′ and b′ ⊆ b′′

≤ is total order.

(c1, c2, ..., cn−1): ordered set of unicorn arcs obtained from aα, bβ

.Definition 3.3 (Unicorn path)..

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P(aα, bβ) := (a = c0, c1, ..., cn−1, cn = b)is called the unicorn path between aα and bβ.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 12 / 29

Example of a unicorn path

P(aα, bβ) := (a, απ3a ∪ π3βb, απ4a ∪ π4βb, απ5a ∪ π5βb, b).Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 13 / 29

Unicorn paths are paths in arc graphs

a, b: two vertices of A(N).Proposition 3.4..

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Unicorn path P(aα, bβ) is a path connecting two vertices a and b inA(N).

Proof

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Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 14 / 29

Unicorn paths are paths in arc graphs

a, b: two vertices of A(N).Proposition 3.4..

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Unicorn path P(aα, bβ) is a path connecting two vertices a and b inA(N).

Proof

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Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 14 / 29

Main theorem

.Theorem 4.1 (K.)........If C(N) is connected, then it is 17-hyperbolic.

Hensel-Przytycki-Webb’s hyperbolicity constant

does not depend on the topological types of surfaces!

NoteC(N) is connected if g = 1, 2 and g + n ≥ 5, and g ≥ 3.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 15 / 29

Main theorem

.Theorem 4.1 (K.)........If C(N) is connected, then it is 17-hyperbolic.

Hensel-Przytycki-Webb’s hyperbolicity constant

does not depend on the topological types of surfaces!

NoteC(N) is connected if g = 1, 2 and g + n ≥ 5, and g ≥ 3.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 15 / 29

Construction of a retraction.Definition 4.2..

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Define a retraction r : AC(N) → C(N) as follows.If a ∈ C(0)(N), then r(a) = a.If a ∈ A(0)(N), we assign a boundary component of a regularneighborhood of its union with ∂N to r(a).

NoteIf there are two boundary components of the regular neighborhood,then we choose the essential one.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 16 / 29

Construction of a retraction.Definition 4.2..

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Define a retraction r : AC(N) → C(N) as follows.If a ∈ C(0)(N), then r(a) = a.If a ∈ A(0)(N), we assign a boundary component of a regularneighborhood of its union with ∂N to r(a).

NoteIf there are two boundary components of the regular neighborhood,then we choose the essential one.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 16 / 29

Construction of a retraction

Similar to the case of orientable surfaces, we can show the followingclaim.

.Claim 4.3..

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r is 2-Lipschitz,i.e. dC(r(a), r(b)) ≤ 2dAC(a, b) for any a, b ∈ AC(N).

The difference fromthe case of orientable surfaces

We have to consider ”twisted” r(a) and r(b).

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 17 / 29

Construction of a retraction

Similar to the case of orientable surfaces, we can show the followingclaim.

.Claim 4.3..

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r is 2-Lipschitz,i.e. dC(r(a), r(b)) ≤ 2dAC(a, b) for any a, b ∈ AC(N).

The difference fromthe case of orientable surfaces

We have to consider ”twisted” r(a) and r(b).

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 17 / 29

Construction of a retraction

Similar to the case of orientable surfaces, we can show the followingclaim.

.Claim 4.3..

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r is 2-Lipschitz,i.e. dC(r(a), r(b)) ≤ 2dAC(a, b) for any a, b ∈ AC(N).

The difference fromthe case of orientable surfaces

We have to consider ”twisted” r(a) and r(b).

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 17 / 29

Construction of a retraction

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 18 / 29

Construction of a retraction

Outline of Proof of Claim 4.3

We prove only for the pairs (a, b) (a, b ∈ AC(N))which fill dAC(a, b) = 1 .

The goal: to show dC(r(a), r(b)) ≤ 2

(1) a, b ∈ C(0)(N)dC(r(a), r(b)) = dC(a, b) = dAC(a, b) = 1 ≤ 2.

(2) a ∈ C(0)(N) and b ∈ A(0)(N)We can take the regular neighborhood of the union of b and boundarycomponents which have endpoints of b without intersecting a.dC(r(a), r(b)) = dC(a, r(b)) ≤ 1 ≤ 2.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 19 / 29

Construction of a retraction

Outline of Proof of Claim 4.3

We prove only for the pairs (a, b) (a, b ∈ AC(N))which fill dAC(a, b) = 1 .

The goal: to show dC(r(a), r(b)) ≤ 2

(1) a, b ∈ C(0)(N)dC(r(a), r(b)) = dC(a, b) = dAC(a, b) = 1 ≤ 2.

(2) a ∈ C(0)(N) and b ∈ A(0)(N)We can take the regular neighborhood of the union of b and boundarycomponents which have endpoints of b without intersecting a.dC(r(a), r(b)) = dC(a, r(b)) ≤ 1 ≤ 2.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 19 / 29

Construction of a retraction

Outline of Proof of Claim 4.3

We prove only for the pairs (a, b) (a, b ∈ AC(N))which fill dAC(a, b) = 1 .

The goal: to show dC(r(a), r(b)) ≤ 2

(1) a, b ∈ C(0)(N)dC(r(a), r(b)) = dC(a, b) = dAC(a, b) = 1 ≤ 2.

(2) a ∈ C(0)(N) and b ∈ A(0)(N)We can take the regular neighborhood of the union of b and boundarycomponents which have endpoints of b without intersecting a.dC(r(a), r(b)) = dC(a, r(b)) ≤ 1 ≤ 2.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 19 / 29

Construction of a retraction

Outline of Proof of Claim 4.3

We prove only for the pairs (a, b) (a, b ∈ AC(N))which fill dAC(a, b) = 1 .

The goal: to show dC(r(a), r(b)) ≤ 2

(1) a, b ∈ C(0)(N)dC(r(a), r(b)) = dC(a, b) = dAC(a, b) = 1 ≤ 2.

(2) a ∈ C(0)(N) and b ∈ A(0)(N)We can take the regular neighborhood of the union of b and boundarycomponents which have endpoints of b without intersecting a.dC(r(a), r(b)) = dC(a, r(b)) ≤ 1 ≤ 2.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 19 / 29

Construction of a retraction

Outline of Proof of Claim 4.3

We prove only for the pairs (a, b) (a, b ∈ AC(N))which fill dAC(a, b) = 1 .

The goal: to show dC(r(a), r(b)) ≤ 2

(1) a, b ∈ C(0)(N)dC(r(a), r(b)) = dC(a, b) = dAC(a, b) = 1 ≤ 2.

(2) a ∈ C(0)(N) and b ∈ A(0)(N)We can take the regular neighborhood of the union of b and boundarycomponents which have endpoints of b without intersecting a.dC(r(a), r(b)) = dC(a, r(b)) ≤ 1 ≤ 2.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 19 / 29

Construction of a retraction

(3) a, b ∈ A(0)(N)The following is all cases of the pairs.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 20 / 29

Construction of a retraction

If r(a) and r(b) are not disjoint, we can take a curve α withoutintersect both r(a) and r(b) as follows.

Example

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 21 / 29

Construction of a retraction

In all cases, there exists an essential curve α which does not intersectboth r(a) and r(b).

Hence, we can show that

dC(r(a), r(b)) ≤ dC(r(a), α) + dC(α, r(b)) ≤ 2.　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　□

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 22 / 29

Outline of proof of Main theorem

.Theorem 4.4 (K.)........If C(N) is connected, then it is 17-hyperbolic.

Outline of Proof of Main theorem

T = abd: any geodesic triangle in C(N) (a, b, d ∈ C(0)(N))a, b, d ∈ A(0)(N):arcs which are adjacent to a, b, d in AC(N).Now we use the following claim..Claim 4.5..

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P : any unicorn path between a and b.G = ab: the edge of T connecting a and b in C(N).c ∈ P : at maximal distance k from G.Then k ≤ 8.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 23 / 29

Outline of proof of Main theorem

.Theorem 4.4 (K.)........If C(N) is connected, then it is 17-hyperbolic.

Outline of Proof of Main theoremT = abd: any geodesic triangle in C(N) (a, b, d ∈ C(0)(N))a, b, d ∈ A(0)(N):arcs which are adjacent to a, b, d in AC(N).

Now we use the following claim..Claim 4.5..

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P : any unicorn path between a and b.G = ab: the edge of T connecting a and b in C(N).c ∈ P : at maximal distance k from G.Then k ≤ 8.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 23 / 29

Outline of proof of Main theorem

.Theorem 4.4 (K.)........If C(N) is connected, then it is 17-hyperbolic.

Outline of Proof of Main theoremT = abd: any geodesic triangle in C(N) (a, b, d ∈ C(0)(N))a, b, d ∈ A(0)(N):arcs which are adjacent to a, b, d in AC(N).Now we use the following claim..Claim 4.5..

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P : any unicorn path between a and b.G = ab: the edge of T connecting a and b in C(N).c ∈ P : at maximal distance k from G.Then k ≤ 8.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 23 / 29

Outline of proof of Main theorem

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 24 / 29

Outline of proof of Main theorem

α, β, δ: one of the endpoints of a, b, d..Lemma 4.6..

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There exist c′ ∈ P(aα, bβ), c′′ ∈ P(bβ, dδ), and c′′′ ∈ P(dδ, aα) suchthat each pair represent adjacent vertices in A(N).

By Claim 4.5 and Lemma 4.6,vertex c′ of AC(N) is at distance ≤ 9 from all sides of T, inparticular from a vertex of G = ab, which is a curve.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 25 / 29

Outline of proof of Main theorem

α, β, δ: one of the endpoints of a, b, d..Lemma 4.6..

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There exist c′ ∈ P(aα, bβ), c′′ ∈ P(bβ, dδ), and c′′′ ∈ P(dδ, aα) suchthat each pair represent adjacent vertices in A(N).

By Claim 4.5 and Lemma 4.6,vertex c′ of AC(N) is at distance ≤ 9 from all sides of T, inparticular from a vertex of G = ab, which is a curve.

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 25 / 29

Outline of proof of Main theorem

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 26 / 29

Outline of proof of Main theorem

Recall the retraction r : AC(N) → C(N).

Since r is 2-Lipschitz,

r(c′) becomes 17-center of the triangle T in C(N).　 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　□

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 27 / 29

Thank you for your attention !

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 28 / 29

Contents

...1 Motivation

...2 Preliminaries

...3 Unicorn pathsDefinition of unicorn pathsUnicorn paths are paths in arc graphs

...4 Curve graphs are hyperbolicMain theoremConstruction of a retractionOutline of proof of Main theorem

Erika Kuno (Osaka university) Uniform hyperbolicity for C(N) December 26, 2014 29 / 29