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

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.

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.

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.

Motivation

.Question 1.3..

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Motivation

.Question 1.3..

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Nonorientable surfaces

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

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

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.

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 ).

.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 !

.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 !

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.

Example

.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.

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

≤ is total order.

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

≤ is total order.

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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).

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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).

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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.

Main theorem

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.

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.

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.

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).

.Claim 4.3..

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.Claim 4.3..

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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.

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

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

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.　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　□

Outline of proof of Main theorem

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.

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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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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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.

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.

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).　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　□

Thank you for your attention !

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

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

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