Equivalent Curves in Surfaces Anja Bankovi c University of - - PowerPoint PPT Presentation

Equivalent Curves in Surfaces

Anja Bankovi´ c

University of Illinois

Equivalent Curves

Fix a closed surface S, genus g ≥ 2.

Equivalent Curves

Fix a closed surface S, genus g ≥ 2. (Horowitz, Randol) ∀n ∈ Z+ ∃γ1, . . . , γn, distinct homotopy classes of closed curves on S with ℓm(γi) = ℓm(γj) ∀i, j ≤ n for every hyperbolic metric m on S.

Equivalent Curves

Fix a closed surface S, genus g ≥ 2. (Horowitz, Randol) ∀n ∈ Z+ ∃γ1, . . . , γn, distinct homotopy classes of closed curves on S with ℓm(γi) = ℓm(γj) ∀i, j ≤ n for every hyperbolic metric m on S. Problem: Do there exist pairs of distinct homotopy classes of curves γ and γ′ which have the same length with respect to every metric in a given family of path metrics?

Equivalent Curves

Fix a closed surface S, genus g ≥ 2. (Horowitz, Randol) ∀n ∈ Z+ ∃γ1, . . . , γn, distinct homotopy classes of closed curves on S with ℓm(γi) = ℓm(γj) ∀i, j ≤ n for every hyperbolic metric m on S. Problem: Do there exist pairs of distinct homotopy classes of curves γ and γ′ which have the same length with respect to every metric in a given family of path metrics? Generic metrics, probably ’no’...

Curves in Flat metrics

Flat(S, q) = {metrics from q–differentials on S}

Curves in Flat metrics

Flat(S, q) = {metrics from q–differentials on S} q-differential ϕ ζ(p) = p

p0

q

• ϕ(z) dz

Preferred coordinates give an atlas of charts on S\{zeros(ϕ)} to C

Curves in Flat metrics

Flat(S, q) = {metrics from q–differentials on S} q-differential ϕ ζ(p) = p

p0

q

• ϕ(z) dz

Preferred coordinates give an atlas of charts on S\{zeros(ϕ)} to C C(S) = {closed curves on S}/homotopy

Curves in Flat metrics

Flat(S, q) = {metrics from q–differentials on S} q-differential ϕ ζ(p) = p

p0

q

• ϕ(z) dz

Preferred coordinates give an atlas of charts on S\{zeros(ϕ)} to C C(S) = {closed curves on S}/homotopy Given γ, γ′ ∈ C(S), define γ ≡q γ′ iff lm(γ) = lm(γ′), ∀m ∈ Flat(S, q)

Curves in Flat metrics

Flat(S, q) = {metrics from q–differentials on S} q-differential ϕ ζ(p) = p

p0

q

• ϕ(z) dz

Preferred coordinates give an atlas of charts on S\{zeros(ϕ)} to C C(S) = {closed curves on S}/homotopy Given γ, γ′ ∈ C(S), define γ ≡q γ′ iff lm(γ) = lm(γ′), ∀m ∈ Flat(S, q)

Theorem

(C. Leininger) γ ≡hyp γ′ ⇒ γ ≡2 γ′

Main results

Main results

Theorem 1: ∀q, k ∈ Z+, ∃γ1, . . . , γk ∈ C(S) such that γi ≡q γj, ∀i, j.

Main results

Theorem 1: ∀q, k ∈ Z+, ∃γ1, . . . , γk ∈ C(S) such that γi ≡q γj, ∀i, j. Define γ ≡∞ γ′ iff lm(γ) = lm(γ′), ∀m ∈ Flat(S, q), ∀q ∈ Z+.

Main results

Theorem 1: ∀q, k ∈ Z+, ∃γ1, . . . , γk ∈ C(S) such that γi ≡q γj, ∀i, j. Define γ ≡∞ γ′ iff lm(γ) = lm(γ′), ∀m ∈ Flat(S, q), ∀q ∈ Z+. Theorem 2: ≡∞ is trivial.

Main results

Theorem 1: ∀q, k ∈ Z+, ∃γ1, . . . , γk ∈ C(S) such that γi ≡q γj, ∀i, j. Define γ ≡∞ γ′ iff lm(γ) = lm(γ′), ∀m ∈ Flat(S, q), ∀q ∈ Z+. Theorem 2: ≡∞ is trivial. Theorem 3: γ ≡1 γ′ ⇔ γ ≡2 γ′

Euclidian Cone Metrics

A metric σ on S is called a Euclidian cone metric if:

Euclidian Cone Metrics

A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric

Euclidian Cone Metrics

A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S\X is Euclidian.

Euclidian Cone Metrics

A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S\X is Euclidian. (c) (∀x ∈ X)(∃ǫ > 0)Bǫ(x) is isometric to some cone.

Euclidian Cone Metrics

A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S\X is Euclidian. (c) (∀x ∈ X)(∃ǫ > 0)Bǫ(x) is isometric to some cone.

Euclidian Cone Metrics

A metric σ on S is called a Euclidian cone metric if: (a) σ is a geodesic metric (b) ∃ finite set X ⊂ S such that σ on S\X is Euclidian. (c) (∀x ∈ X)(∃ǫ > 0)Bǫ(x) is isometric to some cone. Let c(x) ∈ R+ denote a cone angle around x.

Gluing surfaces

A genus 2 surface with Euclidian cone metric:

Gluing surfaces

A genus 2 surface with Euclidian cone metric:

Gluing surfaces

A genus 2 surface with Euclidian cone metric:

Holonomy

∀x ∈ S\X , ρx : π1(S\X, x) → SO(Tx(S\X))

Holonomy

∀x ∈ S\X , ρx : π1(S\X, x) → SO(Tx(S\X))

Holonomy

∀x ∈ S\X , ρx : π1(S\X, x) → SO(Tx(S\X)) Gluing maps: {ρi ◦ τi}k

i=1. Hol ≤ ρ1, ρ2, ..., ρk.

Flat metrics

A cone metric is NPC iff c(x) ≥ 2π, ∀x ∈ S.

Flat metrics

A cone metric is NPC iff c(x) ≥ 2π, ∀x ∈ S. Flat(S) = {m | m is NPC Euclidian cone metric on S inducing the given topology}.

Flat metrics

A cone metric is NPC iff c(x) ≥ 2π, ∀x ∈ S. Flat(S) = {m | m is NPC Euclidian cone metric on S inducing the given topology}. ∀q ∈ Z+, Flat(S, q) = {m ∈ Flat(S) | Hol(m) ⊂ ρ 2π

q }.

Geodesics in m ∈ Flat(S)

≥ π ≥ π ≥ π ≥ π ≥ π ≥ π

Geodesics in m ∈ Flat(S)

≥ π ≥ π ≥ π ≥ π ≥ π ≥ π Examples of geodesics on a genus 2 surface:

Proof of Theorem 1

Theorem 1: ∀q, k ∈ Z+, ∃γ1, . . . , γk ∈ C(S) such that γi ≡q γj, ∀i, j.

Proof of Theorem 1

Theorem 1: ∀q, k ∈ Z+, ∃γ1, . . . , γk ∈ C(S) such that γi ≡q γj, ∀i, j. Step 1: ∀q ∈ Z+, ∃γ ∈ C(S) so that ∀m ∈ Flat(S, q) the geodesic γm contains a cone point and [γ] = 0 in H1(S).

Step 2: Build a rank-2 free group from this curve:

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b

Step 2: Build a rank-2 free group from this curve: a b [a] = 2(q + 2)[γ] = [b]

Step 2: Build a rank-2 free group from this curve: a b [a] = 2(q + 2)[γ] = [b] w0 = (ab)k, w1 = (ab)k−1(ab−1), . . . , wk−1 = (ab)(ab−1)k−1

Step 2: Build a rank-2 free group from this curve: a b [a] = 2(q + 2)[γ] = [b] w0 = (ab)k, w1 = (ab)k−1(ab−1), . . . , wk−1 = (ab)(ab−1)k−1 [wj] = (2k − 2j)[γ]

Proof of Theorem 2

Theorem 2: ≡∞ is trivial.

Proof of Theorem 2

Theorem 2: ≡∞ is trivial. Step 1: ∀γ, γ′ ∈ C(S), ∃m ∈ Flat(S) so that lm(γ) = lm(γ′).

Proof of Theorem 2

Theorem 2: ≡∞ is trivial. Step 1: ∀γ, γ′ ∈ C(S), ∃m ∈ Flat(S) so that lm(γ) = lm(γ′).

Proof of Theorem 2

Theorem 2: ≡∞ is trivial. Step 1: ∀γ, γ′ ∈ C(S), ∃m ∈ Flat(S) so that lm(γ) = lm(γ′). Step 2: Approximate the Flat(S) metrics by some metric in Flat(S, q) by appropriately choosing the metrics on the triangles.