Golden Ratio on Surfaces - CAUcau.ac.kr/~jyham/EIMI.pdf · 2019-10-16 · Golden Ratio on Surfaces...
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Golden Ratio on Surfaces Ji-Young Ham and Joongul Lee Riemann surfaces and Teichmuller theory, Euler International Mathematical Institute, St. Petersburg, Russia July 8-12, 2019 Golden ratio on nonorientable surfaces Theorem [HamLee19] Let k ≥ 3 be an odd natural number. On the Liechti-Strenner surface Σ 2k,k of genus k +2, the mapping class Φ k = r ◦ T c 1 ◦ r k-1 has an invariant subsurface such that the dilatation on the invariant subsurface is the golden ratio. Liechti-Strenner construction of nonorientable surfaces: The graph G 2k,k Let k ≥ 3 be an odd natural number. Let G 2k,k be the graph whose vertices are the vertices of a regular 2k-gon and every vertex v is connected to the k vertices that are the farthest away from v in the cyclic order of the vertices. Figure: The graph G 6,3 . Liechti-Strenner construction of nonorientable surfaces: The surface Σ 2k,k For each G 2k,k , Liechti and Strenner constructed an nonorientable surface Σ 2k,k that contains a collection of curves with intersection graph G 2k,k . Lemma [LiechtiStrenner18] The surface Σ 2k,k is homeomorphic to the nonorientable surface of genus k +2 with k boundary components. 6 1 2 3 4 5 1 2 3 4 5 6 c 1 Figure: The surface Σ 6,3 and the curve c 1 . A collection of filling inconsistently marked curves Figure: A collection of filling inconsistently marked curves. This gives rise to Linear Algebra Theorem [Penner88] The action of S k (C k ) on H admits a faithful representation as a semigroup of invertible (over Z) positive matrices. An invariant surface of Φ 3 whose dilatation is the golden ratio Figure: An invariant surface of Φ 3 whose dilatation is the golden ratio. Golden ratio on orientable surfaces Theorem [HamLee19] Let k ≥ 4 be an even natural number. On the Liechti-Strenner surface Σ k of genus k - 1, the mapping class Φ k = T c 1 ◦ r k-1 ◦ T c 1 ◦ r has an invariant subsurface such that the dilatation on the invariant subsurface is the golden ratio. Liechti-Strenner construction of orientable surfaces: The graph G k Let k ≥ 4 be an even natural number. Let G k be the graph whose vertices are the vertices of a regular 2k-gon and every vertex i is connected to the i + k - 1 and i + k +1 in the cyclic order of the vertices. c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Figure: The graph G 4 . Liechti-Strenner construction of orientable surfaces: The surface Σ k For each G k , Liechti and Strenner constructed an orientable surface Σ k that contains a collection of curves with intersection graph G k . For every k ≥ 4, the surface Σ k is obtained by chaining together 2k annuli in a cycle. Lemma [LiechtiStrenner18] The surface Σ k is homeomorphic to the orientable surface of genus k - 1 with 4 boundary components. Figure: The surface Σ 4 . A collection of filling inconsistently marked curves c 1 c 4 c 7 c 2 c 5 c 8 c 3 c 6 Figure: A collection of filling inconsistently marked curves. This gives rise to Linear Algebra Theorem [Penner88] The action of S k (C k ) on H admits a faithful representation as a semigroup of invertible (over Z) positive matrices. An invariant surface of Φ 4 whose dilatation is 1+ √ 5 2 Figure: An invariant surface of Φ 4 whose dilatation is 1+ √ 5 2 . http://cau.ac.kr/∼jyham [email protected], [email protected]
Transcript of Golden Ratio on Surfaces - CAUcau.ac.kr/~jyham/EIMI.pdf · 2019-10-16 · Golden Ratio on Surfaces...
Golden Ratio on SurfacesJi-Young Ham and Joongul Lee
Riemann surfaces and Teichmuller theory,Euler International Mathematical Institute, St. Petersburg, Russia
July 8-12, 2019
Golden ratio on nonorientable surfaces
Theorem
[HamLee19] Let k ≥ 3 be an odd natural number. On the Liechti-Strennersurface Σ2k,k of genus k + 2, the mapping class Φk = r ◦ Tc1 ◦ rk−1 has aninvariant subsurface such that the dilatation on the invariant subsurface is thegolden ratio.
Liechti-Strenner construction of nonorientable surfaces: The graphG2k,k
Let k ≥ 3 be an odd natural number. Let G2k,k be the graph whose verticesare the vertices of a regular 2k-gon and every vertex v is connected to the kvertices that are the farthest away from v in the cyclic order of the vertices.
Figure: The graph G6,3.
Liechti-Strenner construction of nonorientable surfaces: The surfaceΣ2k,k
For each G2k,k, Liechti and Strenner constructed an nonorientable surface Σ2k,kthat contains a collection of curves with intersection graph G2k,k.
Lemma
[LiechtiStrenner18] The surface Σ2k,k is homeomorphic to the nonorientablesurface of genus k + 2 with k boundary components.
6
1
2
3
4
51
2
3
4
5
6
c1
Figure: The surface Σ6,3 and the curve c1.
A collection of filling inconsistently marked curves
Figure: A collection of filling inconsistently marked curves.
This gives rise to Linear Algebra
Theorem
[Penner88] The action of Sk(Ck) on H admits a faithful representation as asemigroup of invertible (over Z) positive matrices.
An invariant surface of Φ3 whose dilatation is the golden ratio
Figure: An invariant surface of Φ3 whose dilatation is the golden ratio.
Golden ratio on orientable surfaces
Theorem
[HamLee19] Let k ≥ 4 be an even natural number. On the Liechti-Strennersurface Σk of genus k − 1, the mapping class Φk = Tc1 ◦ rk−1 ◦ Tc1 ◦ r hasan invariant subsurface such that the dilatation on the invariant subsurface isthe golden ratio.
Liechti-Strenner construction of orientable surfaces: The graph Gk
Let k ≥ 4 be an even natural number. Let Gk be the graph whose vertices arethe vertices of a regular 2k-gon and every vertex i is connected to thei + k − 1 and i + k + 1 in the cyclic order of the vertices.
c1
c2
c3
c4
c5
c6
c7
c8
Figure: The graph G4.
Liechti-Strenner construction of orientable surfaces: The surface Σk
For each Gk, Liechti and Strenner constructed an orientable surface Σk thatcontains a collection of curves with intersection graph Gk. For every k ≥ 4, thesurface Σk is obtained by chaining together 2k annuli in a cycle.Lemma
[LiechtiStrenner18] The surface Σk is homeomorphic to the orientable surface ofgenus k − 1 with 4 boundary components.
Figure: The surface Σ4.
A collection of filling inconsistently marked curves
c1
c4
c7
c2
c5
c8
c3
c6
Figure: A collection of filling inconsistently marked curves.
This gives rise to Linear Algebra
Theorem
[Penner88] The action of Sk(Ck) on H admits a faithful representation as asemigroup of invertible (over Z) positive matrices.
An invariant surface of Φ4 whose dilatation is 1+√5
2
Figure: An invariant surface of Φ4 whose dilatation is 1+√5
2.
http://cau.ac.kr/∼jyham [email protected], [email protected]
