The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve

Matteo Costantini; André Kappes

doi:10.3934/jmd.2017002

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2017, Volume 11: 17-41. Doi: 10.3934/jmd.2017002

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The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve

Matteo Costantini and
André Kappes

Institut für Mathematik, Goethe-Universität Frankfurt/Main, Robert-Mayer-Str. 6–8, 60325 Frankfurt, Germany

Received: February 21, 2016

Revised: August 14, 2016

Published: November 2016

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Abstract

We compute the algebraic equation of the universal family over the Kenyon-Smillie (2, 3, 4)-Teichmüller curve, and we prove that the equation is correct in two different ways. Firstly, we prove it in a constructive way via linear conditions imposed by three special points of the Teichmüller curve. Secondly, we verify that the equation is correct by computing its associated Picard-Fuchs equation. We also notice that each point of the Teichmüller curve has a hyperflex and we see that the torsion map is a central projection from this point.

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Mathematics Subject Classification: Primary: 14H10, 32G15, 14H30.

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Figure 1. Real points of the curve $S_t$ for $t=3$ near $P_t=(0, 0)$ and the hyperflex $Q_t=(0, -1)$.

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Figure 2. The Kenyon-Smillie $(2, 3, 4)$-lattice surface resulting from unfolding the triangle $\Delta$. Sides are labeled by powers of $\zeta_9 = \exp(2\pi i/9)$, and sides with the same label are identified. The triple (simple) zero is marked by a white (black) dot.

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Figure 3. Dual graphs of the two cusps of the Teichmüller curve. The vertices represent the connected components and the edges correspond to the nodes of the stable curves associated with the cusps.

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Figure 4. Vertical cylinder decomposition of $S_{\infty}$ with cylinders $A$, $B$, $C$, $D$ (from light to dark).

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Figure 5. Horizontal cylinder decomposition of $S_0$ with cylinders $C_1$, $C_2$, $C_3$ (from light to dark).

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Table 1. Degree of $a_{i, j, k}(s_1, s_2)$.

i	4	3	3	2	2	2	1	1	1	1	0	0	0	0	0
j	0	1	0	2	1	0	3	2	1	0	4	3	2	1	0
k	0	0	1	0	1	2	0	1	2	3	0	1	2	3	4
4i+2j+k-7	9	7	6	5	4	3	3	2	1	0	1	0	-1	-2	-3

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