Abstract
Many applications of the dynamics of \(SL(2,\mathbb {R})\) on moduli spaces of translation surfaces to the investigation of translation flows and billiards rely on the features of the closure of certain \(SL(2,\mathbb {R})\)-orbits.
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Notes
- 1.
See Subsection 1.10 above.
- 2.
In fact, it was conjectured by Bainbridge-Möller [8] that it contains exactly two non-arithmetic Teichmüller curves, namely, the algebraically primitive closed \(SL(2,\mathbb {R})\)-orbit generated by the regular 7-gon and the non-algebraically primitive closed \(SL(2,\mathbb {R})\)-orbit generated by the 12-gon.
- 3.
Technically speaking, this parallel transport might be well-defined only on an adequate finite cover of \(\mathcal {C}\) (getting rid of all ambiguities coming from eventual automorphisms of M): see Remark 6. Of course, this minor point does not affect the arguments in this section and, for this reason, we will skip in all subsequent discussion.
- 4.
Recall that Hodge’s decomposition theorem says that \(H^1(M,\mathbb {C}) = H^{1,0}(M)\oplus H^{0,1}(M)\) where \(H^{1,0}(M)\), resp. \(H^{0,1}(M)\), is the space of holomorphic, resp. anti-holomorphic, forms.
- 5.
I.e., this is a \(SL(2,\mathbb {R})\)-equivariant decomposition such that the complexification of each \(\mathbb {L}_j\) is the sum of its (1, 0) and (0, 1) parts: \((\mathbb {L}_j)_{\mathbb {C}}:=\mathbb {L}_j\otimes \mathbb {C}\) equals \(\mathbb {L}_j^{1,0}\oplus \mathbb {L}_j^{0,1}\) where \(\mathbb {L}_j^{a,b} := (\mathbb {L}_j)_{\mathbb {C}}\cap H^{a, b}(X)\).
- 6.
Recall that the Hodge-star operator \(*:H^1(M,\mathbb {R})\rightarrow H^1(M,\mathbb {R})\) is defined by the fact that the form \(c+i(*c)\) is holomorphic for all \(c\in H^1(M,\mathbb {R})\).
- 7.
See Lemma 3.4 of [32] for more details.
- 8.
The Hodge norm \(\Vert .\Vert \) is \(\Vert c\Vert ^2:=\int _M c\wedge *c\).
- 9.
For instance, the set of symplectic planes is open in the Grassmannian of planes while the set of planes whose complexification intersects \(H^{1,0}\) has positive codimension when \(g\ge 3\).
- 10.
Recall that if \(\{\alpha _i,\beta _i: i=1,\dots , g\}\) is a canonical symplectic basis on a genus g translation surface \((M,\omega )\), then \(\Phi (M):=\sum \limits _{i=1}^g(\text {ind}_{\omega }(\alpha _i)+1)(\text {ind}_{\omega }(\beta _i)+1)\) where \(\text {ind}_{\omega }(\gamma )\) is the degree of the Gauss map associated to the tangents of a curve \(\gamma \) not intersecting the set \(\text {div}(\omega )\) of zeroes of \(\omega \).
- 11.
More precisely, we need the fact stated in [44, Corollary 4] that the boundary of any connected component \(\mathcal {C}\) of any stratum of the moduli space \(\mathcal {H}_g\) of translation surfaces of genus g contains a connected component \(\mathcal {C}'\) of the minimal stratum \(\mathcal {H}(2g-2)\).
- 12.
Here, the fact that the angles of the torus do not degenerate (because it is always a square) is crucial.
- 13.
I.e., a point in the boundary of Deligne-Mumford compactification of the moduli space of curves.
- 14.
Here, by “elementary” we mean that, instead of proving the existence of \(M_{\mathcal {C}}\) by indirect methods (including the use of Deligne-Mumford compactification), we will build \(M_{\mathcal {C}}\) directly for certain \(\mathcal {C}\)’s.
- 15.
Our discussion of these particular cases follows an argument described in a blog post entitled “Hodge-Teichmüller planes and finiteness results for Teichmüller curves” in my mathematical blog [17].
- 16.
The “shape” of the covering \(\overline{M}_{*}(d)\) was “guessed” with the help of the computer program Sage. In fact, I tried a few simple-minded finite coverings of \(\overline{M}_{*}\) (including \(\overline{M}_{*}(d)\) for \(d=3, 5, \dots , 13\)) and I asked Sage to determine their connected components. Then, once we got the “correct” connected components (in minimal strata), I looked at the permutations corresponding to these square-tiled surfaces and I found the “partner” leading to the expressions for the permutations \(\overline{h}_{*}(d)\) and \(\overline{v}_{*}(d)\).
- 17.
A Gram-Schmidt orthogonalization process to produce canonical basis of homology modulo 2.
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Matheus Silva Santos, C. (2018). Some Finiteness Results for Algebraically Primitive Teichmüller Curves. In: Dynamical Aspects of Teichmüller Theory. Atlantis Studies in Dynamical Systems, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-92159-4_4
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