To every half-translation surface with marked points, Pan (Int. Math. Res. Not. 2022, published online 2020) associated the saddle-connection graph and its quotient by the automorphism group, which up to an index-two exception is the affine automorphism group of the surface. Veech (lattice) surfaces are known to yield quotients with finitely many vertices, and among translation surfaces only the once-marked torus yields a quotient with finitely many edges. The open problem is the converse and the full characterization: does finiteness of the vertex set of the quotient saddle-connection graph force the surface to be a Veech surface, and which half-translation surfaces have quotient graphs with finitely many edges? No resolution of either part is known in the literature to date.

Background

A half-translation surface (X,ω;Σ)(X,\omega;\Sigma) is a closed Riemann surface XX carrying a meromorphic quadratic differential ω\omega (locally of the form f(z) dz2f(z)\,dz^2, with no poles of order greater than one) together with a finite set Σ\Sigma of marked points containing all zeros and poles of ω\omega. Integrating a square root of ω\omega gives charts whose transition maps are half-translations z↦±z+cz\mapsto\pm z+c, so XX inherits a singular flat metric ∣ω∣|\omega| with cone-type singularities at Σ\Sigma. A saddle connection is a geodesic segment for this flat metric whose interior avoids Σ\Sigma and whose endpoints lie in Σ\Sigma. The saddle connection graph S(X,ω;Σ)S(X,\omega;\Sigma) has saddle connections as vertices, with an edge joining two saddle connections whenever they are interiorly disjoint; it embeds into the arc graph (the graph of isotopy classes of embedded arcs on the underlying topological surface). A homeomorphism of (X,ω;Σ)(X,\omega;\Sigma) that is affine in the flat coordinates away from Σ\Sigma is an affine homeomorphism, and the group Aff(X,ω;Σ)\mathrm{Aff}(X,\omega;\Sigma) of all such homeomorphisms has a linear part, the Veech group, a discrete subgroup of SL(2,R)\mathrm{SL}(2,\mathbb{R}) or PSL(2,R)\mathrm{PSL}(2,\mathbb{R}). The surface is a Veech surface, or lattice surface, when its Veech group is a lattice, that is, has finite covolume. Veech surfaces govern the most rigid flat dynamical systems (the Veech dichotomy: every direction is either completely periodic, meaning the surface decomposes into parallel cylinders along that direction, or uniquely ergodic, meaning the foliation in that direction carries a unique transverse invariant measure) and arise from rational billiards and Teichmüller curves, the closed linear-group orbits in the moduli space of flat surfaces obtained by projecting the Teichmüller disk of a Veech surface, where a Teichmüller disk is the set of flat structures affine-equivalent to a given one.

Pan proved a rigidity theorem: every isomorphism between saddle connection graphs is induced by an affine homeomorphism, and consequently the automorphism group of S(X,ω;Σ)S(X,\omega;\Sigma) is Aff(X,ω;Σ)\mathrm{Aff}(X,\omega;\Sigma) itself, up to an index-two exception when (X,ω;Σ)(X,\omega;\Sigma) is a torus with one marked point. The quotient graph G(X,ω;Σ)=S(X,ω;Σ)/Aut(S)G(X,\omega;\Sigma)=S(X,\omega;\Sigma)/\mathrm{Aut}(S) therefore has as vertices the Aff(X,ω;Σ)\mathrm{Aff}(X,\omega;\Sigma)-orbits of saddle connections. Pan established: (a) if (X,ω;Σ)(X,\omega;\Sigma) is a translation surface (the orientable case, with transition maps z↦z+cz\mapsto z+c), then G(X,ω;Σ)G(X,\omega;\Sigma) has infinitely many edges unless (X,ω;Σ)(X,\omega;\Sigma) is a torus with one marked point; and (b) every Veech surface gives G(X,ω;Σ)G(X,\omega;\Sigma) with finitely many vertices, using the classical fact that a lattice surface has finitely many orbits of triangles, each saddle connection lying in some triangle. He explicitly states that he does not know whether the converse of (b) holds.

The closest prior characterization is due to Smillie and Weiss, who proved that the lattice property is equivalent to a long list of conditions, including: finitely many Aff\mathrm{Aff}-orbits of triangles; having no small triangles (a positive lower bound on the areas of affine triangles); and having no small virtual triangles (a positive lower bound on ∣u∧v∣|u\wedge v| over non-parallel holonomy vectors u,vu,v of saddle connections, where the holonomy vector of a saddle connection records its displacement in flat coordinates). They also introduced the spine, a Veech-group-invariant tree inside the hyperbolic plane parameterizing the affine copies of the surface, and showed that finiteness of the spine quotient forces finite covolume, yielding a compactness characterization of the lattice property. None of the Smillie-Weiss conditions concerns orbit-finiteness of individual saddle connections, a requirement that a priori is much weaker than any of their conditions on pairs or triangles. Pan records a partial reduction in the same spirit: if G(X,ω;Σ)G(X,\omega;\Sigma) has finitely many vertices, the components of the complement of the spine tree fall into finitely many affine orbits, and it would suffice to prove that each component has finite quotient area, equivalently that every saddle connection has a non-trivial affine stabilizer, to conclude that the surface is Veech. Later work on the coarse geometry of the saddle connection graph (Disarlo, Pan, Randecker, and Tang proved that S(X,ω;Σ)S(X,\omega;\Sigma) is a quasitree, meaning quasi-isometric to a tree, where a quasi-isometry distorts distances by at most fixed multiplicative and additive constants, and this holds for every half-translation surface, so the large-scale geometry of SS detects nothing about the Veech property) leaves the quotient-finiteness question untouched.

Problem Statement

Let (X,ω;Σ)(X,\omega;\Sigma) be a half-translation surface with marked points, let S(X,ω;Σ)S(X,\omega;\Sigma) be its saddle connection graph, and let G(X,ω;Σ)G(X,\omega;\Sigma) be the quotient of S(X,ω;Σ)S(X,\omega;\Sigma) by its automorphism group (equivalently, up to the index-two exception of the once-marked torus, by the affine automorphism group Aff(X,ω;Σ)\mathrm{Aff}(X,\omega;\Sigma); the vertices of G(X,ω;Σ)G(X,\omega;\Sigma) are the affine-orbits of saddle connections). Pan's Question 1 asks:

(i) Characterize those half-translation surfaces (X,ω;Σ)(X,\omega;\Sigma) for which G(X,ω;Σ)G(X,\omega;\Sigma) has finitely many edges.

(ii) Is it true that G(X,ω;Σ)G(X,\omega;\Sigma) has finitely many vertices if and only if (X,ω;Σ)(X,\omega;\Sigma) is a Veech surface?

The implication of (ii) that the Veech property implies finitely many vertices is known (Pan's Theorem 1.3(ii)), as is the restriction of (i) to translation surfaces, where only the once-marked torus has a finite-edge quotient. The open content is: prove that finiteness of the vertex quotient implies the Veech (lattice) property, or exhibit a non-Veech half-translation surface whose quotient graph nonetheless has finitely many vertices; and, for (i), give a geometric or dynamical criterion on (X,ω;Σ)(X,\omega;\Sigma) equivalent to finiteness of the edge quotient. Since G(X,ω;Σ)G(X,\omega;\Sigma) is connected, finitely many edges imply finitely many vertices, so an affirmative answer to (ii) would locate the surfaces in (i) among the Veech surfaces.

References

  1. Huiping Pan, Affine equivalence and saddle connection graphs of half-translation surfaces, International Mathematics Research Notices 2022 (2022), no. 4, 2861-2905, DOI: 10.1093/imrn/rnaa195 (arXiv:1809.06248), https://arxiv.org/abs/1809.06248
  2. John Smillie and Barak Weiss, Characterizations of lattice surfaces, Inventiones Mathematicae 180 (2010), no. 3, 535-557, DOI: 10.1007/s00222-010-0236-0 (arXiv:0809.3729), https://arxiv.org/abs/0809.3729
  3. Valentina Disarlo, Huiping Pan, Anja Randecker, and Robert Tang, Large-scale geometry of the saddle connection graph, Transactions of the American Mathematical Society 374 (2021), no. 11, 8101-8129, DOI: 10.1090/tran/8448 (arXiv:2011.12975), https://arxiv.org/abs/2011.12975
  4. Valentina Disarlo, Anja Randecker, and Robert Tang, Rigidity of the saddle connection complex, Journal of Topology 15 (2022), no. 3, 1248-1310, DOI: 10.1112/topo.12242 (arXiv:1810.00961), https://arxiv.org/abs/1810.00961
  5. Ser-Wei Fu and Christopher J. Leininger, Cylinder curves in finite holonomy flat metrics, Journal of Topology and Analysis 15 (2023), no. 2, 445-466, DOI: 10.1142/S1793525321500308 (arXiv:1909.13760), https://arxiv.org/abs/1909.13760
  6. Robert Tang and Richard C. H. Webb, Shadows of Teichmüller discs in the curve graph, International Mathematics Research Notices 2018 (2018), no. 11, 3301-3341, DOI: 10.1093/imrn/rnw318 (arXiv:1510.04259), https://arxiv.org/abs/1510.04259