On a Veech (lattice) translation surface, every completely periodic direction has vanishing Sah-Arnoux-Fathi (SAF) invariant. In genus two, and more generally for every Veech surface of quadratic trace field, the converse holds: after Calta-Smillie normalization the parabolic directions exhaust the slopes in the projective trace field, so every direction of vanishing SAF invariant is parabolic. Arnoux and Schmidt (2009) proved that this equivalence fails for Veech's original regular q-gon surfaces V_q exactly when the trace field has degree at least 3 over Q, and presumed the same failure for every Veech surface of cubic or higher trace field. The problem is to prove or refute that presumption: does every Veech surface whose trace field has degree at least 3 admit a non-parabolic (hence, by the Veech dichotomy, uniquely ergodic) direction of vanishing SAF invariant? The failure is verified for the regular polygons, for the Bouw-Möller and related even-triangle classes, and for the commensurability class of the Hecke group G_18, but is open in general - notably for the Ward surfaces - and the equivalent cusp-set problem for Hecke triangle groups of trace-field degree at least 3 remains, per a 2026 preprint, wide open.

Background

A translation surface is a closed surface built from finitely many Euclidean polygons by gluing parallel sides by translations; equivalently, it is a pair (X,ω)(X,\omega) of a compact Riemann surface and a nonzero holomorphic 1-form. It carries a flat metric with conical singularities, and the straight-line flow in any fixed direction is well defined. An affine diffeomorphism of such a surface is a homeomorphism that is affine in the flat coordinates; its derivative is a constant matrix. Veech proved that the derivatives of the affine diffeomorphisms form a discrete subgroup Γ\Gamma of PSL(2,R)\mathrm{PSL}(2,\mathbb{R}), the Veech group of the surface, and that when Γ\Gamma is a lattice (finite hyperbolic area) the surface - called a Veech surface or lattice surface - enjoys optimal dynamics: every direction is either completely periodic, meaning the surface decomposes into a finite union of cylinders in that direction, or uniquely ergodic. Completely periodic directions are also called parabolic directions, because they correspond (via cotangents of their slopes) to parabolic fixed points of the Veech group.

To each direction one associates an interval exchange transformation (IET), the first-return map of the linear flow to a transverse interval: a map that permutes finitely many subintervals by translations. The Sah-Arnoux-Fathi (SAF) invariant of an IET with interval lengths lil_i and translation lengths tit_i is the element ∑ili∧Qti\sum_i l_i \wedge_{\mathbb{Q}} t_i of the exterior product R∧QR\mathbb{R}\wedge_{\mathbb{Q}}\mathbb{R}; it is a conjugacy-type homomorphism invariant of the IET, and Kenyon and Smillie showed how to derive, from their JJ-invariant of a translation surface (a rational-valued invariant of the singularity structure that detects rational dependence among cone angles), an SAF invariant for each direction. A completely periodic direction has vanishing SAF invariant, since periodic IETs have zero invariant. The converse fails in general: a flow can split into two minimal components whose nonzero invariants cancel.

The trace field of a translation surface is the field obtained by adjoining to Q\mathbb{Q} the traces of the derivatives of all affine diffeomorphisms (Gutkin-Judge); it is constant along the SL(2,R)\mathrm{SL}(2,\mathbb{R})-orbit. Calta and Smillie proved that a translation surface with at least three directions of vanishing SAF invariant can be normalized - put in standard form - so that the slopes of its vanishing-SAF directions form exactly P1(K)\mathbb{P}^1(K) for a field KK, the periodic direction field; for a Veech surface this field coincides with the trace field, and its degree is at most the genus (a bound of Möller, reproved for abelian differentials by Lanneau and Liechti). Hence for a Veech surface in standard form, 'every direction of vanishing SAF invariant is parabolic' is equivalent to 'the cusp set (set of parabolic fixed points) of the Veech group exhausts P1(K)\mathbb{P}^1(K)', where KK is the trace field.

The implication 'vanishing SAF implies parabolic' is a rigidity phenomenon. In genus two it always holds: Calta and McMullen showed that on any genus-two Veech surface all vanishing-SAF directions are parabolic. McMullen further proved that any Veech surface whose trace field is quadratic has the same property - after normalization the parabolic directions fill the whole trace field, with only finitely many cusp orbits. These arguments exploit arithmetic special to quadratic fields (descent and principal ideal domains). Already for genus at least three, Arnoux and Yoccoz had constructed pseudo-Anosov diffeomorphisms with vanishing SAF invariant, but on surfaces that are not Veech surfaces.

Veech's original examples VqV_q come from gluing two copies of a regular qq-gon along opposite sides. Their Veech groups are commensurable with the Hecke groups GqG_q, the triangle Fuchsian groups of signature (2,q,∞)(2,q,\infty) generated by (1λq01)\begin{pmatrix}1&\lambda_q\\0&1\end{pmatrix} and (0−110)\begin{pmatrix}0&-1\\1&0\end{pmatrix} with λq=2cos⁡(π/q)\lambda_q=2\cos(\pi/q); the cusps of GqG_q form the single orbit Gq⋅∞G_q\cdot\infty, studied since Leutbecher through the Rosen λq\lambda_q-continued fractions (a numeration system whose finite expansions are exactly the cusps). A hyperbolic element of a Fuchsian group is called special when its eigenvalues lie in the group's trace field, in which case its fixed points also lie in that field; an affine pseudo-Anosov diffeomorphism (one that is affine in the flat coordinates, with hyperbolic derivative) is special when its derivative is special, and then its invariant directions have vanishing SAF invariant. Arnoux and Schmidt, combining the Leutbecher school's results (Leutbecher, Wolfart) on when Gq⋅∞G_q\cdot\infty fills λqQ(λq2)∪{∞}\lambda_q\mathbb{Q}(\lambda_q^2)\cup\{\infty\} with explicit special hyperbolic matrices, proved: all vanishing-SAF directions on VqV_q are parabolic if and only if the trace field has degree at most two, that is exactly for q∈{3,4,5,6,8,10,12}q\in\{3,4,5,6,8,10,12\}; for all other qq the surface has non-parabolic vanishing-SAF directions, and for q∈{7,9,14,18,20,24}q\in\{7,9,14,18,20,24\} even special affine pseudo-Anosov diffeomorphisms of vanishing SAF invariant.

The picture is therefore sharply split by the degree of the trace field - degree at most two forces all vanishing-SAF directions to be parabolic, while every known Veech surface of degree at least three fails this - but the failure has been verified only family by family, and it is not known whether it is universal. This is the open question that Arnoux and Schmidt state as a presumption.

Problem Statement

Arnoux and Schmidt (2009) presume that no Veech surface whose trace field has degree at least 33 over Q\mathbb{Q} has only parabolic directions of vanishing SAF invariant. Prove or disprove this presumption: does every Veech surface (X,ω)(X,\omega) whose trace field KK satisfies [K:Q]≥3[K:\mathbb{Q}]\ge 3 admit a direction of vanishing Sah-Arnoux-Fathi invariant that is not completely periodic - hence, by the Veech dichotomy, is uniquely ergodic?

Equivalently, in normalized form: after putting the surface in Calta-Smillie standard form, so that the slopes of its vanishing-SAF directions are exactly P1(K)\mathbb{P}^1(K), can the cusp set (set of parabolic fixed points) of its Veech group equal all of P1(K)\mathbb{P}^1(K) when [K:Q]≥3[K:\mathbb{Q}]\ge 3, or must it always be a proper subset?

The contrasting cases are known: for trace fields of degree at most two the cusp set is all of P1(K)\mathbb{P}^1(K) (McMullen), while the failure of this property is proven for the regular qq-gon family VqV_q (Arnoux-Schmidt), for translation surfaces whose Veech groups are commensurable to the relevant even triangle groups (Calta-Schmidt, covering the Bouw-Möller surfaces), and for every surface whose Veech group is commensurable to the Hecke group G18G_{18} (Winsor). The question remains open for Veech surfaces in general, including the Ward surfaces with Veech groups of signature (3,n,∞)(3,n,\infty).

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