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The Teichmueller polynomial of a fibered 3-manifold plays a useful role in the construction of mapping class having small stretch factor. We provide an algorithm that computes this polynomial of the fibered face associated to a pseudo-Anosov mapping class of a disc homeomorphism. As a byproduct, our algorithm allows us to derive all the relevant informations on the topology of the different fibers that belong to the fibered face.
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The minimal stratum in Prym loci have been the first source of infinitely many primitive, but not algebraically primitive Teichmueller curves. We show that the stratum Prym(2,1,1) contains no such Teichmueller curve and the stratum Prym(2,2) at most 92 such Teichmueller curves. This complements the recent progress establishing general -- but non-effective -- methods to prove finiteness results for Teichmueller curves and serves as proof of concept how to use the torsion condition in the non-algebraically primitive case.
This paper is devoted to the classification of the infinite families of Teichmuller curves generated by Prym eigenforms of genus 3 having a single zero. These curves were discovered by McMullen. The main invariants of our classification is the discriminant D of the corresponding quadratic order, and the generators of this order. It turns out that for D sufficiently large, there are two Teichmueller curves when D=1 modulo 8, only one Teichmueller curve when D=0,4 modulo 8, and no Teichmueller curves when D=5 modulo 8. For small values of D, where this classification is not necessarily true, the number of Teichmueller curves can be determined directly. The ingredients of our proof are first, a description of these curves in terms of prototypes and models, and then a careful analysis of the combinatorial connectedness in the spirit of McMullen. As a consequence, we obtain a description of cusps of Teichmueller curves given by Prym eigenforms. We would like also to emphasis that even though we have the same statement compared to, when D=1 modulo 8, the reason for this disconnectedness is different. The classification of these Teichmueller curves plays a key role in our investigation of the dynamics of SL(2,R) on the intersection of the Prym eigenform locus with the stratum H(2,2), which is the object of a forthcoming paper.
In this paper, we investigate the closure of a large class of Teichmuller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratners theorem to unipotent subgroups acting on an ``adequate splitting. This analysis applies for example to all Teichmueller discs stabilized obtained by Thurstons construction with a trace field of degree three which moreover ``obviously not Veech. We produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on L, contrary to the situation in genus two. We also study completely periodic directions on translation surfaces in L. For instance, we prove that completely periodic directions are dense on surfaces obtained by Thurstons construction.
For certain pseudo-Anosov flows $phi$ on closed $3$-manifolds, unpublished work of Agol--Gueritaud produces a veering triangulation $tau$ on the manifold $M$ obtained by deleting $phi$s singular orbits. We show that $tau$ can be realized in $M$ so that its 2-skeleton is positively transverse to $phi$, and that the combinatorially defined flow graph $Phi$ embedded in $M$ uniformly codes $phi$s orbits in a precise sense. Together with these facts we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of $phi$s closed orbits after cutting $M$ along certain transverse surfaces, thereby generalizing work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of $M$. Our work can be used to study the flow $phi$ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the `positive cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.
For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X, omega)$ in the stratum, a matrix is in its Veech group $mathrm{SL}(X,omega)$ if and only if an associated affine automorphism of the infinite surface sends each of a finite set, the ``marked {em Voronoi staples}, arising from orientation-paired segments appropriately perpendicular to Voronoi 1-cells, to another pair of orientation-paired ``marked segments. We prove a result of independent interest. For each real $age sqrt{2}$ there is an explicit hyperbolic ball such that for any Fuchsian group trivially stabilizing $i$, the Dirichlet domain centered at $i$ of the group already agrees within the ball with the intersection of the hyperbolic half-planes determined by the group elements whose Frobenius norm is at most $a$. %When $mathrm{SL}(X,omega)$ is a lattice we use this to give a condition guaranteeing that the full group $mathrm{SL}(X,omega)$ has been computed. Together, these results give rise to a new algorithm for computing Veech groups.
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