We begin an investigation of the action of pseudo-Anosov elements of Out(F2) on the Markoff-type varietiesXκ : x 2 + y 2 + z 2 = xyz + 2 + κ over finite fields Fp with p prime. We first make a precise conjecture about the permutation group generated by Out(F2) on X−2(Fp) that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting 'generically'. We prove that this conjecture is sharp. We show that for a fixed pseudo-Anosov g ∈ Out(F2), there is always an orbit of g of length ≥ C log p + O(1) on Xκ(Fp) where C > 0 is given in terms of the eigenvalues of g viewed as an element of GL2(Z). This improves on a result of Silverman from [24] that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic (p → ∞) behavior of the longest orbit of a fixed pseudo-Anosov g acting on X−2(Fp) is dictated by a dichotomy that we describe both in combinatorial terms and in algebraic terms related to Gauss's ambiguous binary quadratic forms, following Sarnak [21]. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture in Conjecture 1.10.

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Motivated by connections to intersection homology of toric morphisms, the motivic monodromy conjecture, and a question of Stanley, we study the structure of geometric triangulations of simplices whose local h-polynomial vanishes. As a first step, we identify a class of refinements that preserve the local h-polynomial. In dimensions 2 and 3, we show that all geometric triangulations with vanishing local h-polynomial are obtained from one or two simple examples by a sequence of such refinements. In higher dimensions, we prove some partial results and give further examples.i are nonnegative and satisfy i = d−i . Moreover, if the subdivision is regular, then these coefficients are unimodal. Among other applications, Stanley used local h-polynomials to prove that h-polynomials increase coefficientwise under refinement.As discussed in Section 2, the local h-polynomial also behaves predictably with respect to basic operations on subdivisions. It is additive for refinements that nontrivially subdivide only one facet, multiplicative for joins, and vanishes on the trivial subdivision. In particular, if Γ is a refinement of Γ that nontrivially subdivides only Manuscript

Doran, Jensen and Giansiracusa showed a bijection between homogeneous elements in the Cox ring of M 0,n not divisible by any exceptional divisor section, and weighted pure-dimensional simplicial complexes satisfying a zero-tension condition. Motivated by the study of the monoid of effective divisors, the pseudoeffective cone and the Cox ring of M 0,n , we point out a simplification of the zero-tension condition and study the space of balanced complexes. We give examples of irreducible elements in the monoid of effective divisors of M 0,n for large n. In the case of M 0,7 , we classify all such irreducible elements arising from nonsingular complexes and give an example of how irreducibility can be shown in the singular case.

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