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Register a new userGenericity of pseudo-Anosov mapping classes, when seen as mapping classes
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We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.
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Let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$. In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in $text{Mod}(S_g)$. As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cover of $S_g$. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most $2$. We also obtain conditions for the primitivity of a finite-order mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of $text{Mod}(S_g)$ as isometry groups.
Let $S$ be a compact orientable surface, and $Mod(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b in Mod(S)$ are independent (i.e., $[a^n,b^m] ot=1$ for any $n,m ot=0$) pseudo-Anosov elements. Then for any $n,m ge M$, the subgroup $<a^n,b^m>$ is free of rank two, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $<a^n,b^m>$ are pseudo-Anosov. We also show that there exists a constant $N$, which depends on $a,b$, such that $<a^n,b^m>$ is free of rank two and convex-cocompact if $|n|+|m| ge N$ and $nm ot=0$.
We investigate the structure of the characteristic polynomial det(xI-T) of a transition matrix T that is associated to a train track representative of a pseudo-Anosov map [F] acting on a surface. As a result we obtain three new polynomial invariants of [F], one of them being the product of the other two, and all three being divisors of det(xI-T). The degrees of the new polynomials are invariants of [F ] and we give simple formulas for computing them by a counting argument from an invariant train track. We give examples of genus 2 pseudo-Anosov maps having the same dilatation, and use our invariants to distinguish them.
Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted by the morphism. We prove these conjectures for classes of curves or divisors. We also prove that one of these conjectures implies Grothendiecks generalized Hodge conjecture for varieties with Hodge coniveau at least 1.
In this paper we provide a negative answer to a question of Farb about the relation between the algebraic degree of the stretch factor of a pseudo-Anosov homeomorphism and the genus of the surface on which it is defined.
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