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For certain pseudo-Anosov flows $phi$ on closed $3$-manifolds, unpublished work of Agol--Gueritaud produces a veering triangulation $tau$ on the manifold $M$ obtained by deleting $phi$s singular orbits.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.
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The purpose of this paper is to give a new basis for examining the relationships of the Affine Index Polynomial and the Sawollek Polynomial. Blake Mellor has written a pioneering paper showing how the Affine Index Polynomial may be extracted from the Sawollek Polynomial. The Affine Index Polynomial is an elementary combinatorial invariant of virtual knots. The Sawollek polynomial is a relative of the classical Alexander polynomial and is defined in terms of a generalization of the Alexander module to virtual knots that derives from the so-called Alexander Biquandle. The present paper constructs the groundwork for a new approach to this relationship, and gives a concise proof of the basic Theorem of Mellor extracting the Affine Index Polynomial from the Sawollek Polynomial.
We investigate the translation lengths of group elements that arise in random walks on weakly hyperbolic groups. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov and almost every random walk on $mathrm{Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk. We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichmuller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.
We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K with extremal growth at this external point. We use this to find the polynomials of extremal growth for the interval [-1,1] at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by ErdH{o}s in 1947.
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