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Samuel J. Taylor

Publications and source records attributed to Samuel J. Taylor.

At least 19 recordsLinked to original sources
arXiv2026

Simultaneous universal circles and continuous extension

Fenley proved that any foliation almost transverse to a quasigeodesic pseudo-Anosov flow in a closed atoroidal 3-manifold has the continuous extension property, meaning the inclusions of leaves into the universal cover continuously extend to their ideal boundaries. This article gives an alternate proof of an upgraded version of this: the associated Cannon-Thurston map for the flow, constructed by Frankel and Fenley, organizes all of the leafwise continuous extensions. The proof uses the fact that the boundary of the flowspace is naturally a universal circle for the foliation.

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arXiv2026

Constructing depth one laminations transverse to pseudo-Anosov flows

Given a pseudo-Anosov flow $\phi$ on a closed atoroidal $3$--manifold $M$ and a closed surface $S$ almost transverse to $\phi$, we give a homological characterization of when $S$ can be completed to an almost transverse depth one lamination or foliation whose set of compact leaves is $S$. As a consequence, we show that the cone of classes in $H^1(M\backslash \!\! \backslash S)$ that are positive on the closed orbits of $\phi$, when nonempty, is an entire foliation cone of $M\backslash \!\! \backslash S$.

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arXiv2026

A characterization of pseudo-Anosov orbit spaces via bifoliated planes

We characterize the actions on bifoliated planes that arise as orbits spaces of transitive pseudo-Anosov flows on orientable closed 3-manifold. We use branched covers, veering triangulations, and a new compactness criterion to extend previous work that handled the special case where the plane has no odd-prong singularities and the action preserves a leafwise orientation on the foliations.

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arXiv2026

Pseudo-Anosov flows, hyperbolic geometry, and the curve graph

Starting with a pseudo-Anosov flow $\varphi$ on a closed hyperbolic $3$-manifold $M$ and an embedded surface $S \subset M$ that is (almost) transverse to $\varphi$, we relate the hyperbolic geometry of $M$ (e.g. volume, circumference, short geodesics) to dynamical invariants of $\varphi$ encoded by the curve graph of $S$.

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arXiv2025

Universal circles for Anosov foliations

Thurston introduced the notion of a universal circle associated to a taut foliation of a $3$-manifold as a way of organizing the ideal circle boundaries of its leaves into a single circle action. Calegari--Dunfield proved that every taut foliation of an atoroidal $3$-manifold $M$ has a universal circle, but the uniqueness (or lack-thereof) of this structure remains rather mysterious. In this paper, we consider the foliations associated to an Anosov flow $\varphi$ on $M$, showing that several constructions of a universal circle in the literature are typically distinct. Moreover, the underlying action of the Calegari--Dunfield leftmost universal circle is generally not even conjugate to the universal circle arising from the boundary of the flow space of $\varphi$. Our primary tool is a way to use the flow space of $\varphi$ to parameterize the circle bundle at infinity of $\varphi$'s invariant foliations.

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arXiv2025

On fixed points of pseudo-Anosov maps

We give a formula to estimate the number of fixed points of a pseudo-Anosov homeomorphism of a surface. When the homeomorphism satisfies a mild property called strong irreducibility, the log of the number of fixed points is coarsely equal to the Teichmuller translation length. We also discuss several applications, including an inequality relating the hyperbolic volume of a mapping torus to the rank of its Heegaard Floer homology.

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arXiv2024

Simultaneous universal circles

Let phi be a pseudo-Anosov flow on a closed oriented atoroidal 3-manifold M. We show that if F is any taut foliation almost transverse to phi, then the action of pi_1(M) on the boundary of the flow space, together with a natural collection of explicitly described monotone maps, defines a universal circle for F in the sense of Thurston and Calegari-Dunfield.

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arXiv2024

Quadratic-time computations for pseudo-Anosov mapping classes

We give a quadratic-time algorithm to compute the stretch factor and the invariant measured foliations for a pseudo-Anosov element of the mapping class group. As input, the algorithm accepts a word (in any given finite generating set for the mapping class group) representing a pseudo-Anosov mapping class, and the length of the word is our measure of complexity for the input. The output is a train track and an integer matrix where the stretch factor is the largest real eigenvalue and the unstable foliation is given by the corresponding eigenvector. This is the first algorithm to compute stretch factors and measured foliations that is known to terminate in sub-exponential time.

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arXiv2024

Transverse surfaces and pseudo-Anosov flows

Let $φ$ be a transitive pseudo-Anosov flow on an oriented, compact $3$-manifold $M$, possibly with toral boundary. We characterize the surfaces in $M$ that are (almost) transverse to $ϕ$. When $φ$ has no perfect fits (e.g. $φ$ is the suspension flow of a pseudo-Anosov homeomorphism), we prove that any Thurston-norm minimizing surface $S$ that pairs nonnegatively with the closed orbits of $φ$ is almost transverse to $φ$, up to isotopy. This answers a question of Cooper--Long--Reid. Our main tool is a correspondence between surfaces that are almost transverse to $φ$ and those that are relatively carried by any associated veering triangulation. The correspondence also allows us to investigate the uniqueness of almost transverse position, to extend Mosher's Transverse Surface Theorem to the case with boundary, and more generally to characterize when relative homology classes represent Birkhoff surfaces.

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arXiv2023

Endperiodic maps via pseudo-Anosov flows

We show that every atoroidal endperiodic map of an infinite-type surface can be obtained from a depth one foliation in a fibered hyperbolic 3-manifold, reversing a well-known construction of Thurston. This can be done almost-transversely to the canonical suspension flow, and as a consequence we recover the Handel-Miller laminations of such a map directly from the fibered structure. We also generalize from the finite-genus case the relation between topological entropy, growth rates of periodic points, and growth rates of intersection numbers of curves. Fixing the manifold and varying the depth one foliations, we obtain a description of the Cantwell-Conlon foliation cones and a proof that the entropy function on these cones is continuous and convex.

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arXiv2023

Orientable maps and polynomial invariants of free-by-cyclic groups

We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism $φ$ and use it to characterize when the homological stretch factor of $φ$ is equal to its geometric stretch factor.

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arXiv2022

Covers of surfaces, Kleinian groups, and the curve complex

We prove an effective version of a theorem relating curve complex distance to electric distance in hyperbolic 3-manifolds, up to errors that are polynomial in the complexity of the underlying surface. We use this to give an effective proof of a result regarding maps between curve complexes of surfaces induced by finite covers. As applications, we effectively relate the electric circumference of a fibered manifold to the curve complex translation length of its monodromy, and we give quantitative bounds on virtual specialness for cube complexes dual to curves on surfaces.

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arXiv2022

Flows, growth rates, and the veering polynomial

For certain pseudo-Anosov flows $ϕ$ on closed $3$-manifolds, unpublished work of Agol--Guéritaud produces a veering triangulation $τ$ on the manifold $M$ obtained by deleting $ϕ$'s singular orbits. We show that $τ$ can be realized in $M$ so that its 2-skeleton is positively transverse to $ϕ$, and that the combinatorially defined flow graph $Φ$ embedded in $M$ uniformly codes $ϕ$'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 $ϕ$'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 $ϕ$ 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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arXiv2021

Random trees in the boundary of Outer space

We prove that for the harmonic measure associated to a random walk on Out$(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This answers a question of M. Bestvina.

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arXiv2020

A polynomial invariant for veering triangulations

We introduce a polynomial invariant $V_τ\in \mathbb{Z}[H_1(M)/\text{torsion}]$ associated to a veering triangulation $τ$ of a $3$-manifold $M$. In the special case where the triangulation is layered, i.e. comes from a fibration, $V_τ$ recovers the Teichmüller polynomial of the fibered faces canonically associated to $τ$. Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered $3$-manifold. For a general veering triangulation $τ$, we show that the surfaces carried by $τ$ determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is $\textit{equal}$ to the cone over a (generally non-fibered) face of the Thurston norm ball, and that $τ$ computes the norm on this cone in a precise sense. We also give a combinatorial description of $V_τ$ in terms of the $\textit{flow graph}$ for $τ$ and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm determined by $τ$.

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