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Spencer Dowdall

Publications and source records attributed to Spencer Dowdall.

At least 19 recordsLinked to original sources
arXiv2026

Depth of free-by-cyclic groups

For a free group automorphism, we prove that its poset of attracting lamination orbits is a canonical invariant of the associated mapping torus. That is, if a free-by-cyclic group splits as a mapping torus in two different ways, then the corresponding automorphisms have isomorphic posets of lamination orbits. Further, we show that the lamination depth, the size of the largest chain in this poset, is a commensurability invariant of the free-by-cyclic group.

math.GR↗
arXiv2025

Constructing reducibly geometrically finite subgroups of the mapping class group

In this article, we consider qualified notions of geometric finiteness in mapping class groups called parabolically geometrically finite (PGF) and reducibly geometrically finite (RGF). We examine several constructions of subgroups and determine when they produce a PGF or RGF subgroup. These results provide a variety of new examples of PGF and RGF subgroups. Firstly, we consider the right-angled Artin subgroups constructed by Koberda and Clay--Leininger--Mangahas, which are generated by high powers of given elements of the mapping class group. We give conditions on the supports of these elements that imply the resulting right-angled Artin subgroup is RGF. Secondly, we prove combination theorems which provide conditions for when a collection of reducible subgroups, or sufficiently deep finite-index subgroups thereof, generate an RGF subgroup.

math.GT↗
arXiv2024

Extensions of Veech groups I: A hyperbolic action

Given a lattice Veech group in the mapping class group of a closed surface $S$, this paper investigates the geometry of $Γ$, the associated $π_1S$--extension group. We prove that $Γ$ is the fundamental group of a bundle with a singular Euclidean-by-hyperbolic geometry. Our main result is that collapsing "obvious" product regions of the universal cover produces an action of $Γ$ on a hyperbolic space, retaining most of the geometry of $Γ$. This action is a key ingredient in the sequel where we show that $Γ$ is hierarchically hyperbolic and quasi-isometrically rigid.

math.GT↗
arXiv2023

Counting mapping classes by Nielsen-Thurston type

This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichm\"uller space with the Teichm\"uller metric. In that problem the goal is to count the number of mapping classes that send a given point $x$ into the ball of radius $R$ centered about another point $y$. For the action of the entire group, Athreya, Bufetov, Eskin and Mirzakhani have shown this quantity is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichm\"uller space. We refine the problem by considering the action various distinguished subsets of elements and counting these separately. For the set of finite-order elements, we show the associated count grows coarsely at the rate of $e^{hR/2}$, that is, with half the exponent. For the reducible elements, the associated count grows coarsely at the rate of $e^{(h-1)R}$. Finally, for the set of all multitwists, the coarse growth rate is also $e^{hR/2}$. To obtain these quantitative estimates, we introduce a new notion in Teichm\"uller geometry, called complexity length, which reflects some aspects of the negative curvature of curve complexes and also has applications to counting problems.

math.GT↗
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.

math.GR↗
arXiv2022

Discretely shrinking targets in moduli space

We consider the discrete shrinking target problem for Teichmüller geodesic flow on the moduli space of abelian or quadratic differentials and prove that the discrete geodesic trajectory of almost every differential will hit a shrinking family of targets infinitely often provided the measures of the targets are not summable. This result applies to any ergodic $\mathrm{SL}(2,\mathbb{R})$--invariant measure and any nested family of spherical targets. Under stronger conditions on the targets, we moreover prove that almost every differential will eventually always hit the targets. As an application, we obtain a logarithm law describing the rate at which generic discrete trajectories accumulate on a given point in moduli space. These results build on work of Kelmer and generalize theorems of Aimino, Nicol, and Todd.

math.DS↗
arXiv2018

Rank and Nielsen equivalence in hyperbolic extensions

In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of Out$(F_n)$.

math.GT↗
arXiv2018

Contracting orbits in Outer space

We show that strongly contracting geodesics in Outer space project to parameterized quasigeodesics in the free factor complex. This result provides a converse to a theorem of Bestvina--Feighn, and is used to give conditions for when a subgroup of $\mathrm{Out}(\mathbb{F})$ has a quasi-isometric orbit map into the free factor complex. It also allows one to construct many new examples of strongly contracting geodesics in Outer space.

math.GT↗
arXiv2018

Isomorphisms between big mapping class groups

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

math.GR↗
arXiv2017

Endomorphisms, train track maps, and fully irreducible monodromies

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

math.GR↗
arXiv2017

The co-surface graph and the geometry of hyperbolic free group extensions

We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\mathbb{F}$-trees and use this to prove that a finitely generated subgroup of $\mathrm{Out}(\mathbb{F})$ quasi-isometrically embeds into the co-surface graph if and only if it is purely atoroidal and quasi-isometrically embeds into the free factor complex. This answers a question of I. Kapovich. Our earlier work [Hyperbolic extensions of free groups, arXiv:1406.2567] shows that every such group gives rise to a hyperbolic extension of $\mathbb{F}$, and here we prove a converse to this result that characterizes the hyperbolic extensions of $\mathbb{F}$ arising in this manner. As an application of our techniques, we additionally obtain a Scott--Swarup type theorem for this class of extensions.

math.GT↗
arXiv2016

Hyperbolic extensions of free groups

Given a finitely generated subgroup $Γ\le \mathrm{Out}(\mathbb{F})$ of the outer automorphism group of the rank $r$ free group $\mathbb{F} = F_r$, there is a corresponding free group extension $1 \to \mathbb{F} \to E_Γ \to Γ\to 1$. We give sufficient conditions for when the extension $E_Γ$ is hyperbolic. In particular, we show that if all infinite order elements of $Γ$ are atoroidal and the action of $Γ$ on the free factor complex of $\mathbb{F}$ has a quasi-isometric orbit map, then $E_Γ$ is hyperbolic. As an application, we produce examples of hyperbolic $\mathbb{F}$-extensions $E_Γ$ for which $Γ$ has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex. This may be of independent interest.

math.GT↗
arXiv2016

McMullen polynomials and Lipschitz flows for free-by-cyclic groups

Consider a group G and an epimorphism u_0:G\to\Z inducing a splitting of G as a semidirect product ker(u_0)\rtimes_φ\Z with ker(u_0) a finitely generated free group and φ\in Out(ker(u_0)) representable by an expanding irreducible train track map. Building on our earlier work [Dynamics on free-by-cyclic groups, arXiv:1301.7739], in which we we realized G as π_1(X) for an Eilenberg-Maclane 2-complex X equipped with a semiflow ψ, and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant \m for (X,ψ) and investigate its properties. Specifically, \m determines a convex polyhedral cone \C_X in H^1(G;\R), a convex, real-analytic function \H:\C_X\to\R, and specializes to give an integral Laurent polynomial \m_u(ζ) for each integral u\in\C_X. We show that \C_X is equal to the "cone of sections" of (X,ψ) (the convex hull of all cohomology classes dual to sections of of ψ), and that for each (compatible) cross section Θ_u with first return map f_u:Θ_u\toΘ_u, the specialization \m_u(ζ) encodes the characteristic polynomial of the transition matrix of f_u. More generally, for every class u\in\C_X there exists a geodesic metric d_u and a codimension-1 foliation Ω_u of X transverse to ψso that after reparametrizing the flow ψ^u_s maps leaves of Ω_u to leaves via a local e^{s\H(u)}-homothety. Among other things, we additionally prove that \C_X is equal to (the cone over) the component of the BNS-invariant containing u_0 and that each primitive integral u\in\C_X induces a splitting of G as an ascending HNN-extension over a finite-rank free group along an injective endomorphism ϕ_u. For any such splitting, we show that the stretch factor of ϕ_u is exactly given by e^{\H(u)}. In particular, we see that \C_X and \H depend only on the group G and epimorphism u_0.

math.GT↗
arXiv2015

Cannon-Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon--Thurston maps associated to a general class of hyperbolic free group extensions. Let $F_N$ denote a free groups of finite rank $N\ge 3$ and consider a \emph{convex cocompact} subgroup $Γ\le Out(F_N)$, i.e. one for which the orbit map from $Γ$ into the free factor complex of $F_N$ is a quasi-isometric embedding. The subgroup $Γ$ determines an extension $E_Γ$ of $F_N$, and the main theorem of Dowdall--Taylor \cite{DT1} states that in this situation $E_Γ$ is hyperbolic if and only if $Γ$ is purely atoroidal. Here, we give an explicit geometric description of the Cannon--Thurston maps $\partial F_N\to\partial E_Γ$ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon--Thurston map, showing that this map has multiplicity at most $2N$. This theorem generalizes the main result of Kapovich and Lustig \cite{KapLusCT} which treats the special case where $Γ$ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of $Γ$ to the space of laminations of the free group (with the Chabauty topology) is not continuous.

math.GR↗
arXiv2015

Dynamics on free-by-cyclic groups

Given a free-by-cyclic group $G = F_N \rtimes_φ\mathbb{Z}$ determined by any outer automorphism $φ\in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $ε\in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $Θ_u \subset X$ such that: (1) the inclusion $Θ_u \to X$ is $π_1$-injective and $π_1(Θ_u) = \mathrm{ker}(u)$, (2) $u(ε) = χ(Θ_u)$, (3) $Θ_u \subset X$ is a section of the semiflow and the first return map to $Θ_u$ is an expanding irreducible train track map representing $φ_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{φ_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $φ_u$ is precisely $\mathfrak H(u)$, (5) if $φ$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $φ_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.

math.GT↗
arXiv2014

Unbounded asymmetry of stretch factors

A result of Handel-Mosher guarantees that the ratio of logarithms of stretch factors of any fully irreducible automorphism of the free group $F_N$ and its inverse is bounded by a constant $C_N$. In this short note we show that this constant $C_N$ cannot be chosen independent of $N$.

math.GR↗
arXiv2013

Statistical hyperbolicity in Teichmüller space

In this paper we explore the idea that Teichmüller space is hyperbolic "on average." Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichmüller space. We consider several different measures on Teichmüller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichmüller space, showing that "most triangles are mostly thin."

math.GT↗