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Samuel J. Taylor

Publications and source records attributed to Samuel J. Taylor.

At least 37 records · Page 2Linked to original sources
arXiv2020

Central limit theorems for counting measures in coarse negative curvature

We establish central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. Our techniques allow us to remove the usual assumptions of properness and smoothness of the space, or cocompactness of the action. We provide several applications which require our general framework, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds.

math.DS↗
arXiv2020

Weil-Petersson translation length and manifolds with many fibered fillings

We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.

math.GT↗
arXiv2019

Random veering triangulations are not geometric

Every pseudo-Anosov mapping class $φ$ defines an associated veering triangulation $τ_φ$ of a punctured mapping torus. We show that generically, $τ_φ$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.

math.GT↗
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↗
arXiv2017

Counting problems in graph products and relatively hyperbolic groups

We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e. they have full density with respect to counting in balls for the word metric) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including right-angled Artin groups and right-angled Coxeter groups.

math.GT↗
arXiv2017

Fibered faces, veering triangulations, and the arc complex

We study the connections between subsurface projections in curve and arc complexes in fibered 3-manifolds and Agol's veering triangulation. The main theme is that large-distance subsurfaces in fibers are associated to large simplicial regions in the veering triangulation, and this correspondence holds uniformly for all fibers in a given fibered face of the Thurston norm.

math.GT↗
arXiv2017

Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups

We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence.

math.GT↗
arXiv2017

Largest projections for random walks and shortest curves in random mapping tori

We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then use this result to prove Rivin's conjecture that for a random walk $(w_n)$ on the mapping class group, the shortest geodesic in the hyperbolic mapping torus $M_{w_n}$ has length on the order of $1/ \log^2(n)$.

math.GT↗
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↗
arXiv2017

Intersection properties of stable subgroups and bounded cohomology

We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to the ambient group.

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

Counting loxodromics for hyperbolic actions

Let $G \curvearrowright X$ be a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of elements of $G$ which act as loxodromic isometries of $X$ is generic. That is, for any finite generating set of $G$, the proportion of $X$--loxodromics in the ball of radius $n$ about the identity in $G$ approaches $1$ as $n \to \infty$. We also establish several results about the behavior in $X$ of the images of typical geodesic rays in $G$; for example, we prove that they make linear progress in $X$ and converge to the Gromov boundary $\partial X$. Our techniques make use of the automatic structure of $G$, Patterson--Sullivan measure on $\partial G$, and the ergodic theory of random walks for groups acting on hyperbolic spaces. We discuss various applications, in particular to Mod(S), Out($F_N$), and right--angled Artin groups.

math.GT↗
arXiv2016

The geometry of purely loxodromic subgroups of right-angled Artin groups

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(Γ)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In particular, such subgroups are quasiconvex in $A(Γ)$. In addition, we identify a milder condition for a finitely generated subgroup of $A(Γ)$ that guarantees it is free, undistorted, and retains finite generation when intersected with $A(Λ)$ for subgraphs $Λ$ of $Γ$. These results have applications to both the study of convex cocompactness in $\text{Mod}(S)$ and the way in which certain groups can embed in right-angled Artin groups.

math.GR↗
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

Pseudo-Anosovs optimizing the ratio of Teichmüller to curve graph translation length

Given $ϕ$ a pseudo-Anosov map, let $\ell_\mathcal{T}(ϕ)$ denote the translation length of $ϕ$ in the Teichmüller space, and let $\ell_\mathcal{C}(ϕ)$ denote the stable translation length of $ϕ$ in the curve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function of Euler characteristic $χ(S)$, the minimal possible ratio $τ(ϕ) = \frac{\ell_\mathcal{T}(ϕ)}{\ell_\mathcal{C}(ϕ)}$ is $\log(|χ(S)|)$, up to uniform additive and multiplicative constants. In this short note, we introduce a new construction of such ratio optimizers and demonstrate their abundance in the mapping class group. Further, we show that ratio optimizers can be found arbitrarily deep into the Johnson filtration as well as in the point pushing subgroup.

math.GT↗