arXiv subjects

Carolyn R. Abbott

Publications and source records attributed to Carolyn R. Abbott.

11 recordsLinked to original sources

arXiv2026

A new construction of subgroups of big mapping class groups

We explicitly construct new subgroups of the mapping class groups of an uncountable collection of infinite-type surfaces, including, but not limited to, free groups, Baumslag-Solitar groups, mapping class groups of other surfaces, and a large collection of wreath products. For each such subgroup $H$ and surface $S$, we show that there are countably many non-conjugate embeddings of $H$ into $\textrm{Map}(S)$; in certain cases, there are uncountably many such embeddings. The images of each of these embeddings cannot lie in the isometry group of $S$ for any hyperbolic metric and are not contained in the closure of the compactly supported subgroup of $\textrm{Map}(S)$. In this sense, our construction is new and does not rely on previously known techniques for constructing subgroups of mapping class groups. Notably, our embeddings of $\textrm{Map}(S')$ into $\textrm{Map}(S)$ are not induced by embeddings of $S'$ into $S$. Our main tool for all of these constructions is the utilization of special homeomorphisms of $S$ called shift maps, and more generally, multipush maps.

math.GT↗

arXiv2023

Higher rank confining subsets and hyperbolic actions of solvable groups

Recent papers of the authors have completely described the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace, Cornulier, Monod, and Tessera, which applies, in particular, to solvable groups with virtually cyclic abelianizations. In this paper, we extend this machinery and give a correspondence between the hyperbolic actions of certain solvable groups with higher rank abelianizations and confining subsets of these more general groups. We then apply this extension to give a complete description of the hyperbolic actions of generalized solvable Baumslag-Solitar groups and to reprove a result of Sgobbi-Wong computing their Bieri-Neumann-Strebel invariants.

math.GR↗

arXiv2022

Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G,X,H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich--Rafi and Dowdall--Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.

math.GR↗

arXiv2022

Valuations, completions, and hyperbolic actions of metabelian groups

Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian-by-cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations.

math.GR↗

arXiv2022

Largest hyperbolic actions and quasi-parabolic actions in groups

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a largest hyperbolic action. While hyperbolic groups admit largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for non-hyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have largest hyperbolic actions, including for instance many 3-manifold groups and most mapping class groups. Our proofs use the quasi-trees of metric spaces of Bestvina--Bromberg--Fujiwara, among other tools. In addition, we give a complete characterization of the poset of hyperbolic actions of Anosov mapping torus groups, and we show that mapping class groups of closed surfaces of genus at least two have hyperbolic actions which are comparable only to the trivial action.

math.GR↗

arXiv2021

Actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces

We give a complete list of the cobounded actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces up to a natural equivalence relation. The set of equivalence classes carries a natural partial order first introduced by Abbott-Balasubramanya-Osin, and we describe the resulting poset completely. There are finitely many equivalence classes of actions, and each equivalence class contains the action on a point, a tree, or the hyperbolic plane.

math.GR↗

arXiv2021

Infinite-type loxodromic isometries of the relative arc graph

An infinite-type surface $Σ$ is of type $\mathcal{S}$ if it has an isolated puncture $p$ and admits shift maps. This includes all infinite-type surfaces with an isolated puncture outside of two sporadic classes. Given such a surface, we construct an infinite family of intrinsically infinite-type mapping classes that act loxodromically on the relative arc graph $\mathcal{A}(Σ, p)$. J. Bavard produced such an element for the plane minus a Cantor set, and our result gives the first examples of such mapping classes for all other surfaces of type $\mathcal{S}$. The elements we construct are the composition of three shift maps on $Σ$, and we give an alternate characterization of these elements as a composition of a pseudo-Anosov on a finite-type subsurface of $Σ$ and a standard shift map. We then explicitly find their limit points on the boundary of $\mathcal{A}(Σ,p)$ and their limiting geodesic laminations. Finally, we show that these infinite-type elements can be used to prove that Map$(Σ,p)$ has an infinite-dimensional space of quasimorphisms.

math.GT↗

arXiv2021

Boundaries of coned-off hyperbolic spaces

Coning off a collection of uniformly quasiconvex subsets of a Gromov hyperbolic space leaves a new space, called the cone-off. Kapovich and Rafi generalized work of Bowditch to show this space is still Gromov hyperbolic. We show that the Gromov boundary of cone-off embeds in the boundary of the original hyperbolic space. (A stronger version of this result was previously obtained by Dowdall and Taylor; see Note in text.) Moreover, under some acylindricity assumptions we give a precise description of the image. As an application, we are able to characterize the elliptic and loxodromic elements of groups acting on certain cone-offs of acylindrical actions.

math.GR↗

arXiv2019

The geometry of generalized loxodromic elements

We explore geometric conditions which ensure a given element of a finitely generated group is, or fails to be, generalized loxodromic; as part of this we prove a generalization of Sisto's result that every generalized loxodromic element is Morse. We provide a sufficient geometric condition for an element of a small cancellation group to be generalized loxodromic in terms of the defining relations and provide a number of constructions which prove that this condition is sharp.

math.GR↗

arXiv2018

Property $P_{naive}$ for acylindrically hyperbolic groups

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, \dots, h_k$, there exists another element $γ\neq 1$ such that for all $i$, $\langle h_i, γ\rangle = \langle h_i \rangle* \langle γ\rangle$. We also obtain that if a collection of subgroups $H_1, \dots, H_k$ is a hyperbolically embedded collection, then there is $γ\neq 1$ such that for all $i$, $\langle H_i, γ\rangle = H_i * \langle γ\rangle$.

math.GR↗

arXiv2016

Not all finitely generated groups have universal acylindrical actions

The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and $\operatorname{Out}(\mathbb F_n)$ for $n\geq 2$. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Osin asks whether every finitely generated group has an acylindrical action on a hyperbolic space for which all generalized loxodromic elements are loxodromic. We answer this question in the negative, using Dunwoody's example of an inaccessible group as a counterexample.

math.GR↗