arXiv subjects

Alexander J. Rasmussen

Publications and source records attributed to Alexander J. Rasmussen.

14 recordsLinked to original sources

arXiv2026

Random quotients preserve acylindrical and hierarchical hyperbolicity

We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main tools relate the theories of spinning families and projection complexes to random walks. In the presence of a hierarchical hyperbolic structure on the group, we leverage the fine control of projections to show that this structure is preserved in the quotient asymptotically almost surely. The same techniques yield that random quotients of a non-elementary hyperbolic group (relative to any finite collection of finitely generated peripheral subgroups) are asymptotically almost surely hyperbolic (relative to commensurable peripheral subgroups). Finally, we also prove that any two groups that are both acylindrically and hierarchically hyperbolic have a common quotients which is itself acylindrically and hierarchically hyperbolic. This produces "exotic" hierarchically hyperbolic groups with strong fixed point properties, such as Kazhdan's property (T).

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arXiv2025

Disintegrating the curve complex

We study a finite sequence of graphs, beginning with the curve graph and ending with a graph quasi-isometric to a tree. There is a Lipschitz map from one graph in the sequence to the next. This sequence was first introduced by Hamenstädt. We prove (as conjectured by Hamenstädt) that the graphs in this sequence are hyperbolic and that the coarse fibers of the maps in the sequence are quasi-trees. This gives an upper bound on the asymptotic dimension of each graph in the sequence and as a result, an upper bound on the asymptotic dimension of the curve graph. Additionally, we show that the action of the mapping class group on each graph in the sequence is acylindrical, and classify the boundary and actions of individual mapping classes for each graph in the sequence.

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arXiv2025

Irrational rotations and 2-filling rays

We study a skew product transformation associated to an irrational rotation of the circle [0,1]/~. This skew product keeps track of the number of times an orbit of the rotation lands in the two complementary intervals of {0,1/2} in the circle. We show that under certain conditions on the continued fraction expansion of the irrational number defining the rotation, the skew product transformation has certain dense orbits. This is in spite of the presence of numerous non-dense orbits. We use this to construct laminations on infinite type surfaces with exotic properties. In particular, we show that for every infinite type surface with an isolated planar end, there is an infinite clique of 2-filling rays based at that end. These 2-filling rays are relevant to Bavard--Walker's loop graphs.

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arXiv2023

Transverse measures to infinite type laminations

We study the cone of transverse measures to a fixed geodesic lamination on an infinite type hyperbolic surface. Under simple hypotheses on the metric, we give an explicit description of this cone as an inverse limit of finite-dimensional cones. We study the problem of when the cone of transverse measures admits a base and show that such a base exists for many laminations. Moreover, the base is a (typically infinite-dimensional) simplex (called a Choquet simplex) and can be described explicitly as an inverse limit of finite-dimensional simplices. We show that on any fixed infinite type hyperbolic surface, every Choquet simplex arises as a base for some lamination. We use our inverse limit description and a new construction of geodesic laminations to give other explicit examples of cones with exotic properties.

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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.

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arXiv2023

Largest hyperbolic actions of 3--manifold groups

The set of equivalence classes of cobounded actions of a group G on different hyperbolic metric spaces carries a natural partial order. Following Abbott--Balasubramanya--Osin, the group G is H--accessible if the resulting poset has a largest element. In this paper, we prove that every non-geometric 3--manifold has a finite cover with H--inaccessible fundamental group and give conditions under which the fundamental group of the original manifold is H--inaccessible. We also prove that every Croke--Kleiner admissible group (a class of graphs of groups that generalizes fundamental groups of 3--dimensional graph manifolds) has a finite index subgroup that is H--inaccessible.

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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.

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arXiv2022

Stable commutator length on big mapping class groups

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.

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arXiv2022

Laminations and 2-filling rays on infinite type surfaces

The loop graph of an infinite type surface is an infinite diameter hyperbolic graph first studied in detail by Juliette Bavard. An important open problem in the study of infinite type surfaces is to describe the boundary of the loop graph as a space of geodesic laminations. We approach this problem by constructing the first examples of 2-filling rays on infinite type surfaces. Such rays accumulate onto geodesic laminations which are in some sense filling, but without strong enough properties to correspond to points in the boundary of the loop graph. We give multiple constructions using both a hands-on combinatorial approach and an approach using train tracks and automorphisms of flat surfaces. In addition, our approaches are sufficiently robust to describe all 2-filling rays with certain other basic properties as well as to produce uncountably many distinct mapping class group orbits.

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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.

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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.

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arXiv2020

WWPD elements of big mapping class groups

We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-Fujiwara's weak proper discontinuity and is useful for constructing non-trivial quasimorphisms. We use this classification to give a sufficient criterion for subgroups of big mapping class groups to have infinite-dimensional second bounded cohomology and use this criterion to give simple proofs that certain natural subgroups of big mapping class groups have infinite-dimensional second bounded cohomology.

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arXiv2020

Geometry of the graphs of nonseparating curves: covers and boundaries

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparating curves to the nonseparating curve graph of the cover span quasiconvex subgraphs which are infinite diameter and not coarsely equal to the nonseparating curve graph of the cover. In the finite type case, we also reprove a theorem of Hamenstädt identifying the Gromov boundary with the space of ending laminations on full genus subsurfaces. We introduce several tools based around the analysis of bicorn curves and laminations which may be of independent interest for studying the geometry of nonseparating curve graphs of infinite type surfaces and their boundaries.

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arXiv2020

Uniform hyperbolicity of the graphs of nonseparating curves via bicorn curves

We show that the graphs of nonseparating curves for oriented finite type surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonseparating. As demonstrated by Aramayona-Valdez, this proves also that the graph of nonseparating curves for any oriented infinite type surface with finite positive genus is hyperbolic.

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