 Award ID(s):
 1906487
 NSFPAR ID:
 10184632
 Date Published:
 Journal Name:
 Transactions of the American Mathematical Society
 ISSN:
 00029947
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

We consider two manifestations of nonpositive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, rightangled Coxeter groups, most 3 3 –manifold groups, rightangled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that rightangled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and rightangled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an “almost hierarchically hyperbolic space” is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.more » « less

The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious rightangled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious rightangled Artin subgroup. This alleged rightangled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of locally reducible Artin groups, which includes all 2dimensional Artin groups, and for spherical Artin groups of any type other than 𝐸₆, 𝐸₇, 𝐸₈. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.more » « less

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 the largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for nonhyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have the largest hyperbolic actions, including for instance many 3manifold groups and most mapping class groups. Our proofs use the quasitrees 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.more » « less

We provide a new method of constructing nonquasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings’ foldings. The hyperbolic groups constructed are in the natural class of rightangled Coxeter groups (RACGs for short) and can be chosen to be
$2$ dimensional. More specifically, given a nonquasiconvex subgroup of a (possibly nonhyperbolic) RACG, our construction gives a corresponding nonquasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of nonquasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are as short as possible (length two words), finitely generated free subgroups, nonfinitely presentable subgroups, and subgroups of fundamental groups of square complexes of nonpositive sectional curvature. 
We construct the first examples of normal subgroups of mapping class groups that are isomorphic to nonfree rightangled Artin groups. Our construction also gives normal, nonfree rightangled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudoAnosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudoAnosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.more » « less