Quasiflats in hierarchically hyperbolic spaces

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups, the maximal rank of a free abelian subgroup for right-angled Coxeter and Artin groups, and, for the Weil--Petersson metric, the rank is the integer part of half the complex dimension of Teichm\"{u}ller space. We prove that any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a mild condition satisfied by all natural examples). This resolves outstanding conjectures when applied to various examples. For mapping class group, we verify a conjecture of Farb; for Teichm\"{u}ller space we answer a question of Brock; for CAT(0) cubical groups, we handle special cases including right-angled Coxeter groups. An important ingredient in the proof is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. We deduce a number of applications. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. We give a new proof of quasi-isometric rigidity of mapping class groups, which, given our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.Comment: 58 pages, 6 figures. Revised according to referee comments. This is the final pre-publication version; to appear in Duke Math. Jou

Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to at most quadratic in the complexity of the surface. We also apply the main result to various other hierarchically hyperbolic groups and spaces. We also prove a small-cancellation result namely: if $G$ is a hierarchically hyperbolic group, $H\leq G$ is a suitable hyperbolically embedded subgroup, and $N\triangleleft H$ is "sufficiently deep" in $H$, then $G/\langle\langle N\rangle\rangle$ is a relatively hierarchically hyperbolic group. This new class provides many new examples to which our asymptotic dimension bounds apply. Along the way, we prove new results about the structure of HHSs, for example: the associated hyperbolic spaces are always obtained, up to quasi-isometry, by coning off canonical coarse product regions in the original space (generalizing a relation established by Masur--Minsky between the complex of curves of a surface and Teichm\"{u}ller space).Comment: Minor revisions in Section 6. This is the version accepted for publicatio

A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). Under residual finiteness assumptions, we construct many non-elementary hyperbolic quotients of mapping class groups. Using these quotients, we reduce questions of Reid and Bridson-Reid-Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.Comment: Revised according to comments from reader

Hierarchically hyperbolic spaces I: curve complexes for cubical groups

In the context of CAT(0) cubical groups, we develop an analogue of the theory of curve complexes and subsurface projections. The role of the subsurfaces is played by a collection of convex subcomplexes called a \emph{factor system}, and the role of the curve graph is played by the \emph{contact graph}. There are a number of close parallels between the contact graph and the curve graph, including hyperbolicity, acylindricity of the action, the existence of hierarchy paths, and a Masur--Minsky-style distance formula. We then define a \emph{hierarchically hyperbolic space}; the class of such spaces includes a wide class of cubical groups (including all virtually compact special groups) as well as mapping class groups and Teichm\"{u}ller space with any of the standard metrics. We deduce a number of results about these spaces, all of which are new for cubical or mapping class groups, and most of which are new for both. We show that the quasi-Lipschitz image from a ball in a nilpotent Lie group into a hierarchically hyperbolic space lies close to a product of hierarchy geodesics. We also prove a rank theorem for hierarchically hyperbolic spaces; this generalizes results of Behrstock--Minsky, Eskin--Masur--Rafi, Hamenst\"{a}dt, and Kleiner. We finally prove that each hierarchically hyperbolic group admits an acylindrical action on a hyperbolic space. This acylindricity result is new for cubical groups, in which case the hyperbolic space admitting the action is the contact graph; in the case of the mapping class group, this provides a new proof of a theorem of Bowditch.Comment: To appear in "Geometry and Topology". This version incorporates the referee's comment

Development of the Vibration Isolation System for the Advanced Resistive Exercise Device

This paper describes the development of the Vibration Isolation System for the Advanced Resistive Exercise Device from conceptual design to lessons learned. Maintaining a micro-g environment on the International Space Station requires that experiment racks and major vibration sources be isolated. The challenge in characterizing exercise loads and testing the system in the presence of gravity led to a decision to qualify the system by analysis. Available data suggests that the system is successful in attenuating loads, yet there has been a major component failure and several procedural issues during its 3 years of operational use

A harvestman (Arachnida: Opiliones) from the Early Devonian Rhynie cherts, Aberdeenshire, Scotland

A harvestman (Arachnida: Opiliones) is described from the Early Devonian (Pragian) Rhynie cherts, Aberdeenshire, Scotland. Eophalangium sheari gen. et sp. nov. is the oldest known harvestman. The material includes both males and a female preserving, respectively, a cuticle-lined penis and ovipositor within the opisthosoma. Both these structures are of essentially modern appearance. The Rhynie fossils also show tracheae which are, again, very similar to those of living harvestmen. This is the oldest unequivocal record of arachnid tracheal respiration and indicates that E. sheari was terrestrial. An annulate, setose ovipositor in the female suggests that it can be excluded from the clades Dyspnoi and Laniatores, in which the ovipositor lacks such annulations. However, the penis shows evidence of two muscles, a feature of uncertain polarity seen in modern Troguloidea (Dyspnoi). The presence of median eyes and long legs excludes Cyphophthalmi, and thus, E. sheari is tentatively referred to the suborder Eupnoi. Therefore, this remarkable material is implicitly a crown-group harvestman and is one of the oldest known crown-group chelicerates. It also suggests an extraordinary degree of morphological stasis within the eupnoid line, with the Devonian forms differing little in gross morphology – and perhaps in reproductive behaviour – from their modern counterparts.Peer Reviewe