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We give a proof that there exists a universal constant K such that the disc graph associated to a surface S forming a boundary component of a compact, orientable 3-manifold M is K-quasiconvex in the curve graph of S. Our proof does not require the use of train tracks.
These are notes from a mini-course lectured by Denis Osin on acylindrically hyperbolic groups given at Beyond Hyperbolicity in Cambridge in June 2016. In this course we will be looking at acylindrically hyperbolic groups. Such groups make up a large portion of the "universe of finitely presented groups" described in Martin Bridson's course. Through examples of such groups we shall see that this class is indeed a large portion of the universe, but we shall also see that there is much we can prove. In particular, we will discuss results on small-cancellation theory and Dehn fillings, as well as some newer results.
We introduce the concept of a standard form for two embedded maximal sphere systems in the doubled handlebody, and we prove an existence and uniqueness result. In particular, we show that pairs of maximal sphere systems in the doubled handlebody (up to homeomorphism) bijectively correspond to square complexes satisfying a set of properties. This work is a variant on Hatcher's normal form.
The goal of this survey is to define and explain the Morse boundary of a geodesic space, a generalisation of the Gromov boundary of a hyperbolic space that records the “hyperbolic directions” in the space. This article relates Morse boundaries to the notion of stability, which generalises to arbitrary geodesic spaces the notion of quasiconvexity in hyperbolic spaces.
We construct a family of right-angled Coxeter groups which provide counter-examples to questions about the stable boundary of a group, one-endedness of stable subgroups, and the commensurability types of right-angled Coxeter groups.
A finitely generated subgroup H of a torsion-free hyperbolic group G is called immutable if there are only finitely many conjugacy classes of injections of H into G. We show that there is no uniform algorithm to recognize immutability, answering a uniform version of a question asked by the authors.
We study the acylindrical hyperbolicity of groups acting by isometries on CAT(0) cube complexes, and obtain simple criteria formulated in terms of stabilisers for the action. Namely, we show that a group acting essentially and non-elementarily on a finite dimensional irreducible CAT(0) cube complex is acylindrically hyperbolic if there exist two hyperplanes whose stabilisers intersect along a finite subgroup. We also give further conditions on the geometry of the complex so that the result holds if we only require the existence of a single pair of points whose stabilisers intersect along a finite subgroup.
The first part of this survey is a heuristic, non-technical discussion of what an HHS is, and the aim is to provide a good mental picture both to those actively doing research on HHSs and to those who only seek a basic understanding out of pure curiosity. It can be read independently of the second part, which is a detailed technical discussion of the axioms and the main tools to deal with HHSs.
Since the notion was introduced by Gromov in the 1980s, hyperbolicity of groups and spaces has played a significant role in geometric group theory; hyperbolic groups have good geometric properties that allow us to prove strong results. However, many classes of interest in our exploration of the universe of finitely generated groups contain examples that are not hyperbolic. Thus we wish to go 'beyond hyperbolicity' to find good generalisations that nevertheless permit similarly strong results. This book is the ideal resource for researchers wishing to contribute to this rich and active field. The first two parts are devoted to mini-courses and expository articles on coarse median spaces, semihyperbolicity, acylindrical hyperbolicity, Morse boundaries, and hierarchical hyperbolicity. These serve as an introduction for students and a reference for experts. The topics of the surveys (and more) re-appear in the research articles that make up Part III, presenting the latest results beyond hyperbolicity.
In late summer, sometime between cal a.d. 340–405, a hoard of tightly packed, stacked copper-alloy vessels was deposited in the Vale of Pewsey, Wiltshire. The corrosion of the vessels allowed for the preservation of delicate plant macrofossils and pollen. Analysis of this material has provided insights into the date, season and context of this act of structured deposition. A second hoard of similar vessels was deposited in the fourth or fifth century only a few miles away at Wilcot. The hoards and their deposition relate to Romano-British lifeways, at a time when the region was on the cusp of a dramatic period of change. The distribution of late Roman coins and belt fittings offers further insights into the social and economic character of Wiltshire at their times of deposition.
A fine-grained, up to 3-m-thick tephra bed in southwestern Saskatchewan, herein named Duncairn tephra (Dt), is derived from an early Pleistocene eruption in the Jemez Mountains volcanic field of New Mexico, requiring a trajectory of northward tephra dispersal of ~1500 km. An unusually low CaO content in its glass shards denies a source in the closer Yellowstone and Heise volcanic fields, whereas a Pleistocene tephra bed (LSMt) in the La Sal Mountains of Utah has a very similar glass chemistry to that of the Dt, supporting a more southerly source. Comprehensive characterization of these two distal tephra beds along with samples collected near the Valles caldera in New Mexico, including grain size, mineral assemblage, major- and trace-element composition of glass and minerals, paleomagnetism, and fission-track dating, justify this correlation. Two glass populations each exist in the Dt and LSMt. The proximal correlative of Dt1 is the plinian Tsankawi Pumice and co-ignimbritic ash of the first ignimbrite (Qbt1g) of the 1.24 Ma Tshirege Member of the Bandelier Tuff. The correlative of Dt2 and LSMt is the co-ignimbritic ash of Qbt2. Mixing of Dt1 and Dt2 probably occurred during northward transport in a jet stream.