Acylindrical hyperbolicity of cubical small-cancellation groups

We provide an analogue of Strebel's classification of geodesic triangles in classical $C'(\frac16)$ groups for groups given by Wise's cubical presentations satisfying sufficiently strong metric cubical small cancellation conditions. Using our classification, we prove that, except in specific degenerate cases, such groups are acylindrically hyperbolic.Comment: Added figures. Exposition improved in Section 3, correction/simplification in Section 5, background added and citations updated in Section

Growth Tight Actions

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of $G$ on $\mathcal{X}$ is a growth tight action. It follows that if $\mathcal{X}$ is a cocompact, relatively hyperbolic $G$--space, then the action of $G$ on $\mathcal{X}$ is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some infinite, infinite index normal subgroups is relatively hyperbolic, and, conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that many CAT(0) groups, including flip-graph-manifold groups and many Right Angled Artin Groups, and snowflake groups admit cocompact, growth tight actions. These provide first examples of non-relatively hyperbolic groups admitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmueller space with the Teichmueller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements.Comment: 29 pages, 4 figures v2 added references v3 40 pages, 6 figures, expanded preliminary sections to make paper more self-contained, other minor improvements v4 updated bibliography, to appear in Pacific Journal of Mathematic

Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction

We introduce and begin a systematic study of sublinearly contracting projections. We give two characterizations of Morse quasi-geodesics in an arbitrary geodesic metric space. One is that they are sublinearly contracting; the other is that they have completely superlinear divergence. We give a further characterization of sublinearly contracting projections in terms of projections of geodesic segments.Comment: 24 pages, 5 figures. v2: 22 pages, 5 figures. Correction in proof of Thm 7.1. Proof of Prop 4.2 revised for improved clarity. Other minor changes per referee comments. To appear in Documenta Mathematic

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ and not strongly contracting with respect to another. In the case of classical $C'(1/6)$ small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. We show that many graphical $Gr'(1/6)$ small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. In the course of our analysis we show that if the defining graph of a graphical $Gr'(1/6)$ small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.Comment: 40 pages, 14 figures, v2: improved introduction, updated statement of Theorem 4.4, v3: new title (previously: "Contracting geodesics in infinitely presented graphical small cancellation groups"), minor changes, to appear in Groups, Geometry, and Dynamic

Metrics on diagram groups and uniform embeddings in a Hilbert space

"Vegeu el resum a l'inici del document del fitxer adjunt"

Testing Cayley graph densities

We present a computer-assisted analysis of combinatorial properties of the Cayley graphs of certain finitely generated groups: Given a group with a finite set of generators, we study the density of the corresponding Cayley graph, that is, the least upper bound for the average vertex degree (= number of adjacent edges) of any finite subgraph. It is known that an m-generated group is amenable if and only if the density of the corresponding Cayley graph equals to 2m. We test amenable and non-amenable groups, and also groups for which amenability is unknown. In the latter class we focus on Richard Thompson's group F

On quasiconvex subgroups of word hyperbolic groups

We prove that a quasiconvex subgroup H of in¢nite index of a torsion free word hyperbolic group can be embedded in a larger quasiconvex subgroup which is the free product of H and an in¢nite cyclic group. Some properties of quasiconvex subgroups ofword hyperbolic group are also discussed

A dichotomy for finitely generated subgroups of word hyperbolic groups

Given L > 0 elements in a word hyperbolic group G, there exists a number M = M(G, L) > 0 such that at least one of the assertions is true: (i) these elements generate a free and quasiconvex subgroup of G; (ii) they are Nielsen equivalent to a system of L elements containing an element of length at most M up to conjugation in G. The constant M is given explicitly. The result is generalized to groups acting by isometries on Gromov hyperbolic spaces. For proof we use a graph method to represent finitely generated subgroups of a group