We investigate several quantitative generalizations of the notion of quasiconvex subsets of (Gromov) hyperbolic spaces to arbitrary geodesic metric spaces. Some of these, such as the Morse property, strong contraction, and superlinear divergence, had been studied before in more specialized contexts, and some, such as contraction, we introduce for the first time. In general, we prove that quasiconvexity is the weakest of the properties, strong contraction is the strongest, and all of the others are equivalent. However, in hyperbolic spaces all are equivalent, and we prove that in CAT(0) spaces all except quasiconvexity are equivalent. Despite the fact that many of these properties are equivalent, they are useful for different purposes. For instance, it is easy to see that the Morse property is quasiisometry invariant, but the contraction property gives good control over the divagation behavior of geodesic rays with a common basepoint. We exploit this control to define a boundary for arbitrary finitely generated groups that shares some properties of the boundary of a hyperbolic group. Our boundary is a metrizable topological space that is invariant under quasiisometries of the group, and the group acts on it with simple dynamics. We investigate the geometry of infinitely presented graphical small cancellation groups. Such groups include the so-called ‘Gromov Monsters’, which were introduced as a source of counter-examples to the Baum-Connes conjecture. We give a local-to-global characterization of contracting geodesics in these groups, which we think of as defining ‘hyperbolic directions’. Our characterization depends on a beautiful interplay between combinatorial and geometric versions of negative curvature. The result shows that the geometry of these groups is reminiscent of the geometry of relatively hyperbolic groups in the sense that there are certain well-defined non-hyperbolic regions, and geodesics that avoid these regions behave like hyperbolic geodesics. However, the groups are in general not relatively hyperbolic. Armed with our understanding of geodesics in graphical small cancellation groups, we construct examples of the wide varieties of contraction behaviors that occur: we show that every degree of contraction can be achieved by a periodic geodesic in some finitely generated group, that there are groups in which every element has a strongly contracting axis even though the group is not hyperbolic, and that there are examples of finitely generated groups in which the existence of a strongly contracting axis for a given element depends on the choice of generating set for the group. Since the Morse property is invariant under quasiisometries, we can say that a subgroup of a finitely generated group is Morse (or equivalently, contracting, divergent,...) if it has this property as a subset in some/any Cayley graph of the group. However, we could also let the group act on some other geodesic metric space and ask which elements have a contracting/Morse/strongly contracting axis for that particular action. In particular, we explore actions that are not cocompact. To preserve a connection with the geometry of the group, we require the action to be metrically proper. We also introduce a condition known as ‘complementary growth gap’ that says that there is an orbit of the group in the space that, while metrically distorted, is not too badly distorted from a growth-theoretic point of view. Our condition generalizes the ‘parabolic growth gap’ condition for Kleinian groups, and includes additional examples such as the action of the mapping class group of a hyperbolic surface on its Teichmüller space. We prove growth and cogrowth results for the orbit pseudometric induced on the group by such an action under the hypothesis the group has one element that acts with a strongly contracting axis. Our results generalize results that were known for word metrics on hyperbolic groups to far more general situations. The generality of our results is even more striking considering that there are contemporaneous papers that achieve similar results only for actions on hyperbolic spaces.

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