Abstract. 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 X , we prove that if G contains a strongly contracting element and if G is not too badly distorted in X , then the action of G on X is a growth tight action. It follows that if X is a cocompact, relatively hyperbolic G-space, then the action of G on 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 Teichmüller space with the Teichmüller 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.
The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.2010 Mathematics Subject Classification. 20F65, 20F67.1 In even more recent developments, Behrstock [6] produces interesting examples of right-angled Coxeter groups whose Morse boundaries contain a circle, and Charney and Murray [13] give conditions that guarantee that a homeomorphism between Morse boundaries of CAT(0) spaces is induced by a quasi-isometry.
We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space. We also give an algorithm to detect such cut pairs. 20F65; 20E05
Abstract. Let G be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from G to Z have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing G as the mapping torus of a free group automorphism. This is similar to the case for 3-manifold groups, and different from the case of mapping tori of exponentially growing free group automorphisms. The proof uses a hierarchical decomposition of G and requires determining the Bieri-Neumann-Strebel invariant of the fundamental group of certain graphs of 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.
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