Christopher H. Cashen

Line patterns in free groups

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.

Quasi-isometries Between Tubular Groups

We give a method of constructing maps between tubular groups inductively ac- cording to a finite set of strategies. This map will be a quasi-isometry exactly when the set of strategies satisfies certain consistency criteria. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for building a quasi-isometry between them. For two given tubular groups there are only finitely many candidate sets of strategies to consider, so it is possible in finite time to either produce a consistent set of strategies or decide that such a set does not exist. Consequently, there is an algorithm that in finite time decides whether or not two tubular groups are quasi-isometric.

Mapping tori of free group automorphisms, and the Bieri–Neumann–Strebel invariant of graphs of groups

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.

Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology

Abstract We consider a ‘contracting boundary’ of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space.We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.

Virtual geometricity is rare

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.

Growth tight actions of product groups

A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elements are growth tight. Examples of non-growth tight actions are product groups acting on the

Characterizations of Morse Quasi-Geodesics via Superlinear Divergence and Sublinear Contraction

L^1

Quasi-isometries between groups with two-ended splittings

products of Cayley graphs of the factors. In this paper we consider actions of product groups on product spaces, where each factor group acts with a strongly contracting element on its respective factor space. We show that this action is growth tight with respect to the

Negative curvature in graphical small cancellation groups.

L^p

A metrizable topology on the contracting boundary of a group

metric on the product space, for all