Generalizations of Hyperbolic Boundaries

We study aspects of geometry that can still be unambiguously described when all measurements are allowed to have a certain amount of error. One strategy for dealing with such errors is to pass to a boundary at infinity, which is a space that parameterizes all the possible directions one can travel in a straight line starting from some fixed point. There are several different versions of such boundaries. In particular, we study Gromov boundaries, contracting boundaries, Floyd boundaries, and Martin boundaries. These all agree when the original space has a property called `hyperbolicity`, and boundaries for such hyperbolic spaces have proven to be a very useful tool. The novelty of this project is to develop these boundary theories when only some of the directions to infinity are hyperbolic. This will allow us to extend some of the results for hyperbolic spaces that are proved via boundary methods to spaces of interest that are not hyperbolic.

We study the coarse geometry of finitely generated groups and associated metric spaces via boundary methods. There are various ways to define a boundary at infinity for a group, sometimes requiring some hypothesis on the structure of the group. These include Gromov, Floyd, Bowditch, and Martin boundaries, which all agree when the group is hyperbolic, and in this case boundary methods have proven to be a very powerful. Recently a new construction, the contracting boundary, has been introduced, which agrees with all the others when the group is hyperbolic. In general, however, the contracting boundary does not compactify the group, but instead parameterizes all of the hyperbolic directions, in the hope that restricting to the hyperbolic directions will allow us to recapture some of the strong conclusions drawn from hyperbolic boundaries that do not generalize directly when applied to non-hyperbolic groups. In this project we introduce a new topology on the contracting boundary of a group that is better behaved topologically and more intimately connected to the geometry of the group than that which was previously studied, but still agrees with the Gromov boundary when the group is hyperbolic. These improvements will allow us to generalize results from Gromov boundaries of hyperbolic groups to non-hyperbolic groups that contain some hyperbolic directions. Interesting examples of such groups include many CAT(0) groups, relatively hyperbolic groups, mapping class groups, and outer automorphism groups of free groups. The concept of a 'hyperbolic direction' is quantified by a condition called 'contraction', first introduced by Bestvina and Fujiwara. The concept has been generalized and developed in previous work of the author and will be the main technical tool.