Group Theory Seminar
Sierpiński Carpets and relatively hyperbolic groups
Speaker: Lucas Souza (Universidad Federal de Minas Gerais)Date: Thursday, 06 February 2025 - 11:00Place: Aula Gris 2, ICMAT
Abstract:
The topology of a Bowditch boundary of a relatively hyperbolic group (with respect to a set of parabolic subgroups) has its importance, not only as an invariant for the group, but as a way to encode algebraic information about it, such as decompositions of the group as graphs of groups or if the group is a group of isometries of a manifold, for example. We present some examples of topological spaces that can appear as boundaries of relatively hyperbolic groups.
Whyburn showed that if we take a 2-sphere and remove an infinite collection of open disks satisfying some properties, then we get the 1-dimensional Sierpiński Carpet. After that, Cannon generalized it for n-dimensional Sierpiński Carpets. Recently, Tshishiku and Walsh gave another characterization of the Sierpiński Carpet: if we take a 2-sphere, remove a countable dense set and replace each point by a circle, then we get a 1-dimensional Sierpiński Carpet. We generalized their result for a n-dimensional Sierpiński Carpet. Then, we are able to show that some groups have a Bowditch boundary homeomorphic to the n-dimensional Sierpiński Carpet.
