Pathological topology in boundaries of hyperbolic groups

In geometric group theory we study groups by their actions on metric spaces. Although a given group might admit many actions on different metric spaces, on a large scale these spaces will all look similar, and so the large scale properties of a space on which a group acts are intrinsic to the group. One particularly natural example of a large scale property used in this way is the Gromov boundary of a hyperbolic metric space. This is a topological space that can be thought of as compactifying the metric space at infinity. 

In this talk I will describe some constructions of spaces occurring in this way with nasty, fractal-like properties. On the other hand, there are limits to how pathological these spaces can be: theorems of Bestvina and Mess, Bowditch and Swarup imply that boundaries of hyperbolic groups are locally path connected whenever they are connected. I will discuss these results and some generalisations.