Local Connectivity of the Boundary of a Relatively Hyperbolic Group

Explore a deep dive into the local connectivity properties of boundaries in relatively hyperbolic groups through this 56-minute lecture by Chris Hruska. Delve into the historical context of Gromov hyperbolic groups and their locally connected boundaries, a significant result from the 1990s. Examine Bowditch's introduction of a natural boundary for relatively hyperbolic group pairs and his conjecture about their local connectivity. Discover the breakthrough finding that the boundary of every one-ended relatively hyperbolic group pair is indeed locally connected, without any restrictions. Learn how this result extends to a general setting, encompassing groups that are not necessarily finitely generated or even countable. Gain insights into the implications of this work for JSJ decompositions, boundary classification problems, and boundary analysis in geometric group theory.