Thursday, April 01, 2021, 12:30pm - 01:30pm
It is a fundamental principle in geometric group theory that large scale geometric properties of metric spaces on which a group acts should be considered properties of the group itself. If a given group acts on a hyperbolic metric space, then a particularly natural example of such a large-scale property is the Gromov boundary of that space: this is a topological space that compactifies the hyperbolic space at infinity. Connectivity properties of this boundary tell us how we can cut the group up into simpler pieces: a famous theorem of Stallings says that the group splits (as an amalgamated product or HNN extension) over a finite group if and only if the boundary is disconnected. In fact, one can say more: work of Brian Bowditch tells us that the structure of the collection of local cut points in the boundary determines a canonical JSJ decomposition for the group. In this talk I'll describe how Bowditch's JSJ decomposition is built from the structure of the boundary.
Location: Zoom