### Higher dimensional local connectedness of boundaries of hyperbolic groups

Algebra and Geometry Seminar

19th December 2018, 2:30 pm – 3:30 pm

Howard House, 4th Floor Seminar Room

It is an fundamental principle of 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. Few explicit topological spaces are known to arise as boundaries of hyperbolic groups, and I will begin by surveying some of them. I will then describe a new result that restricts the topological pathologies of a Gromov boundary by giving a condition for higher dimensional local connectedness. This generalises a theorem of Bestvina and Mess.

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