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Study Group "Pat­ter­son-Sul­li­van The­ory": Ben­jamin Hin­richs, "Hy­per­bol­ic groups"

Ort: D2.314

Abstract: We study an important class of discrete groups in detail following [Gu ́e+17, Chapter 2.1]. First we
consider the definition of a hyperbolic group Γ (also called Gromov hyperbolic or word hyperbolic),
focusing on the group itself rather than its Cayley graph, and not wasting time on the general theory
of hyperbolic metric spaces. Then we study the notion of the boundary ∂Γ, its topology, and the most
important properties of the Γ-action on ∂Γ (this is a main goal of the talk). Furthermore, we study the
following topics:
ˆClassification of elements of Γ according to their action on ∂Γ
ˆThe Γ-equivariant homeomorphism ∂Γ ∼= Λ(Γ). In the case Γ ⊂Iso(Hn+1), see [Bow93]
ˆExamples and non-examples of hyperbolic groups. Illustration of the above concepts in some
examples
If time allows:
ˆDetailed study of the example of the fundamental group of a closed Riemannian manifold of negative
curvature
ˆThe Gromov flow space