Proper actions on ℓ p -spaces for relatively hyperbolic groups

Abstract

We show that for any group G that is hyperbolic relative to subgroups that admit a proper affine isometric action on a uniformly convex Banach space, then G acts properly on a uniformly convex Banach space as well.arXiv:1801.08047v3 [math.GR]

Acylindrically hyperbolic groups with exotic properties

“…On the other hand, Yu [36] proved that every hyperbolic group admits a proper affine action on an pspace for large enough p, which is a strong negation of the property F L p . A generalization of Yu's result to relatively hyperbolic groups has been recently obtained by Chatterji and Dahmani in [10]. It is also known that every acylindrically hyperbolic group G admits an unbounded quasi-cocycle G → p (G) for all p ∈ [1, +∞) (see [16,18]), which can be seen as a violation of the "quasified" version of F L p .…”