Proper actions and weak amenability of classical relatively hyperbolic groups

Gromov introduced a notion of hyperbolicity for discrete groups (and general metric spaces) as an abstraction of the properties of universal covers of closed, negatively curved manifolds and their fundamental groups. The fundamental group of a manifold with pinched negative curvature and a cusp is not hyperbolic, but it is relatively hyperbolic with respect to the cusp subgroup, which has polynomial growth. We introduce a thinning technique which allows to reduce questions about these classical relatively hyperbolic groups to the case of bounded geometry hyperbolic graphs. As applications, we show that such groups admit a proper affine action on an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -space and are weakly amenable in the sense of Cowling–Haagerup. These results generalize earlier work of G. Yu and N. Ozawa, respectively, from the setting of hyperbolic groups to classical relatively hyperbolic groups.