The large scale geometry of Gromov hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the linear isoperimetric filling inequality for 1–cycles, the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. In this talk, I will describe a number of closely analogous results for spaces of rank n > 1 in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. A central role is played by a suitable class of n–dimensional surfaces of polynomial growth of order n, which serve as a substitute for quasi-geodesics.
I will talk about joint work with Özlem Beyarslan (the last version of our paper is available here: http://www.math.uni.wroc.pl/~pkowa/mojeprace/vfree4.pdf). We showed that for a finitely generated virtually free group G, the theory of actions of G on fields has a model companion, which we call G-TCF. We also gave an algebraic condition on G, which is equivalent to simplicity of the theory G-TCF. Recently, we learnt from Ehud Hrushovski an argument showing that if the group Z × Z embeds into G, then the theory of G-actions on fields does NOT have model companion. I will present this argument as well.