Controlled Floyd Separation and Non Relatively Hyperbolic Groups

We introduce the notion of controlled Floyd separation between geodesic rays starting at the identity in a finitely generated group G. Two such geodesic rays are said to be Floyd separated with respect to quasigeodesics if the (Floyd) length of c-quasigeodesics (for fixed but arbitrary c) joining points on the geodesic rays is asymptotically bounded away from zero. This is always satisfied by Morse geodesics. The main purpose of this paper is to furnish an example of a finitely generated group $G$ such that 1) all finitely presented subgroups of G are hyperbolic, 2) G has an uncountable family of geodesic rays that are Floyd separated with respect to quasigeodesics, 3) G is not hyperbolic relative to any collection of proper subgroups. 4) G is a direct limit of hyperbolic CAT(0) cubulated groups. 5) G has trivial Floyd boundary in the usual sense. On the way towards constructing $G$, we construct a malnormal infinitely generated (and hence non-quasiconvex) subgroup of a free group, giving negative evidence towards a question of Swarup and Gitik.