It is a well known fact that an infinite hyperbolic group contains an element of infinite order (see e.g. Bridson, Haefliger, Metric spaces of non-positive curvature, Prop. 2.22 on p. 458)
I am thinking about this in the case of relatively hyperbolic groups. I know that Osin proved the corresponding statement for "hyperbolic elements in relatively hyperbolic groups (w.r.t proper subgroups)". (see Osin, Elementary subgroups of relatively hyperbolic groups and bounded generation)
In Bridson and Haefliger's book this statement (for hyperbolic groups) is proven with "cone-types". More precisely one shows that a hyperbolic group has only finitely many cone types (this relies, besides hyperbolicity, on the local finiteness of the Cayley graph) and groups with finitely many cone types must contain an element of infinite order. The proof of Osin is done with quite a different approach.
Now I'm asking if there is some concept of "relative cone type" which could be applied for (not necessarely f.g.) groups (which are finitely presented w.r.t some collection of subgroups)?