"Relative cone types" for groups relative to some collection of subgroups

I think that the main interesting points of having finitely many cone types is that it gives that the growth series is rational and the set of geodesics in the Cayley graph is a regular language. As far as I know, it is open if having finitely many cone types is equivalent to the language of geodesic being regular.

Also note that having finitely many cone types depend on the generating set.

A related concept to cone types is the falsification by fellow traveler property (fftp). This property implies that there are finitely many cone types. (see A Short course in geometric group theory by Neumann and Shapiro). Related to your question, I proved with Laura Ciobanu that groups hyperbolic relative to groups having fftp, they have fftp.

http://arxiv.org/abs/1402.2985

In general, the number of cone types is not necessarily finite in a relatively hyperbolic groups. However, there exists indeed some kind of relative cone types, for which Cannon's result holds. In the paper "Patterson-Sullivan measures and growth of relatively hyperbolic groups"(http://arxiv.org/abs/1308.6326), Wen-yuan Yang defined a "partial" cone and showed that there are finitely many such partial cone types. See Lemma 5.14 therein.

As the name indicates, the partial cone at an element is a subset in a normal cone, where the elements to be excluded are the ones having a long intersection with a peripheral coset near the base element.

For your application to existence of hyperbolic elements, as far as I see, Yang's result could not be applied as the case in hyperbolic groups.

stephen's accepted answer is certainly very good, but here is a complete answer.

First, as you say, citing Osin's paper, you really do not need to use a notion of relative cone type to study the existence of non-torsion elements in relatively hyperbolic groups.

Besides Wang formulation of partial cone types, in his above-mentioned paper, you can define a notion of relative cone types, which is the exact equivalent of cone types but for relative geodesics, i.e. geodesics in the relative Cayley graph. You can then prove that there is a finite number of relative cone types. This defines an automaton with finite number of vertices (but infinite number of edges) encoding relative geodesics. The complete construction is performed in Theorem 4.2 here : https://arxiv.org/abs/2004.12777.

A group having such an automaton is called relatively automatic, so in other words, relative hyperbolicity implies relative automaticity. Now, the existence of this automaton easily yields an infinite order element. You just need to take a finite loop in the automaton. The resulting word will be infinite order.

So this slightly generalizes the existence of infinite order elements to all relatively automatic groups. However, this notion of relative automaticity is only used for thermodynamical formalism purpose, and so there is no given example of a group which would be relatively automatic but not relatively hyperbolic.