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)?



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


