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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)?

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  • $\begingroup$ This doesn't answer the question, but there is a more general result that infinite automatic groups have elements of infinite order. That follows easily from the fact that any repeating string in the word-acceptor automaton must have infinite order in the group. I guess that's a generalization of the cone-type argument for hyperbolic groups. Note that some relatively hyperbolic groups, such as geometrically finite hyperbolic groups, are automatic, so the argument applies to them. $\endgroup$ – Derek Holt Apr 21 '15 at 20:41
  • $\begingroup$ In Gromov's original paper, the fact that infinite hyperbolic group have an element of infinite order is a particular case of a statement about arbitrary isometric group actions on hyperbolic spaces, which applies equally to relatively hyperbolic groups (at least when relatively hyperbolic is suitably defined/characterized in terms of actions on hyperbolic spaces). $\endgroup$ – YCor Dec 3 '20 at 8:42
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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.

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    $\begingroup$ Regarding your final paragraph, surely there are trivial examples? For instance, $\mathbb{Z}^2$ should be automatic relative to the trivial subgroup, but of course it is not (properly) relatively hyperbolic. Or am I being too optimistic? $\endgroup$ – HJRW Dec 3 '20 at 13:23
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    $\begingroup$ @HJRW Yes, you're absolutely right ! Actually, any group with finitely many cone types has a finite number of cone types relatives to the trivial subgroup. $\endgroup$ – M. Dus Dec 3 '20 at 15:21
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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

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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.

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