It is well known that in a given $\delta$-hyperbolic group there are only finitely many conjugacy classes of finite subgroups. This is clearly false for relatively hyperbolic groups since we have no control over the parabolic subgroups. Fix a relatively hyperbolic group $\Gamma$ with parabolic subgroups $P\_i$. Are there only finitely many conjugacy classes of finite subgroups $F<\Gamma$ with the property that $F$ is contained in a two-ended subgroup $H<\Gamma$ which is not conjugate into any $P\_i$? Perhaps weaker: are there only finitely many isomorphism types of two-ended subgroups which are not conjugate into any parabolic? Weaker still: Is there an upper bound to the orders of elements in such subgroups $F$?


(Motivation: I am trying to find a law obeyed by non-parabolic two-ended subgroups of relatively hyperbolic groups, or at least a partition of them into finitely many families, each of which obeys a law.)



Finite subgroups of relatively hyperbolic groups
Here's an idea for a proof in hyperbolic groups which generalizes to relatively hyperbolic groups. Actually, this will prove something slightly weaker ~~but which easily strengthens to what you want~~: namely, that there are only finitely many conjugacy classes of finite subgroups $F < \Gamma$ such that $F$ is contained in a 2-ended subgroup $H$, $H$ is not parabolic, *and* $F$ fixes each end of $H$.


Take $C$ to be the Cayley graph of a hyperbolic group. For every 2-ended group $H$ with fixed points $\xi,\eta$ at infinity, take its ``axis bundle'' consisting of all geodesics in $C$ connecting $\xi$ to $\eta$. This axis bundle has a uniform width, a number $W$ independent of $H$ such that for any point $p$ in the axis bundle, its $W$ neighborhood separates each geodesic in the axis bundle. Then take $N(p)$ to be, say, the $101W$ neighborhood of $p$, and so $N(p)$ separates each axis by removing a subsegment of length $100W$. It follows that any element of $H$ that fixes $\xi$ and $\eta$ and that moves $N(p)$ off itself has infinite order. So, every element of the subgroup $F$ moves $N(p)$ so that the image has nontrivial intersection with $N(p)$, which places $F$ amongst finitely many conjugacy classes of finite subgroups.


If $C$ is the Cayley graph of a relatively hyperbolic group, you can still define the axis bundle of a nonparabolic 2-ended group, but it does not have uniform width. The width can bloat out where the axis bundle runs through a parabolic coset. Nonetheless, there is still a uniform number $W$ such that there exists a point $p$ in the axis bundle whose $W$ neighborhood separates every geodesic in the axis bundle. The point $p$ will be located in the region where the axis bundle is passing from one parabolic coset to another. The proof above then goes through, using that choice of $p$.


