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In a project I am working on I need to find a pair of elements $a,b$ of a one-ended $\delta$-hyperbolic group $G$ such that the subgroup $\langle a,b \rangle$ is not virtually cyclic. Since the group $G$ is one-ended, and therefore not virtually cyclic, such pair must exist. I am interested in putting an upper bound on their length.

Question: suppose that $G = \langle S \mid R\rangle$ is a one-ended $\delta$-hyperbolic group, what is the smallest possible $N$ (in terms of $|S|$, $|R|$, and $\delta$) such that there exist elements $a,b \in B_G(1,N)$ such that the elements $a, b, ab, ab^{-1}$ are all of infinite order and the subgroup $\langle a,b \rangle \leq G$ is not virtually cyclic?

I was able to show that in a $\delta$-hyperbolic group one can use the size of a ball of radius $K \delta$ (the constant $K$ depends on which definition of $\delta$-hyperbolicity is used - slim/thin triangles, Gromov product...) as an upper bound on the maximal order of an element, and then write down a set of (in)equations over $G$, whose solutions are pairs of elements that don't generate virtually cyclic subgroups, then use the Ciobanu-Elder algorithm (https://arxiv.org/abs/2001.09591) to get an upper bound on the word-length of the solutions. What I get is exponential in the size of the ball of radius $K\delta$, giving me an upper bound that is doubly exponential in $\delta$.

So I guess my updated question is: can I do better than $2^{|S|^{K\delta}}$?

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    $\begingroup$ From experience of similar problems, I suspect that this is difficult. Wouldn't this question be more suitable for MathOverflow? $\endgroup$
    – Derek Holt
    Commented Jun 11 at 7:59
  • $\begingroup$ Did you check the literature on uniform exponential growth in hyperbolic groups? The main argument in such proofs is finding short generators of rank 2 free subgroups. (Or free subsemigroup which suffices for your purposes.) $\endgroup$ Commented Jun 11 at 12:38

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