# Short generators of non-elementary subgroups of hyperbolic groups

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 $$2^{|S|^{K\delta}}$$?

• From experience of similar problems, I suspect that this is difficult. Wouldn't this question be more suitable for MathOverflow? Commented Jun 11 at 7:59
• 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.) Commented Jun 11 at 12:38