I've come across the following question in my research, which seems elusive but is almost surely decidable.

Let $H$ be a quasi-convex subgroup of the hyperbolic group $G$. Given $x, y \in G$, we wish to decide whether $HxH = HyH$. This is equivalent to asking whether there exists $h, h' \in H$ such that $hx = yh'$.

It is easy to see that the question $x \in yH$ is easily decidable, since $H$ has a solvable membership problem and this reduces to checking whether $y^{-1} x \in H$.

The double coset problem seems harder, but the instinct is that this might solved by bounding the lengths of $h, h'$, akin to the solution to the conjugacy problem.

Does this problem appear in the literature anywhere? Any references or thoughts are appreciated.

Thanks!