Let $G$ be a one-ended hyperbolic group. Can $G$ contain an $\infty$-ended group? If it can, are there any conditions on $G$ beyond hyperbolic which makes it impossible?
As a particular example, if $G$ is one-ended, linear, and hyperbolic, can $G$ contain a non-abelian free group?
It is clear that, in general, the number of ends of a group will not necessarily become smaller when taking subgroups. Indeed, $\mathbb{Z} \leq \mathbb{Z}^2$, but the former is $2$-ended while the latter is $1$-ended. However, $\mathbb{Z}^2$ is not hyperbolic. I do not have a clear picture how to proceed from here.
In fact every one-ended hyperbolic group contains a free group of rank $\ge 2$. This was proved by Gromov in his first monograph on hyperbolic groups.