Does a group with exponential growth always have a hyperbolic subgroup which has exponential growth?

Question

Does a group with exponential growth always have a hyperbolic subgroup which has exponential growth?

There is a heuristic theorem. When the group has a hyperbolic subgroup, by Tits' Alternative theorem, the subgroup if doesn't contain degree 2 free group then it is virtually solvable.

Exponential growth and hyperbolicity have overlap to some extent.

I wonder to what extent can this question be answered.

Answer 1

Polycyclic groups that are not virtually nilpotent, such as $\langle x,y,z \mid xy=yz, x^y=z, z^y=yz \rangle$ have exponential growth, but they have no non-elementary hyperbolic subgroups (i.e. all of their hyperbolic subgroups are virtually cyclic).

Answer 2

I totally agree with what Derek said. I think it is a good answer which helps me understand better polynomial growth. The first example I learn in geometric group theorem which is polycyclic but not nilpotent is $Z^2 \rtimes _ {\varphi}Z$ where ${\varphi}$ is an isomorphism of ${Z}^2$ that has an eigenvalue bigger than 1.