trefoniu1

2022-01-29

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

Jason Olsen

Expert

There is a heuristic theorem. When the group has a hyperbolic subgroup, by Tits

Expert

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