I'm starting to study at the elementary level the relationship between topology and geometry of a Riemannian manifold of negative curvature. The first two theorems, simple and interesting in this direction are:


$\bf{Preissmann~~ Theorem}:$ Let  be $M$ a Riemannian manifold with sectional curvature $K<0$, then every non trivial Abelian subgroup of the fundamental group $\pi_1(M)$ is  cyclic infinite.


$\bf{Byers~~ Theorem}:$  Let  be $M$ a Riemannian manifold with sectional curvature $K<0$, then every no trivial solvable subgroup of the fundamental group $\pi_1(M)$ is  cyclic infinite, and $\pi_1(M)$ have no cyclic subgroup of finite index.

$\bf{My ~~Question}:$ I'm looking for nontrivial examples (counterexamples) for  Byers  Theorem, i.e, non trivial  examples of a Riemannian manifold that:

1. Has a no solvable fundamental group.


EDIT:


2  Has a cyclic $\bf{infinite}$ subgroup of finite index of the fundamental group.(In this case the trivial examples are welcome.)