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 wich sectional curvature $K<0$, then every non trivial subgroup of the fundamental group $\pi_1(M)$ is ciclic infinite.

$\bf{Byers~~ Theorem}:$ Let be $M$ a riemannian manifold wich sectional curvature $K<0$, then every no trivial solvable subgroup of the fundamental group $\pi_1(M)$ is ciclic 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 riemann manifold such that:

1. Have a no solvable fundamental group.

2. Have a ciclic subgroup of finite indice.(In this case the trivial examples are welcome.)