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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.

2. Has a cyclic subgroup of finite index of the fundamental group.(In this case the trivial examples are welcome.)

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.)

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.

2. Has a cyclic subgroup of finite index.(In this case the trivial examples are welcome.)

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.

2. Has a cyclic subgroup of finite index of the fundamental group.(In this case the trivial examples are welcome.)