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0. Finite groups.

1. ...

2. ...

3. Fundamental groups of closed connected hyperbolic three-manifolds.

There is a proof using far too many tools from kleinian groups. Finite index subgroups are again fundamental groups of closed connected hyperbolic three-manifolds, so are quasi-isometric to hyperbolic three-space, so are Gromov hyperbolic. Suppose that $N$ is an infinite cover of $M$ with $\pi_1(N)$ finitely generated. Applying the tameness theorem (Agol, and also Calegari-Gabai) and the covering theorem (Canary) we have that $N$ is either completely degenerate (and homotopic to a closed surface) or has flaring ends. In the latter case $\pi_1(N)$ is quasi-convex in $\pi_1(M)$ and so is Gromov hyperbolic.

0. Finite groups.

1. ...

2. ...

3. Fundamental groups of closed connected hyperbolic three-manifolds.

Here is a proof of the latter using far too many tools from kleinian groups. Fundamental groups of closed connected hyperbolic three-manifolds are quasi-isometric to hyperbolic three-space and thus are Gromov hyperbolic. This deals with all finite index subgroups of $\pi_1(M)$.

Suppose instead that $N$ is an infinite cover of $M$, with $\pi_1(N)$ finitely generated. Applying the tameness theorem (Agol, and also Calegari-Gabai) and the covering theorem (Canary) we have that $N$ is either completely degenerate (and homotopic to a closed surface) or has flaring ends. In the latter case $\pi_1(N)$ is quasi-convex in $\pi_1(M)$ and thus is Gromov hyperbolic.

0. Finite groups.

1. ...

2. ...

3. Fundamental groups of closed connected hyperbolic three-manifolds.

0. Finite groups.

1. ...

2. ...

3. Fundamental groups of closed connected hyperbolic three-manifolds.

There is a proof using far too many tools from kleinian groups. Finite index subgroups are again fundamental groups of closed connected hyperbolic three-manifolds, so are quasi-isometric to hyperbolic three-space, so are Gromov hyperbolic. Suppose that $N$ is an infinite cover of $M$ with $\pi_1(N)$ finitely generated. Applying the tameness theorem (Agol, and also Calegari-Gabai) and the covering theorem (Canary) we have that $N$ is either completely degenerate (and homotopic to a closed surface) or has flaring ends. In the latter case $\pi_1(N)$ is quasi-convex in $\pi_1(M)$ and so is Gromov hyperbolic.

0. Finite groups.

1. ...

2. ...

3. Fundamental groups of closed connected hyperbolic three-manifolds.

0. Finite groups.

1. ...

2. ...

3. Fundamental groups of closed connected hyperbolic three-manifolds.