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No, they are always different. Mostow Rigidity tells you that complete, finite-volume hyperbolic manifolds are determined by their fundamental groups. These groups never contain a $\mathbb Z \oplus \mathbb Z$, and any group acting on $\mathbb E^2$ contains this as a subgroup of finite index. Groups acting on the three-sphere are finite. This generalizes to all $n$: Mostow Rigidity remains true (and these fundamental groups never contain $\mathbb Z^n$; they are Gromov hyperbolic groups), Bieberbach Theorems say that the groups acting on $\mathbb E^n$ are virtually abelian, and the groups acting on the $n$-sphere are finite.

EDIT: I forgot to say compact above. Peripheral subgroups of a nonuniform group lattice acting on $\mathbb H^n$ are abelian of rank $n-1$, so they aren't Gromov hyperbolic. However, they contain nonabelian free subgroups, so they still don't act nicely on $\mathbb E^n$ or the sphere.

No, they are always different. Mostow Rigidity tells you that complete, finite-volume hyperbolic manifolds are determined by their fundamental groups. These groups never contain a $\mathbb Z \oplus \mathbb Z$, and any group acting on $\mathbb E^2$ contains this as a subgroup of finite index. Groups acting on the three-sphere are finite. This generalizes to all $n$: Mostow Rigidity remains true (and these fundamental groups never contain $\mathbb Z^n$; they are Gromov hyperbolic groups), Bieberbach Theorems say that the groups acting on $\mathbb E^n$ are virtually abelian, and the groups acting on the $n$-sphere are finite.

EDIT: I forgot to say compact above. Peripheral subgroups of a nonuniform group acting on $\mathbb H^n$ are abelian of rank $n-1$, so they aren't Gromov hyperbolic. However, they contain nonabelian free subgroups, so they still don't act nicely on $\mathbb E^n$ or the sphere.

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No, they are always different. Mostow Rigidity tells you that complete, finite-volume hyperbolic manifolds are determined by their fundamental groups. These groups never contain a $\mathbb Z \oplus \mathbb Z$, and any group acting on $\mathbb E^2$ contains this as a subgroup of finite index. Groups acting on the three-sphere are finite. This analogy generalizes to all $n$: Mostow Rigidity remains true (and these fundamental groups never contain $\mathbb Z^n$)Z^n$; they are Gromov hyperbolic groups), Bieberbach Theorems say that the groups acting on$\mathbb E^n$are virtually abelian, and the groups acting on the$n\$-sphere are finite.

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