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The question is phrased very generally, so I'm not sure if the following is what you're looking for. However, my answer to Does any surface of constant curvature admit a cocompact group action? might be useful to you, since you are interested in the case of constant negative curvature. The result can be summarised as follows:

Theorem A: If $M$ is a 2-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected.

Since a complete, simply connected, hyperbolic manifold is isometric to the hyperbolic plane, this necessary condition is of course also sufficient.

In higher dimensions, comparable results are much more difficult and delicate. For instance, using the tameness theorem of Agol and Calegari--Gabai, one can conclude the following:

Theorem B: If $M$ is a 3-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected or homotopy equivalent to a closed surface.

In this case, the necessary condition is no longer sufficient: for instance, the quotient of hyperbolic 3-space by a surface subgroup of an isometrically embedded copy of the hyperbolic plane cannot admit a cocompact action.

Above dimension 3, very little is known. For instance, it is an open question whether or not a closed hyperbolic 4-manifold can have a normal surface subgroup. (Note that the above results follow from classification theorems for normal subgroups of hyperbolic manifold groups.) However, you might be interested in the following result, which can be deduced from deep work of Paulin and others:

Theorem C: Let $M$ be any non-compact manifold of negative sectional curvature. If $M$ is homotopy equivalent to a closed negatively curved manifold of dimension greater than 2 then $M$ does not admit a properly discontinuous, cocompact group action by isometries.

I'm happy to provide further details if these kinds of results are what you are looking for.

The question is phrased very generally, so I'm not sure if the following is what you're looking for. However, my answer to Does any surface of constant curvature admit a cocompact group action? might be useful to you, since you are interested in the case of constant negative curvature. The result can be summarised as follows:

Theorem A: If $M$ is a 2-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected.

Since a complete, simply connected, hyperbolic manifold is isometric to the hyperbolic plane, this necessary condition is of course also sufficient.

In higher dimensions, comparable results are much more difficult and delicate. For instance, using the tameness theorem of Agol and Calegari--Gabai, one can conclude the following:

Theorem B: If $M$ is a 3-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected or homotopy equivalent to a closed surface.

In this case, the necessary condition is no longer sufficient: for instance, the quotient of hyperbolic 3-space by a surface group preserving an isometrically embedded copy of the hyperbolic plane cannot admit a cocompact action.

Above dimension 3, very little is known. For instance, it is an open question whether or not a closed hyperbolic 4-manifold can have a normal surface subgroup. (Note that the above results follow from classification theorems for normal subgroups of hyperbolic manifold groups.) However, you might be interested in the following result, which can be deduced from deep work of Paulin and others:

Theorem C: Let $M$ be any non-compact manifold of negative sectional curvature. If $M$ is homotopy equivalent to a closed negatively curved manifold of dimension greater than 2 then $M$ does not admit a properly discontinuous, cocompact group action by isometries.

I'm happy to provide further details if these kinds of results are what you are looking for.

The question is phrased very generally, so I'm not sure if the following is what you're looking for. However, my answer to Does any surface of constant curvature admit a cocompact group action? might be useful to you, since you are interested in the case of constant negative curvature. The result can be summarised as follows:

Theorem A: If $M$ is a 2-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected.

Since a complete, simply connected, hyperbolic manifold is isometric to the hyperbolic plane, this necessary condition is of course also sufficient.

In higher dimensions, comparable results are much more difficult and delicate. For instance, using the tameness theorem of Agol and Calegari--Gabai, one can conclude the following:

Theorem B: If $M$ is a 3-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected or homotopy equivalent to a closed surface.

In this case, the necessary condition is no longer sufficient: for instance, the quotient of hyperbolic 3-space by a surface subgroup of an isometrically embedded copy of the hyperbolic plane cannot admit a cocompact action.

Above dimension 3, very little is known. For instance, it is an open question whether or not a closed hyperbolic 4-manifold can have a normal surface subgroup. (Note that the above results follow from classification theorems for normal subgroups of hyperbolic manifold groups.) However, you might be interested in the following result, which can be deduced from deep work of Paulin and others:

Theorem C: Let $M$ be any non-compact manifold of negative sectional curvature. If $M$ is homotopy equivalent to a closed manifold of dimension greater than 2 then $M$ does not admit a properly discontinuous, cocompact group action by isometries.

I'm happy to provide further details if these kinds of results are what you are looking for.

The question is phrased very generally, so I'm not sure if the following is what you're looking for. However, my answer to Does any surface of constant curvature admit a cocompact group action? might be useful to you, since you are interested in the case of constant negative curvature. The result can be summarised as follows:

Theorem A: If $M$ is a 2-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected.

Since a complete, simply connected, hyperbolic manifold is isometric to the hyperbolic plane, this necessary condition is of course also sufficient.

In higher dimensions, comparable results are much more difficult and delicate. For instance, using the tameness theorem of Agol and Calegari--Gabai, one can conclude the following:

Theorem B: If $M$ is a 3-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected or homotopy equivalent to a closed surface.

In this case, the necessary condition is no longer sufficient: for instance, the quotient of hyperbolic 3-space by a surface subgroup of an isometrically embedded copy of the hyperbolic plane cannot admit a cocompact action.

Above dimension 3, very little is known. For instance, it is an open question whether or not a closed hyperbolic 4-manifold can have a normal surface subgroup. (Note that the above results follow from classification theorems for normal subgroups of hyperbolic manifold groups.) However, you might be interested in the following result, which can be deduced from deep work of Paulin and others:

Theorem C: Let $M$ be any non-compact manifold of negative sectional curvature. If $M$ is homotopy equivalent to a closed negatively curved manifold of dimension greater than 2 then $M$ does not admit a properly discontinuous, cocompact group action by isometries.

I'm happy to provide further details if these kinds of results are what you are looking for.

The question is phrased very generally, so I'm not sure if the following is what you're looking for. However, this answer might be useful to you, since you are interested in the case of constant negative curvature. The result can be summarised as follows:

Theorem A: If $M$ is a 2-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected.

Since a complete, simply connected, hyperbolic manifold is isometric to the hyperbolic plane, this necessary condition is of course also sufficient.

In higher dimensions, comparable results are much more difficult and delicate. For instance, using the tameness theorem of Agol and Calegari--Gabai, one can conclude the following:

Theorem B: If $M$ is a 3-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected or homotopy equivalent to a closed surface.

In this case, the necessary condition is no longer sufficient: for instance, the quotient of hyperbolic 3-space by a surface subgroup of an isometrically embedded copy of the hyperbolic plane cannot admit a cocompact action.

Above dimension 3, very little is known. For instance, it is an open question whether or not a closed hyperbolic 4-manifold can have a normal surface subgroup. (Note that the above results follow from classification theorems for normal subgroups of hyperbolic manifold groups.) However, you might be interested in the following result, which can be deduced from deep work of Paulin and others:

Theorem C: Let $M$ be any non-compact manifold of negative sectional curvature. If $M$ is homotopy equivalent to a closed manifold of dimension greater than 2 then $M$ does not admit a properly discontinuous, cocompact group action by isometries.

I'm happy to provide further details if these kinds of results are what you are looking for.

The question is phrased very generally, so I'm not sure if the following is what you're looking for. However, my answer to Does any surface of constant curvature admit a cocompact group action? might be useful to you, since you are interested in the case of constant negative curvature. The result can be summarised as follows:

Theorem A: If $M$ is a 2-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected.

Since a complete, simply connected, hyperbolic manifold is isometric to the hyperbolic plane, this necessary condition is of course also sufficient.

In higher dimensions, comparable results are much more difficult and delicate. For instance, using the tameness theorem of Agol and Calegari--Gabai, one can conclude the following:

Theorem B: If $M$ is a 3-dimensional, complete, non-compact hyperbolic manifold and $\pi_1M$ is finitely generated then $M$ admits a discrete, cocompact group action only if $M$ is simply connected or homotopy equivalent to a closed surface.

In this case, the necessary condition is no longer sufficient: for instance, the quotient of hyperbolic 3-space by a surface subgroup of an isometrically embedded copy of the hyperbolic plane cannot admit a cocompact action.

Above dimension 3, very little is known. For instance, it is an open question whether or not a closed hyperbolic 4-manifold can have a normal surface subgroup. (Note that the above results follow from classification theorems for normal subgroups of hyperbolic manifold groups.) However, you might be interested in the following result, which can be deduced from deep work of Paulin and others:

Theorem C: Let $M$ be any non-compact manifold of negative sectional curvature. If $M$ is homotopy equivalent to a closed manifold of dimension greater than 2 then $M$ does not admit a properly discontinuous, cocompact group action by isometries.

I'm happy to provide further details if these kinds of results are what you are looking for.