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Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and cocompactly on a finite-dimensional CAT(0) space.

So is there a group I have to leave out?

Not every CAT(0) space with a proper isometric cocompact group action is finite-dimensional. For example the trivial group acts on the compact CAT(0)-space $[0;1]^\mathbb{N}$.

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Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and cocompactly on a finite-dimensional CAT(0) groupspace.

So is there a group I have to leave out?

Not every CAT(0) space with a proper isometric cocompact is finite-dimensional. For example the trivial group acts on the compact CAT(0)-space $[0;1]^\mathbb{N}$.

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# infinite dimensional CAT(0) groups

Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and cocompactly on a finite-dimensional CAT(0) group.

So is there a group I have to leave out?

Not every CAT(0) space with a proper isometric cocompact is finite-dimensional. For example the trivial group acts on the compact CAT(0)-space $[0;1]^\mathbb{N}$.