I recently came across an interesting result of Kobayashi [Corollary 5.5], a special case of which is the following: Suppose $\Gamma$ is a discrete torsion free subgroup of $SL_n(\mathbb{R})$ which acts properly discontinuously on the homogeneous space $X=SL_n(\mathbb{R})/SL_{n-1}(\mathbb{R})$. Then the cohomological dimension of $\Gamma$ is less than or equal to $n$.

The homogeneous space $X$ above is diffeomorphic to a fiber bundle with base space $S^{n-1}$ and fibers $\mathbb{R}^n$. This motivates my question:

Suppose $G$ is a discrete torsion free group acting properly discontinuously on $X M \times \mathbb{R}^n$ where $X$ M$ is a compact manifold. What can be said about the cohomological dimension of $G$? Is it less than or equal to $n$?

It seems like the answer should be yes

Kobayashi's proof uses spectral sequences, but a tool which I am not sure how one would approach this problemfamiliar with. So before spending time learning about these objects, I was wondering if there is an obvious obstruction to the generalization of Kobayashi's result mentioned above.

Suppose $G$ is a discrete torsion free group acting properly discontinuously and freely on $X \times \mathbb{R}^n$ where $X$ is a compact manifold. What can be said about the cohomological dimension of $G$? Is it less than or equal to $n$?

It seems like the answer should be yes, but I am not sure how one would approach this problem.

Suppose $G$ is a discrete group acting properly discontinuously and freely on $X \times \mathbb{R}^n$ where $X$ is a compact manifold. What can be said about the cohomological dimension of $G$? Is it less than or equal to $n$?

It seems like the answer should be yes, but I am not sure how one would approach this problem.