For a permutation $\pi: [n]\rightarrow [n]$, we define the displacement of $\pi$ to be $\sum_{i\in [n]} |i-\pi(i)|$.
For given $k$, prove that the number of even permutations of $[n]$ with displacement $2k$ minus the number of odd permutations of $[n]$ with displacement $2k$ is $(-1)^{k}\binom{n-1}{k}$.

In the problem 2019-08 (https://mathsci.kaist.ac.kr/pow/2019/2019-08-group-action/), we considered a group G acting by isometries on a proper geodesic metric space X properly discontinuously and cocompactly. Such an action is called a geometric action. The conclusion was that a geometric action leads to that G is finitely generated.

Would this conclusion still hold in the case the space X is not necessarily proper?