Let $X$ be a manifold on which a discrete group $\Gamma$ acts properly such that the quotient $\Gamma\backslash X$ is compact. For $g\in\Gamma$, let $X^g$ denote the set of points in $X$ fixed by $g$. Then $X^g$ is a submanifold of $X$.
The centralizer $Z_\Gamma(g)$ acts naturally on $X^g$, since for any $x\in X^g$, $h\in Z_\Gamma(g)$, we have $$g(hx)=h(gx)=hx,$$ whence $hx\in X^g$.
I would like to better understand the action of $Z_\Gamma(g)$ on $X^g$.
Question 1: Does $Z_\Gamma(g)$ always act effectively on $X^g$?
Question 2: Are there natural conditions one can impose so that $Z_\Gamma(g)$ acts freely on $X^g$ (besides requiring the action of $\Gamma$ on $X$ to be free)?