TITLE: center of fundamental group of finite volume-hyperbolic orbifold
QUESTION [5 upvotes]: Is center of a fundamental group of finite volume-hyperbolic n-orbifold trivial?
Is there a good reference that the proof is wriiten?

REPLY [6 votes]: Let $G$ be the fundamental group of your orbifold, and recall that $G$ is a lattice in the group of isometries of hyperbolic space.
An element in the center of $G$  has to fix every fixed point of any loxodromic and any parabolic element of $G$. The fixed points of such elements are dense in the boundary at infinity of hyperbolic space (this is a consequence of the fact that the orbifold has finite volume, I think that a proof of this fact can be found in Ratcliffe's book, for example). 
This implies that any element in the center of $G$ acts as the identity on the boundary of hyperbolic space, so the center of $G$ is trivial.