ON MACBEATH’S FORMULA FOR HYPERBOLIC MANIFOLDS

Abstract

Around 1973, Macbeath provided a formula for the number of fixed points of an element in a group $G$ of conformal automorphisms of a closed Riemann surface $S$ of genus at least two. Such a formula was initially used to obtain the character of the representation associated to the induced action of $G$ on the first homology group of $S$, and later turned out to be extremely useful in many other contexts. By using a simple counting procedure, we provide a similar formula for the number of connected components of an element in a finite group of isometries of a hyperbolic manifold.