Reflection groups of geodesic spaces and Coxeter groups

H.S.M. Coxeter showed that a group $\mathrm{\Gamma }$ is a finite reflection group of an Euclidean space if and only if $\mathrm{\Gamma }$ is a finite Coxeter group. In this paper, we define {\it reflections} of geodesic spaces in general, and we prove that $\mathrm{\Gamma }$ is a cocompact discrete reflection group of some geodesic space if and only if $\mathrm{\Gamma }$ is a Coxeter group.