Homotheties and Isometries of Metric Spaces

Let $(M,d)$ be a locally compact metric space. Since the work of van Dantzig and van der Waerden, it is well known that if $M$ is connected then its group of isometries $I(M,d)$ is locally compact with respect to the compact-open topology. In this paper we prove some extensions of this result to the group of homotheties $H(M,d)$. It is proved that when $(M,d)$ is aHeine-Borel metric space, its group of homotheties $H(M,d)$ is also a Heine-Borel metric space. We also prove that, when $(M,d)$ is a Heine-Borel ultrametric space, its group of isometries is an increasing union of compact subgroups and if $G$ is a finitely generated subgroup of $I(M,d)$ then $G$ is compact. It is also proved that when the space $\Sigma(M)$ of the connected components of $M$ is quasi-compact with respect to quotient topology, its group of homotheties $H(M,d)$ is locally compact with respect to the compact-open topology. An other main result is a generalization of a classical result in Riemannian geometry for Finsler manifolds: if $(M,d)$ is a connected Finsler manifold then its group of homotheties is a Lietransformation group with respect to the compact-open topology.