Mauro Patrão, *Homotheties and Isometries of Metric
Spaces*

** Abstract**

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 a

Heine-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 Lie

transformation group with respect to
the compact-open topology.

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February 18, 2004