A group of isometries with non-closed orbits

TL;DR

This paper provides a counterexample of a one-dimensional manifold with a complete metric where the isometry group has a non-closed orbit, addressing a question in the field of geometric group actions.

Contribution

It constructs a specific example of a manifold with a complete metric exhibiting a non-closed orbit under its isometry group, answering an open question.

Findings

Existence of a one-dimensional manifold with non-closed isometry orbit

Counterexample to a previously open question

Illustrates complex behavior of isometry groups on manifolds

Abstract

In this note we give an example of a one-dimensional manifold with two connected components and a complete metric whose group of isometries has an orbit which is not closed. This answers a question of S. Gao and A. S. Kechris.