A group of isometries with non-closed orbits

In [3, p. 35] S. Gao and A. S. Kechris asked the following question. Let (X, d) be a locally compact complete metric space with finitely many pseudo-components or connected components. Does its group of isometries have closed orbits? This is the case if X is connected since then the group of isometries acts properly by an old result of van Dantzig and van der Waerden [1] and hence all of its orbits are closed. The above question arose in the following context. Suppose a locally compact group with a countable base acts on a locally compact space with a countable base. Then the action has locally closed orbits (i.e. orbits which are open in their closures) if and only if there exists a Borel section for the action (see [4], [2]) or, in other terminology, the corresponding orbit equivalence relation is smooth. For isometric actions it is easy to see that an orbit is locally closed if and only if it is closed. In this note we give a negative answer to the question of Gao and Kechris. Our space is a one-dimensional manifold with two connected components, one compact isometric to S 1 , and one non-compact, the real line with a locally Euclidean metric. It has a complete metric whose group of isometries has non-closed dense orbits on the compact component. In the course of the construction we give an example of a 2-dimensional 2000 Mathematics Subject Classification. Primary 37B05, 54H15; Secondary 54H20.

Key words and phrases. Proper action, group of isometries, smooth orbit equivalence relation.

During this research the second author was fully supported by SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik” at the University of Bielefeld, Germany. He is grateful for its generosity and hospitality. manifold with two connected components one compact and one noncompact and a complete metric whose group G of isometries also has non-closed dense orbits on the compact component. The difference is that G contains a subgroup of index 2 which is isomorphic to R.

Let (Y, d 1 ) be a metric space. Later on Y will be a torus with a flat Riemannian metric. Let Z = Y ∪ (Y × R). We fix two positive real numbers R and M. We endow Z with the following metric d depending on R and M. • g, where id denotes the identity map, is an isometry of Z which fixes Y , hence maps every line {y} × R to itself, by b). Let h y : R → R be defined by g(y, t) = (y, h y (t)). Then h y is an isometry of the Euclidean line R for every y ∈ Y and all the h y ’s are the same, by c), say h y = g R . Thus g = (g Y , g R ).

1.2. Let now Y be a 2-dimensional torus with a flat Riemannian metric.

Y is also an abelian Lie group whose composition we write as multiplication. Every translation

1.3. a) If g R is an isometry of the Euclidean line R then there is a unique isometry g of X such that g(y, t) ∈ Y × {g R (t)}. If g R is the translation by a, so g R = L a with L a (t) = t+ a, then g is the restriction of g(L g(a) , L a ) to X. If g R is the reflection at O, g R = -1, then g is the restriction of g(inv , -1) to X, where inv :

Every isometry of X is of the form in a). It follows that the group of isometries of X has dense non-closed orbits on Y and the other component H is one orbit. c) H is locally isometric to the real line with the Euclidean metric, actually d((g(t), t), (g(s), s)) = (1+ • g(0) ) |t-s| for small |t-s|, where • g(0) is the tangent of the one-parameter group g(t), t ∈ R, and • is the norm on the tangent space of Y at the identity element derived from the Riemannian tensor.

Proof. c) follows from the definition of the metric d on Y ×R. The maps given in a) are isometries of Z and map X to X, hence are isometries of X. To prove the uniqueness claim in a) it suffices to prove it for g R = id . But then g is the identity on the image of the one-parameter group g(t), t ∈ R, by 1.1 a) and hence on all of Y . Hence g has the form given by 1.1 d). To show b) it suffices to show that every isometry h of H is of the form given in a). This follows from c).

1.4 Remark. In our example the space has dimension 2 and the group of orientation preserving isometries is of index 2 in the group of all isometries and is isomorphic to R. We can reduce the dimension of our space to 1 to obtain a group of isometries with closed orbits on the non-compact component, which is diffeomorphic and locally isometric to R, and non-closed dense orbits on the compact component, which isometric to S 1 . The example is as follows. Take a one-dimensional subtorus Y 1 of Y containing the identity element of Y . Define X 1 = Y 1 ∪ H ⊂ Y ∪ H. Then the group of isometries of Y 1 consists of those maps g a = g(L g(a) , L a ) restricted to Y 1 with g(a) ∈ Y 1 , and of the maps g(inv • L g(2a) , -1 • L a ) restricted to Y 1 with g(2a) ∈ Y 1 . The proof follows from the proof of 1.3.

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