A group of isometries with non-closed orbits

A GR OUP OF ISOMETRIES WITH NON-CLOSED ORBITS H. ABELS AND A. MANOUSSOS Abstract. In this note we give an exa mple of a o ne-dimensional manifold with t wo connected co mpo nen ts and a complete metric whose group of isometrie s has an orbit whic h is not closed. This answers a question of S. Gao and A. S. Kechris. 1. Preliminaries and the construction of the example In [3 , p. 35] S. Gao a nd A. S. Kec hris ask ed the follow ing question. Let ( X , d ) b e a lo cally compact complete metric space with finitely man y pseudo-compo nen ts or connected comp o nen ts. Do es its group of isome tr ies ha v e closed orbits? This is t he case if X is connected since then the group of isometries acts prop erly b y an old re sult of v an Dan tzig and v an der W aerden [1] and hence all of its orbits a r e close d. The ab o ve question arose in the follow ing conte xt. Supp ose a lo cally compact g roup with a countable base acts on a lo cally compact space with a coun table base. The n the action has lo cally closed orbits (i.e. orbits whic h are op en in their closures) if and only if there exists a Borel section for the action (see [4 ], [2]) or, in o ther terminology , the corre- sp onding orbit equiv alence relation is smo oth. F or isometric actions it is easy to see that a n orbit is lo cally c losed if and only if it is closed. In this note we giv e a negative answ er to t he question of Gao and Kec hris. Our space is a o ne- dimensional manifold with tw o connected comp o- nen ts, one compact isometric to S 1 , and one non-compact, the real line with a lo cally Euclidean metric. It has a complete metric whose g roup of isometries has non-closed dense orbits on the compact component. In the course o f the construction we giv e an example of a 2-dimensional 2000 Mathematics S u bje ct Classific ation. Primary 37B05, 54H15; Secondary 54H20. Key wor ds and phr ases. Prop er action, group of isometries , smoo th orbit equiv- alence relation. During this r esearch the seco nd author was fully s upp or ted by SFB 7 01 “Sp ek- trale Strukturen und T op ologische Methoden in der Mathematik” at the University of Biele feld, Ger many . He is g rateful fo r its generosit y and hospitality . 1 2 H. ABELS AND A. MANOU SSOS manifold with tw o connected comp onen ts o ne compact and one non- compact and a complete metric whose group G of isometries also has non-closed dense orbits o n the compact comp onen t. The difference is that G con ta ins a subgroup of index 2 whic h is isomorphic to R . Let ( Y , d 1 ) b e a metric space. Later on Y will b e a torus with a flat Riemannian metric. Let Z = Y ∪ ( Y × R ). W e fix tw o p ositive real n um b ers R and M . W e endo w Z with the fo llo wing metric d dep ending on R and M . d ( y 1 , y 2 ) = d 1 ( y 1 , y 2 ) d (( y 1 , t 1 ) , ( y 2 , t 2 )) = d 1 ( y 1 , y 2 ) + min( | t 1 − t 2 | , M ) d ( y 1 , ( y 2 , t 2 )) = d (( y 2 , t 2 ) , y 1 ) = d ( y 1 , y 2 ) + R, for y 1 , y 2 ∈ Y and t 1 , t 2 ∈ R . It is easy to c hec k that d is a metric on Z if 2 R ≥ M . The metric space Z has the follo wing pro p erties 1.1. a) F or a giv en p o int ( y , r ) ∈ Y × R there is a unique p oin t in Y whic h is closest to ( y , r ), namely y . b) Given a p oint y ∈ Y the set of p oin ts in Y × R whic h are closest to y is the line { y } × R . c) F or eve ry p oin t ( y , r ) ∈ Y × R and ev ery y ′ ∈ Y there is a un ique p oin t o n the line { y ′ } × R whic h is closest to ( y , r ), namely ( y ′ , r ). d) Let g Y b e an isometry of Y and let g R b e an isometry of the Euclidean line R . Define a map g = g ( g Y , g R ) : Z → Z b y g | Y := g Y and g ( y , r ) = ( g Y ( y ) , g R ( r )) for ( y , r ) ∈ Y × R . Then g is an isometry of Z . e) Ev ery isometry of Z is of the form giv en in d) if Y is compact. Pr o of. a) through d) are easily c hec ked. T o pro v e e) let g b e an isometry of Z . Then g ( Y ) = Y a nd g ( Y × R ) = Y × R , since Y is compact and Y × R consists of non- compact components . Then g Y := g | Y is an isometry of Y . The map g ( g Y , id ) − 1 ◦ g , where id denotes the iden tity map, is an isometry of Z whic h fix es Y , hence maps ev ery line { y } × R to itself, b y b). Let h y : R → R b e defined b y g ( y , t ) = ( y , h y ( t )). T hen h y is an isometry of the Euclidean line R for eve ry y ∈ Y a nd all the h y ’s are t he same, b y c), say h y = g R . Th us g = ( g Y , g R ).  1.2. Let no w Y b e a 2-dimensional torus with a flat R iemannian metric. Y is also a n ab elian Lie g roup whose comp o sition w e write as m ultipli- cation. Ev ery translation L x of Y , L x ( y ) = x · y , is an isometry . Let g ( t ), t ∈ R , b e a dense one parameter subgroup of Y . Let H ⊂ Y × R b e its graph, H = { ( g ( t ) , t ) ; t ∈ R } . O ur example is X = Y ∪ H with the metric induced from Z = Y ∪ ( Y × R ). A GROUP OF ISOMETRIES WI TH NON -CLOSED ORBITS 3 1.3. a) If g R is an isometry of the Euclidean line R then there is a unique isometry g of X s uch 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 r estriction of g ( L g ( a ) , L a ) to X . If g R is the reflec tio n at O , g R = − 1 , t hen g is the restriction o f g ( in v , − 1 ) to X , where in v : Y → Y , inv ( y ) = y − 1 . The reflection in a ∈ R is the comp osition L − 2 a ◦ ( − 1 ) = − 1 ◦ L 2 a . b) Ev ery isometry of X is o f the form in a). It follows that the group of isometries of X has dense no n- closed orbits on Y and the other comp onen t H is o ne orbit. c) H is lo cally isometric to the real line with the Euclidean metric, actually d (( g ( t ) , t ) , ( g ( s ) , s )) = (1 + k • g (0) k ) | t − s | for small | t − s | , where • g (0) is the tangen t o f the one-parameter group g ( t ), t ∈ R , and k · k is the norm on the tangen t space of Y at the iden tity elemen t derive d from the Riemannian tensor. Pr o of. c) follows from t he definition of t he metric d on Y × R . The maps giv en in a) are isometries o f Z and map X to X , hence are isometries of X . T o pro ve the uniqueness claim in a) it suffic es to prov e it fo r g R = id . But then g is the iden tit y on the image of the one-par a meter group g ( t ), t ∈ R , by 1.1 a) and hence on all of Y . Hence g has the form giv en b y 1.1 d). T o sho w b) it suffices t o sho w that ev ery isometry h of H is of the fo r m g iv en in a). This follow s from c).  1.4 R e mark. In our example the space has dimension 2 and the group of orien tation preserving isometries is of index 2 in the group o f all isometries and is is omo r phic to R . W e can reduce the dimension of our space to 1 to obtain a group o f isometries with clos ed orbits on the non-compact comp onen t, whic h is diffeomorphic and lo cally isometric to R , and non-closed dense orbits o n the compact comp onen t, whic h isometric to S 1 . The example is as follow s. T ake a one-dimensional subtorus Y 1 of Y containing the iden tit y ele ment of Y . Define X 1 = Y 1 ∪ H ⊂ Y ∪ H . Then the group of isometries of Y 1 consists o f those maps g a = g ( L g ( a ) , L a ) restricted to Y 1 with g ( a ) ∈ Y 1 , and of the maps g ( i nv ◦ L g (2 a ) , − 1 ◦ L a ) r estricted to Y 1 with g (2 a ) ∈ Y 1 . The pro of follo ws from the pro o f of 1.3. Reference s [1] D. v an Dant zig and B. L. v an der W aerden, ¨ Ub er metrisch homo gene R¨ aume , Abh. Math. Seminar Hamburg 6 (1928), 367-3 76. [2] E. G. Effros. T ra nsformation gr oups and C ∗ -algebr as , Ann. o f Ma th. (2) 81 (1965), 38- 5 5. [3] S. Gao and A. S. Kec hr is, On the classific ation of Polish metric sp ac es up to isometry , Mem. Amer. Math. So c. 161 (200 3), no. 766. 4 H. ABELS AND A. MANOU SSOS [4] J. Glimm, L o c al ly c omp act tra nsformation gr oups , T rans. Amer. Ma th. So c. 101 (1961), 1 24-13 8. F akul t ¨ at f ¨ ur Ma thema tik, Universit ¨ at Bielefeld, Postf ach 100131, D-33501 Bielefeld, Germany E-mail ad dr ess : abel s@math .uni-biele feld.de F akul t ¨ at f ¨ ur Ma thema tik , SFB 701, Universit ¨ at Bielefeld, Post- f a ch 100131, D-33501 Bielefeld, Germany E-mail ad dr ess : aman ouss@m ath.uni-bi elefeld.de