Nov 27, 1998 - A number of examples of non-conjugate and non-isomorphic stability groups is presented. Introduction. The recent progress in studying the ...
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The Erwin Schrodinger International Institute for Mathematical Physics
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On the Conjugacy and Isomorphism Problems for Stabilizers of Lie Group Actions
Valentin Ya. Golodets Sergey D. Sinel'shchikov
Vienna, Preprint ESI 635 (1998)
Supported by Federal Ministry of Science and Transport, Austria Available via http://www.esi.ac.at
November 27, 1998
ON THE CONJUGACY AND ISOMORPHISM PROBLEMS FOR STABILIZERS OF LIE GROUP ACTIONS VALENTIN Ya. GOLODETS
SERGEY D. SINEL'SHCHIKOV
B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Kharkov, Ukraine Abstract. The spaces of subgroups and Lie subalgebras with the group actions by conjugations are considered for real Lie groups. Our approach allows one to apply the properties of algebraically regular transformation groups to nding the conditions when those actions turn out to be type I. It follows, in particular, that in this case the stability groups for all the ergodic actions of such groups are conjugate (e.g. when the stability groups are compact). The isomorphism of the stability groups for ergodic actions is also established under some assumptions. A number of examples of non-conjugate and non-isomorphic stability groups is presented.
Introduction The recent progress in studying the measured equivalence relations and measure groupoids raised some problems concerning the non-free group actions. For a Borel group action on a measure space, its stability groups form a Borel eld of subgroups [23]. Such elds arise, in particular, in classi cation of 1-cocycles of an amenable equivalence relation in terms of the associated Mackey actions [10], when the Mackey action in question is non-free. Also it is applied to produce a cocycle with a given non-free Mackey action [8]. Moreover, it turned out that the amenability properties of a group action are completely determined by the amenability of the associated equivalence relation, together with the amenability of the stability groups [27, 8, 5, 1]. Nevertheless, many natural questions about non-free actions have no answers so far. For example, if the group acts transitively, then all the stability groups are conjugate. One may ask in what way this property transforms when passing to properly ergodic actions. Is it possible to describe the cases when all the stability groups of such actions are conjugate or at least isomorphic a.e.? Does there exist an ergodic group action with non isomorphic a.e. stability groups? These questions are discussed in the present work for actions of real Lie groups. In fact, they form a part of another important problem about the properties of families of closed subgroups for a given group. We prove that the compact stability groups for ergodic Lie group actions are always conjugate (Theorem 3.6). This is done by using the properties of the Fell factor for ergodic group actions [6]. Speci cally, our approach is as follows: we assign to each point x of a G-space its stability group Gx and the Lie subalgebra Hx of Gx. The rst map was shown to be Borel by L.Auslander, C.C.Moore [2] and A.Ramsay [20]. The second map is introduced 1
here by the authors; it is also shown to be Borel in the present work. It is worthwhile to note that the space of Lie subalgebras of a Lie algebra equipped with the Fell topology is homeomorphic to an algebraic subvariety of a Grassmann variety. This permits one to apply the properties of algebraically regular actions (the local closeness of orbits) [28], in particular to studying a conjugacy phenomenon for Lie subalgebras. It also turns out that the action of an algebraic group by conjugations is also regular when restricted to the part of the Fell factor formed by nite subgroups. This fact combined with the techniques related to the Lie subalgebras as above, as well as the structure theory of Lie groups [26] and their cohomology [18], together with the results on induced actions by R. Zimmer [28], allows us to establish that the action of a real Lie group on the space of its compact subgroups is type I, and hence our conjugacy result. The isomorphism of nite stability groups is also proved in the general case (Theorem 5.2). Furthermore, counterexamples show that the above conjugacy is no longer true for ergodic actions of general locally compact second countable (l.c.s.c.) groups (Example 3.8), nor in the case of non-compact stability groups (Example 3.7). It seems reasonable to note that the conjugacy of stability groups for an ergodic group action already implies that this action is induced from some ergodic action of the normalizer of a stability group. This is due to the existence of a transitive Fell factor [28]. Note that the extensive study of some special classes of stability groups for ergodic actions of semisimple Lie groups was also made in the work by G. Stuck and R. Zimmer [24]. Our next step is intended to generalize the conjugacy result for compact stabilizers of ergodic Lie group actions to the case of almost connected stabilizers (i.e., in the Lie group case those ones having only nitely many connected components). For that, we rst prove under some special assumptions that the action of a real Lie group by conjugations on the space of its almost connected subgroups is of type I. A very important step here, being also of some independent interest, is to prove that for an action of a real Lie group on the space of subalgebras of its Lie algebra via the adjoint representation is also of type I. All that is established below in the following cases: the topological component G0 of a given real Lie group G is a nite index subgroup in a real algebraic group, or G0 semisimple; G is almost connected, and G0 is simply connected with real eigenvalues of all operators of its adjoint representation. G is almost connected with G0 nilpotent; It should be noted that an exponential real Lie group may have an ergodic action with nonconjugate isotropies if its adjoint representation operators have non-real eigenvalues (Example 4.5). Such action is also constructed for Mautner group (Example 3.7). It follows that those groups are not in the class of groups described above. The isomorphism of almost connected stability groups is proved for ergodic actions of matricial (e.g. Mautner group) and simply connected real Lie groups. However, we show that isomorphism of stability groups for ergodic group actions is not a general property, even for the case of Lie groups. We produce such actions of amenable and non-amenable real Lie groups (see section 6). The authors are grateful to J.-P. Thouvenot for helpful discussions during the research. 2
1 Preliminaries Let G be a l.c.s.c. group and X a standard Borel G-space with ergodic quasi-invariant measure . The stability group at a point x 2 X is a subgroup Gx = fg 2 G : gx = xg in G. It follows from [14], [25] that Gx is a closed subgroup at a.e. x 2 X . A Borel map F : X ! Y between two nonsingular G-spaces (X; ) and (Y; ) is called a factor G-map if F is equivalent to , and F (gx) = gF (x) for all g 2 G at a.e. x 2 X . In this case (Y; ) will be termed a factor (G-space) of (X; ), and (X; ) an extension of (Y; ). The ergodicity of a G-space clearly implies the ergodicity for any of its factors. We use the space S = S (G) of closed subgroups of G. In the topology introduced by J. M. G. Fell [6] this is a closed subset of a compact Hausdor space C (G) of closed subsets of T G. This topology is determined by the basis of open sets U(C; F ) = fK 2 C (G) : K C = ; and K \ A 6= ; for any A 2 Fg, where C is compact in G and F is a nite collection of open sets. By our assumption on G; S = S (G) becomes a compact metric space [6] and hence a standard Borel G-space with G acting on S via conjugations. Furthermore, the Borel map X ! S given by x 7! Gx [2], [20] is clearly a factor G-map with S being given the image of for a measure. Thus, for a given G-space (X; ) we call S a Fell factor. Proposition 1.1. Let G be a l.c.s.c. group, H a closed normal subgroup, : G ! G=H the natural projection map. Then the associated map : S (G) ! S (G=H ) is Borel with respect to the Fell topologies. Proof. Note rst that
U(C ; U1 ; : : :; Un) = U(C ; ;)
\
U(;; U1)
\
:::
\
U(;; Un)
for C compact and U1; : : :; Un open subsets of G,Swhile if U is open, then one can nd compact sets C1; C2; : : :, with union U so that U(;; U ) = 1 n=1 (S (G) n U(Cn; ;)). Thus the sets U(C ; ;) generate the Borel -algebra in S (G), and hence of course a similar statement for S (G=H ); so we need only to see that ?1(U(C ; ;)) is Borel in S (G). S1 Q ; then Let Qn be anTincreasing sequence of compact sets in G with G = n=1 n T ?1 ?1 (U(C ; ;))) = 1 n=1 U( (C ) Qn ; ;). 2 Together with the Fell topology, one can impose on S (G) another topology introduced in the paper by Michael [17]. This topology, just like that of Fell, is generated by the family of open sets of the form U(C ; A1; : : :; An ) with C being allowed to be closed instead of being compact. Note that Sc (G) equipped with Michael topology is Hausdor and locally compact separable G-space, while possessing the same Borel structure as when being equipped with the Fell topology. Lemma 1.2. Let G be a l.c.s.c. group and Sc (G) the collection of compact subgroups of G. Then Sc (G) is a Borel subset of S (G) with respect to the Fell topology. S 1 Qn . Then Proof. Let Qn be an increasing sequence of compact sets with G = n=1 T 1 Sc(G) = S (G) n n=1 U(;; G n Qn). 2 Proposition 1.3. Suppose we have a Fell convergent sequence of subsets Hn 2 C (G) with a limit subset H , and xn 2 Hn a convergent sequence of points with limit x. Then T x 2 H. Proof. If x 62 H , then there is a compact neighborhood Ux of x with Ux H = ;, and hence U(Ux ; ;) is an open Fell neighborhood of H . Since xn ! x (n ! 1), we have 3
xn 2 Ux for n large enough, which is a contradiction with the Fell convergence Hn ! H and the choice of the neighborhood U(Ux ; ;). 2
2 The space of subalgebras of a Lie algebra.
Let G be a Lie algebra. One may treat G as a vector group, and hence the collection of its subgroups S (G ) is closed in the Fell topology. We are also interested in considering the collection LA(G ) of Lie subalgebras of G . For this we need rst Proposition 2.1. The collection of vector subspaces of L(G ) is closed in the Fell topology. Proof. Note that the vector subspaces are exactly those subgroups of S (G ) which are invariant with respect to multiplication of their elements by scalar multiples. Let Hn be a Fell convergent sequence of subspaces of G with a limit subgroup H. Suppose we are given v 2 H and 2 R. Since Hn converges to H, we can consider a decreasing sequence Un of neighborhoods of v together with the associated neighborhoods U(;; Un) of H to nd a sequence vn 2 Hn with vn ! v . Hence vn converges to v , which is in H by Proposition 1.3. 2 Proposition 2.2. The collection LA(G ) of Lie subalgebras of G is closed in S (G ) in the Fell topology. Proof. Let Hn be a Fell convergent sequence of subalgebras of G with a limit subspace H. Given any u; v 2 H, then the Fell convergence Hn ! H permits one to nd un ; vn 2 Hn with un ! u; vn ! v, and hence [un; vn] converges to [u; v]. Now an application of Proposition 1.3 yields [u; v ] 2 H. 2 Proposition 2.3. The collection LAn (G ) of all n-dimensional Lie subalgebras of G is closed in the Fell topology for each n. Proof. In virtue of the previous proposition it suces to prove the statement for vector subspaces. Let Hk be a sequence of n-dimensional subspaces of G convergent in the Fell topology to a subspace p-dimensional subspace H. Note rst that a convergence of vk 2 Hk to some v already impliesTv 2 H. In fact, if we assume v 62 H, then there is some compact neighborhood Uv of v; Uv H = ;, and U(Ux; ;) is a neighborhood of H which yields a contradiction with the convergence Hk ! H. Let fek1 ; : : :; ekng be an orthonormal basis in Hk . After picking out a subsequence one may assume that eki ! ei 2 H as k ! 1. Clearly fei gni=1 form an orthonormal system in H, hence p > n. On the other hand, if fei gpi=1 is any orthonormal basis in H, then one can nd open neighborhoods U1; : : :; Up, respectively, of e1 ; : : :; ep small enough that the Gram matrix of any collection of vectors v1 ; : : :; vp with vi 2 Ui , is nondegenerate. Form a neighborhood U(;; U1; : : :; Up); it contains Hk for k large enough, and hence one may nd in Hk a collection v1 ; : : :; vp as above, which implies p n. 2 Lemma 2.4. The collection Ln (G ) of n-dimensional vector subspaces of G , given the Fell topology, is homeomorphic to a Grassmann manifold Gnn+k . Proof. The Grassmann manifoldV Gnn+k admits a natural embedding as a Zariski closed subvariety of a projective space P1 ( n Rn+k) over the n-th exterior power of Rn+k via (linear span of e1 ; : : :; en ) 7! (line containing the vector e1 ^ : : : ^ en ) for any linear independent 4
vector system e1 ; : : :; en [12]. This embedding furnishes Gnn+k with a topology induced by the natural topology of the projective space. Since this topology is clearly metrizable as well as the Fell topology, everything may be done through considering the convergent sequences. Recall also that v1 ^ : : : ^ vn = e1 ^ : : : ^ en for some constant i each vi is in the linear span of e1; : : :; en . Suppose we have a convergence of n-dimensional subspaces Hk ! H in the Fell topology. Let fe1 ; : : :; en g be an orthonormal basis in H , then by the Fell convergence of Hk for any neighborhoods U1 ; : : :; Un respectively of e1 ; : : :; en there are ek1 ; : : :; ekn 2 Hk with eki 2 Ui when k is large enough. By decreasing Ui one may also arrange that eki ! ei as k ! 1. This readily implies ek1 ^ : : : ^ ekn ! e1 ^ : : : ^ en , and hence the desired convergence in the projective space. Conversely, suppose we are given n-dimensional subspaces Hk ; H respectively with the orthonormal basis fek1 ; : : :; ekn g and fVe1; : : :; en g such that the line containing ek1 ^ : : : ^ ekn converges in the projective space P1( n Rn+k ) to the line containing e1 ^ : : : ^ en . Thus after multiplying if necessary ek1 by 1 we have ek1 ^ : : : ^ ekn ! e1 ^ : : : ^ en as k ! 1. Replace each eki by some vector vik from Hk so that v1k ^ : : : ^ vnk = ek1 ^ : : : ^ ekn , and vik converge to ei (k ! 1). Let U(C; U1 ; : : :; Up ) be some neighborhood of H with respect to the Fell topology. If u = Pi iei 2 Um for some 1 m p, Tthen since vik ! ei as k ! 1; uk = Pi i vik is also in Um for k large enough, that is Hk Um 6= ;. Suppose also that there is some sequence ukj 2 Hkj such that ukj 2 C . After picking out a subsequence we may assume that ukj converges to some u 2 C as j ! 1. On the other hand, it follows from Proposition 1.3 that u 2 H . This contradicts the de nition of T U(C; U1 ; : : :; Up ) as a neighborhood of H , and thus shows that Hk C = ; for k large enough.
2
By an algebraic group G we mean a Zariski closed subgroup of GL(n; R). Its Lie algebra G may be treated as a subalgebra of the matrix algebra Mat(n; R), consisting of all tangent vectors to G at the identity. The adjoint representation Ad : G ! Aut G is given by the conjugation of matrices: (Ad g )(x) = gxg ?1; g 2 G; x 2 G ; it is clearly algebraically regular. This representation generates in a natural way an action of G on LA(G ). We are going to establish that this is an algebraically regular action on an algebraic variety. Recall that LAn (G ), while being a subset of a Grassmann variety Gnn+k , becomes already V n 1 a subset of a projective space P ( G ) via the embedding of Gnn+k as a Zariski closed subset V n 1 of P ( G ) [12]. V Proposition 2.5. LA(G ) is a Zariski closed subset of P1 ( G ), and hence an algebraic variety. V Proof. It suces to see that LAn (G ) for each n is Zariski closed in P1 ( n G ). Let e1 ; : : :; em be a basis in G , then the vectors ei1 ^ : : : ^ ein ; i1 < : : : < in, form a basis V n n 1 in ^ G . Denote by U an ane chart in P ( G ) distinguished by nonzero homogeneous coordinate corresponding to the vector e1 ^ : : : ^ en , and assume that D0 will stand for the linear span of e1 ; : : :; en . Then a n-dimensional plane D is in U i the natural projection map P of G onto D0 is one-to-one when restricted to D, and the inverse images vi = ei + j>n aij vj of ei ; i = 1; : : :; n, with respect to that restricted projection form a basis in D. De ne the maps i : U ! G ; i (D) = vi ; they are regular since aij is the coordinate of D with respect to the vector v1 ^ : : : ^ vj ^ : : : ^ vn , where vj (j > n) is written in the above exterior product instead of the i-th factor vi . Let also ij : U ! ^n+1 G be given 5
by ij (DT) = [ Ti (D); j (D)] ^T 1 (D) ^ : : : ^ n (D). These are also regular, andV hence LAn (G ) U = i;j (?ij1(f0g) Gnn+k ) is a Zariski closed subset of U . Since P1( n G ) is covered by only a nite collection of ane charts like U , we conclude that LAn (G ) is Zariski closed by the ane closeness criterion [12]. 2 Recall that a group action is said to be of type I i its orbit space is countably separated. Lemma 2.6. Let G be an algebraic group. The G-action on LA(G ) is type I. Proof. It suces to see that the action is type I when restricted to each LAn (G ). But since V n 1 the latter is a G-invariant algebraic subvariety of P ( G ) (Proposition 2.3) with respect to the natural G-action given by g (Rv1 ^ : : : ^ vn ) = Rgv1g ?1 ^ : : : ^ gvn g ?1, the matters reduce to proving the algebraic nature of the last action, which seems to be obvious. Now recall that the orbits of regular actions of algebraic groups on algebraic varieties are all Zariski locally closed [28], hence locally closed in the Fell topology (see propositions 2.2 { 2.4), which implies type I for a group action [22]. 2
3 Spaces of compact subgroups of Lie groups Lemma 3.1. Let G be a real algebraic group. The G-action on the space Sf (G) of nite
subgroups by conjugations is type I. Proof. It suces to prove that the G-action is type I when restricted to the space Sn (G) of n-element subgroups. Consider the direct product Gn equipped with the G-action
h(g1 ; : : :; gn) = (hg1h?1 ; : : :; hgnh?1 ) and the action of a permutation group Sn :
(g1; : : :; gn) = (g(1); : : :; g(n)); these actions clearly commute, so that we have a G Sn -action. Form also the algebraically regular maps k ; ik ; pjk : Gn ! G given by k (g1; : : :; gn) = gk (the projection), ik (g1; : : :; gn) = gk?1, and pjk (g1; : : :; gn) = gj gk . Let [n A = f(g1; : : :; gn) 2 Gn : k (g1; : : :; gn) = eg;
B=
\n [n k=1 j =1
k=1
f(g1; : : :; gn) 2 Gn : ik(g1; : : :; gn) = j (g1; : : :; gn)g; C=
\n [n [n j =1 k=1 r=1
f(g1; : : :; gn) 2 Gn :
pjk (g1; : : :; gn) = r (g1; : : :; gn)g; T T then the Zariski closed set Q = A B C is invariant with respect to the action of (a real algebraic group) G Sn . This action has locally closed orbits and hence is type I. Moreover, after discarding from Q a Zariski closed G Sn -invariant set, we get an open set Qe consisting of (g1; : : :; gn ) with all gi di erent. The G Sn -action on Qe is also type I. 6
e n is also type I. In fact, let T Qe be a We claim that the G-action on the quotient Q=S Borel section with respect to the G Sn -action (i.e. a Borel set which meets each orbit exactly e n, once), and q : Qe ! Qe =Sn the natural projection. One can easily verify that Gq (T ) = Q=S and each G-orbit meets the Borel set q (T ) only nitely many times, so the G-action induces on q (T ) a nite equivalence relation. The latter admits again a Borel section T1 q (T ), which e n. This assures the above action to be also works as a Borel section for the G-action on Q=S type I. On the other hand, the map Sn (g1; : : :; gn ) 7! (the subgroup generated by g1; : : :; gn) e n and the space Sn(G) of provides a Borel G-isomorphism between the quotient space Q=S n-element subgroups, which was to be proved. 2 Lemma 3.2. Let G be a l.c.s.c. group, X and Y locally compact Hausdor topological G-spaces, F : X ! Y a Borel G-equivariant map. Suppose that the G-action on Y is type I, and for each y 2 F (X ), the restriction of the equivalence relation on X generated by the G-action, to F ?1 (y), is type I. Then the G-action on X is type I. Proof. We are in the conditions of the Glimm theorem [7]; that is, to prove type I-ness for the G-action on X , it suces to verify that any ergodic quasiinvariant probability measure is supported on a single orbit. So let be any such measure on X . Then F is an ergodic quasiinvariant measure for the factor G-space Y , and since the latter is type I, it contains a conull orbit. We claim that at F -a.a. y 2 Y the induced equivalence relation on F ?1 (y ) (which is actually given by the action of the stability group Gy ) is ergodic with respect to the Z corresponding conditional measure. To see this, consider a decomposition = y d (y ) of Y with respect to F ; it follows from [21, lemma 3.1] that the family of conditional probability measures y can be chosen so that y g ?1 gy for all g 2 G; y 2 F (X ). If the action of the stability group Gy on F ?1 (y ) for some y is not ergodic, one can nd a Gy -invariant Borel subset B F ?1 (y ) with 0 < y (B ) < 1 (actually B can be chosen to be strictly invariant with respect to each g 2 Gy ). Then its saturation GB is an analytic (in fact Borel) invariant set in X , and now one can use the transitivity of the factor G-space Y and the above property of the conditional measures to deduce that 0 < (GB ) < 1. This clearly contradicts to the ergodicity of the G-space X , thus proving that y is Gy -ergodic. Being also type I, it is in fact essentially transitive. It remains now to observe that, due to the quasiinvariance of , the equivalence classes of conditional measures over the conull orbit in Y are twisted with each other by the G-action, and so is itself supported on a single orbit. 2 In the subsequent observations we require some facts concerning the space LA(G ) of subalgebras of a Lie algebra G of a real Lie group G. De ne the G-equivariant map L : Sc(G) ! LA(G ) by setting L(H ) to be the Lie subalgebra of G , corresponding to the subgroup H. Proposition 3.3. The map L : S (G) ! LA(G ) is Borel with respect to the Fell topologies. Proof. Since the sets U(C ; ;) generate the Borel -algebra in S (G) (see proof of propoS theU((exp C )1=k; ;), sition 1.1), it suces for our purposes to prove that ?1 (U(C ; ;)) = 1 k=1 where C G is compact, and (exp C )1=k = fexp( kt v ) : exp(tv ) 2 exp C g. Suppose a subgroup H G is an element of L?1 (U(C ; ;), then for any vectorT v 2 C the one-parameter group fexp(tv ) : t 2 Rg is not contained in H , and hence (U n feg) fexp(tv ) : 7
t 2 Rg T H = ; for small enough neighborhood U of the identity e in G. On the other hand (exp C )1=k U for k large enough, which already implies H 2 U((exp C )1=k ; ;). Conversely, suppose H 2 U((exp C )1=k ; ;) for some k. This clearly implies that H does not contain the entire one-parameter subgroup fexp(tv ) : t 2 Rg for any v 2 C , and hence its Lie algebra may not contain the vector v . That is, H 2 L?1 (U(C ; ;)). 2 Lemma 3.4. Let G be a real Lie group. The G-action on Sc (G) by conjugations is type I. Proof. Let G0 be the topological component of the identity in G, and R G0 the radical. Then R is a connected solvable Lie group, and the quotient G0 =R is a semisimple Lie group. We begin with proving that the R-action on Sc (G) is type I. This may be done by induction in the dimensionality of R. If R is zero dimensional (the identity) group, then the action is
of course trivial, and hence type I. Otherwise R admits a chain of connected normal subgroups R = R0 R1 : : : Rp Rp+1 = f0g with all the quotients Ri=Ri+1 being connected Abelian Lie groups. Since such groups are all isomorphic to a direct product of the form Rm Tn, and Tn is a normal (in fact central) subgroup of R=Ri+1 , one can arrange matters so that each quotient Ri=Ri+1 is isomorphic either to a nitely dimensional vector space Rm or to a nitely dimensional torus Tn. It follows from the induction hypothesis that the action of the quotient group R=Rp, on Sc(G=Rp) is type I, that is, there exists a countable family of R=Rp-invariant Borel sets Bj Sc (G=Rp) which separate the R=Rp-orbits. Clearly, this R=Rp-action may be also treated as a R-action, which has a natural extension to Sc (G). We denote this extension by p : Sc (G) ! Sc (G=Rp) (this map is Borel by Proposition 1.1; it is associated with the natural projection p : G ! G=Rp). m. Let K0; K G be two compact subgroups with We rst consider the T case Rp =R T p(K0) = p(K ). Since K0 Rm = K Rm = f0g, we deduce that for each k 2 K0 there is the unique element (k) 2 Rm with k (k) 2 K . One can also verify that : K0 ! Rm is a Borel (hence continuous) 1-cocycle (that is, (k1k2 ) = k2?1 (k1)k2 (k2)) [18]. By [18, theorem 2.3] every such cocycle is trivial, which easily implies that K and K0 are conjugate. In terms of the above notation we deduce that p?1 (Bj ) is a sequence of R-invariant Borel subsets of Sc(G) which separate the R-orbits, and so the R-space Sc (G) is type I. Now turnTto the caseTRp = Tn. Let K0; K G be two compact subgroups with p(K0) = p(K ) and K Tn = K0 Tn = C . Clearly C is a normal subgroup of both K0 and K . It follows from p(K0) = p(K ) that for each k 2 K0=C there exists a unique (k) 2 Tn=C with k(k) 2 K=C . One can also verify as above that : K=C ! Tn=C is a continuous 1cocycle, and the cohomologous cocycles 1 and 2 determine the conjugate subgroups K1 =C and K2 =C of N (C )=C . In fact we have the conjugacy of K1 and K2 . By [18, theorem 2.2] the rst cohomology group H 1(K0=C; Tn=C ) is countable. This together with the fact that the family of closed subgroups of Tn is countable imply that for any compact subgroup K G; p?1 (p(K )) (hence also p?1 (p(fg ?1Kg : g 2 Gg))) meets at most countable family of conjugacy classes of compact subgroups of G. This means, in particular, that we are in the conditions of Lemma 3.2, which implies in this case also that the R-space Sc (G) is type I. Observe that the above induction argument doesn't regard the solvability properties of the quotient group R=Rp; it is only intrinsic that R=Rp-action on Sc (G=Rp) is type I. Therefore, all that we need now is to establish that the action of G=R on Sc (G=R) is type I. This is a subject of Proposition 3.5. Let G be a real Lie group whose topological component of the identity 8
G0 is semisimple. The G-action on Sc (G) by conjugations is type I. Proof. Let C (G0 ) be the centralizer of G0 . Clearly this is a normal subgroup. It is also discrete countable since its topological component is obviously in G0 and hence is a subgroup of the center of G0, which is discrete by our assumptions on G0 . Let G be the Lie algebra of G = G=C (G0). Denote also as above Ad the adjoint representation of G by the automorphisms of G , and LA(G ) the space of Lie subalgebras of G
equipped with the Fell topology. The group Aut G is clearly algebraic, and hence its action on LA(G ) is type I [12]. On the other hand, since G0 = G0 =C (G0) = Ad G0 is semisimple and centerfree, G0 = Ad G0 is the component of the identity of Aut G [11]. In particular, G0 is a subgroup of nite index in Aut G [12], and hence so is G. Now it follows from [27, lemma 5.6] that the action of G via Ad on LA(G ) is type I. Consider the G-equivariant map L : Sc (G) ! LA(G ) by setting L(H ) to be the Lie subalgebra of G , corresponding to the subgroup H , then L is Borel by Proposition 3.3, and so LA(G ) becomes a type I factor G-space of Sc (G). Since there is a one-to-one correspondence between the Lie subalgebras of G and connected subgroups of G, L(H ) is a complete invariant of H0 ; in particular, the topological components of subgroups from L?1 (H) are the same for each Lie subalgebra H G . Note that NG (H0) is exactly the (closed) subgroup of G whose action leaves L?1 (H) invariant. Furthermore, since G0 is a subgroup of nite index in Aut G , one can easily deduce that NG (H0) is also a nite index subgroup in NAut G (H0 ). The group NAut G (H0) is clearly algebraic due to the compactness of H0 . Since all the subgroups in L?1 (H) contain H0 as a topological component of the identity, we have in fact an action of the algebraic quotient group NAut G (H0)=H0 on L?1 (H), and the latter may be identi ed with the subcollection Sf (NAut G (H0)=H0) of nite subgroups of NAut G (H0)=H0. Clearly this identi cation is Borel, and hence the NAut G (H0)=H0-action on L?1 (H) is type I by Lemma 3.1. Note that the above action may be also treated as a NAut G (H0)-action, and so [27, lemma 5.6] implies that the action of the nite index subgroup NG (H0) on L?1 (H) is also type I. On the other hand, the associated equivalence relation is exactly one given by the restriction to L?1 (H) of the relation generated by the G-action on Sc(G). Thus an application of lemma 3.2 establishes that the G-action on Sc(G) is type I. Let p : G ! G be the natural projection, and p : Sc (G) ! Sc (G) the associated Borel map (Proposition 1.1), which is clearly a G-equivariant map. We claim p is countable-to-one. In fact, since p is locally di eomorphic, for Q 2 Sc(G) all K 2 p?1 (Q) have the same Lie subalgebra in G , and hence the same topological component of the identity, say K0. Now given any connected components qQ0 of Q, then any two connected components g1K0 and g2 K0 of (possibly di erent) groups from p?1 (Q) may di er only by an element of a countable group C (G0), and so there may be only a countable family of those. Since also Q, being compact, has only nitely many connected components, we conclude that p?1 (Q) is at most countable. In particular, for every Q 2 Sc (G), the restriction of the equivalence relation, generated by the G-action on Sc (G), to p?1 (Q), is type I. Now the statement of our proposition follows from Lemma 3.2. 2 Theorem 3.6. [9] Let G be a real Lie group, and (X; ) an ergodic G-space with all the stability groups being compact. Then all the stability groups over a conull subset of X are conjugate in G. 2 Example 3.7. The compactness condition concerning the stability groups turns out to be essential in Theorem 3.6, as one can see from the following observations. 9
Consider the Mautner group
80 eit 0 z 1 9 1 < = H = :@ 0 ei t z2 A : t 2 R; z1 ; z2 2 C ; ;
0 0 1 for some xed irrational real . Clearly H is a dense subgroup in 80 1 0 z1 1 9 < = G = :@ 0 2 z2 A : 1; 2 2 T; z1 ; z2 2 C ; : 0 0 1 For the one-parameter subgroup 80 1 0 t 1 9 < = A = :@ 0 1 t A : t 2 R ; 0 0 1 form the homogeneous space G=A, which possess an ergodic H -action. The stability group of this action at a coset gA with 0 1 0 z1 1 g = @ 0 2 z2 A ; 0 0 1 is the subgroup 80 1 0 t 1 9 < 1 = gAg ?1 = :@ 0 1 2t A : t 2 R; ; 0 0 1 where the conjugation by g is certainly in the group G. However, this subgroup is not conjugate in the Mautner group H to A unless (1; 2) = (eit ; ei t) for some t 2 R. In particular, the domains of conjugacy of H -stability groups are just the H -orbits in G=A, which are all the null sets due to the proper ergodicity of the H -action on G=A. Example 3.8. It also turns out that even nite stabilizers may appear to be non-conjugate for an ergodic action of a general l.c.s.c. group. Consider the compact Abelian group K = (Z2)Z with Haar measure , the Lebesgue space X = (K K; ), and let Q be the Bernoulli shift on K; (Qx)n = xn?1 . Then (X; ) admits the measure preserving transformations Q(x; y ) = (Qx; Qy ); S (x; y ) = (x; y + x) and a K -action (k)(x; y ) = (x; y + k); k 2 K . These generate an ergodic solvable totally disconnected l.c.s.c. transformation group G on (X; ), which is the direct product of two subgroups. The rst one is Z2 (given by S ), and the second is the semidirect product of its normal subgroup K by a subgroup Z given by Q. This G-action is non-free, and the stability group at (x; y ) 2 X is Z2 generated by the transformation S (x). Clearly there are a continuum of di erent stability groups over any subset with positive measure in X . On the other hand the set of inner automorphisms of G is only countable, and hence the stability groups are not conjugate over a conull subset.
10
4 The spaces of almost connected subgroups of Lie groups Proposition 4.1. Let G be a Lie group. The subset Sac(G) of almost connected subgroups (that is, those ones containing only nitely many connected components) is Borel in S (G) with respect to the Fell topology. Proof. It clearly suces to prove that the collection Qn of (closed) subgroups possessing at least n connected components is open with respect to the Michael topology. Let n = 2, and H 2 Q2 . Since G is a normal topological space, there Texist open neighborhoods V1 and V2 of closed sets H0 andSH n H0 respectively such that V1 V2 = ;. Now form the Michael neighborhood US(G n (V1 V2 ); V1; V2) of H in S (G). Since V1 and V2 are disjoint, we conclude that U(G n (V1 V2 ); V1; V2) Q2. An obvious generalization of the above simple argument proves that Qn is open also for n > 2: 2 Lemma 4.2. Let G be a real Lie group whose topological component of the identity G0 is a nite index subgroup in a real algebraic group, or G0 semisimple. The G-action on Sac (G) by conjugations is type I. Proof This is very close to that of Proposition 3.5, except that, in the case of the rst assumption, the group Aut G is to be replaced by the algebraic group in which G0 is embedded as a nite index subgroup, and there is no longer necessity to divide by C (G0). Of course, Sc(G) is everywhere replaced by Sac(G). The only essential di erence is that the groups NAut G (H0) and NAut G (H0)=H0 need not to be algebraic since H0 may be non-compact. However, these are real Lie groups, so to prove the type I property of action of NAut G (H0) on L?1 (H), one should replace the reference to Lemma 3.1 by one to Lemma 3.4. The details are left to the reader. 2 Corollary 4.3. Let G be a real Lie group whose topological component of the identity G0 is a nite index subgroup in a real algebraic group, or G0 semisimple. Suppose that (X; ) is an ergodic G-space whose stability groups are all almost connected. Then all the stability groups over a conull subset of X are conjugate in G. Proof. We have a Borel G-map : X ! Sac (G) given by (x) = Gx (the stability group at x 2 X ). Thus, (Sac (G); ) becomes an ergodic G-space. Since it is also type I by Lemma 4.2, it is essentially transitive, which is exactly our statement. 2 Lemma 4.4. Let G be a real almost connected Lie group whose topological component G0 is nilpotent. Then the G-action on Sac (G) by conjugations is type I. Proof. In view of [27, lemma 5.6] it suces to prove the lemma in the case of connected G. We rst claim that the action of G on LA(G ) via the adjoint representation Ad G is type I. In fact, let Ge be the universal covering group of G, then Ge is a simply connected nilpotent Lie group with the same Lie algebra G as G. In view of that one may treat the above action as an Ad Ge -action on LA(G ) which is algebraically regular since Ge is algebraic [15, 16], and hence type I by Lemma 2.6. De ne just as in section 3 the Borel G-equivariant map L : Sac(G) ! LA(G ) by setting L(H ) to be the Lie subalgebra of G corresponding to the subgroup H . Since there is a oneto-one correspondence between the Lie subalgebras of G and connected subgroups of G, L(H ) is a complete invariant of H0 ; in particular, the topological components of subgroups from L?1 (H) are the same for each Lie subalgebra H G . Note that NG (H0) is exactly the (closed) subgroup of G whose action leaves L?1 (H) invariant. Since all the subgroups from L?1 (H) contain H0 as a topological component of the identity, we have in fact an action of the quotient group NG (H0)=H0 on L?1 (H), and the latter
11
may be identi ed with the subcollection Sf (NG (H0)=H0) of nite subgroups of NG (H0)=H0. Clearly this identi cation is Borel, and hence the NG (H0)=H0-action on L?1 (H) is type I by Lemma 3.4. On the other hand, the associated equivalence relation is exactly the one given by the restriction to L?1 (H) of the relation generated by the G-action on Sac(G). Thus an application of Lemma 3.2 establishes that the G-action on Sac (G) is type I. 2 However, an attempt of transferring of Lemma 4.4 to the class of exponential Lie groups fails, as one can see from Example 4.5. It follows from the above observations that it suces to produce an ergodic action of an exponential Lie group, whose stability groups are all almost connected and non-conjugate a.e. Consider the real Lie group 80 teit 0 w 1 9 < = G = :@ 0 tei t z A : t 2 R; w; z 2 C ; ; 0 0 1 for xed > 0; 2 R with =2 2= Q. Clearly, G is simply connected, and one can also quite easily verify that for each g 2 G, the operator Ad g has no eigenvalues with modulus 1 other than 1. This implies that G is an exponential Lie group [3, 19]. G is a closed subgroup in the real Lie group 80 0 w 1 9 < 1 = H = :@ 0 2 z A : > 0; 1; 2; w; z 2 C ; j1 j = j2j = 1; : 0 0 1 Consider also the subgroup 80 r 0 s 1 9 < = A = :@ 0 r s A : r; s 2 R; 0 0 1 of H with > 0 being the same as in the de nition of G. A routine veri cation shows that G acts ergodically on the homogeneous space H=A. A direct calculation also shows that the stability group of this action at the coset 0 1 0 w 1 @ 0 2 z A A 0 0 1 is the subgroup 80 r 0 ?r w + s + w 1 9 1 < = r ?r z + 2 s + z A : r; s 2 R : @ 0 : 0 0 ; 1 Each such subgroup has a normal closed one-parameter subgroup 80 1 0 s 1 9 1 < = @ A 0 1 s : s 2 R ; 2 : 0 0 1 ; 12
the two of those, determined by ; i and 0 ; i0 respectively, are conjugate in G i 10 = eit1 ; 20 = ei t2 for some t 2 R. This clearly implies that the stability groups of the above action are conjugate in G only along the G-orbits, and hence non-conjugate a.e. 2 Now we shall also see that an almost connectedness assumption in Lemma 4.4 is essential. Exampe 4.6. Consider a Lie group 80 1 r a 1 9 < = G = :@ 0 1 v A : r 2 Q; a; v 2 R; : 0 0 1 It is clearly nilpotent, with in nitely many connected components. A routine calculation shows that each subgroup from the family of connected subgroups 80 1 0 v 1 9 < = H = :@ 0 1 v A : v 2 R; ; 0 0 1
2 R, has the conjugacy class fH +r : r 2 Qg. This conjugacy class is clearly not locally closed in Sac(G), and thus Sac(G) is not a type I G-space. 2 Proposition 4.7. Let G be an almost connected real Lie group whose topological component G0 is exponential, and a linear representation of G in a real vector space V such that for each g 2 G0 , all the eigenvalues of the linear map ( g ) are real. Then the partition into orbits in V is type I. Proof. Clearly it suces to prove our statement for G connected, that is we may assume G0 = G [27]. We proceed by induction in dim V . If dim V = 1, then the action in question reduces to a multiplication by a character, and hence is clearly type I. In the general case note that under our assumption on G, there exists a complete Ginvariant ag f0g = V0 V1 : : : Vn?1 Vn = V with dim Vk = k. Consider the quotient representation of G in the space V=V1, where the eigenvalues are also real, and hence the partition into orbits is type I by the induction hypothesis. Let p : V ! V=V1 be the projection map, and v 2 V=V1, then the reduction of the equivalence relation in V to a (Borel) subset p?1 (v) is generated by the action of the (closed) subgroup Gv = fg 2 G : (g)v = vg. We claim that Gv is connected. In fact, since G is exponential, any g 2 Gv can be written as g = exp X for some vector X of the Lie algebra of G. Consider the representation of the one-parameter group exp(tX ) in the invariant subspace generated by exp(tX )v. Clearly, (exp(X )) is the identity of this subspace, and hence we actually deal with the representation of a circle group. This representation may be treated as orthogonal, and hence there exists a basis of eigenvectors. Each such eigenvector determines some character of a circle group, which by our assumption should be real-valued and hence trivial. This implies, in particular, that (exp(tX ))v = v for all t 2 [0; 1], and hence our statement. Thus the equivalence relation in p?1 (v) is generated by an action of a connected Lie group Gv , and so each orbit is either a point or a 1-dimensional submanifold. Since the line p?1 (v) can contain at most countable family of 1-dimensional orbits, the equivalence relation in p?1 (v) is type I. Now we are in the conditions of Lemma 3.2, which implies the statement of the proposition. 2 13
Proposition 4.8. Let G be an almost connected real Lie group whose topological component G0 is exponential, and a linear representation of G in a real vector space V such that for each g 2 G0 , all the eigenvalues of the linear map ( g ) are real. Then the partition into orbits in the projective space P1(V ) is type I. Proof. Consider an action of G R on V given by (g; )v = (g )v . We conclude from Proposition 4.7 that this action is type I. Let B V be a Borel set which meets each G R-orbit exactly once, and p : V ! P1(V ) the natural projection. Since the restriction of p onto B is one-to-one, p(B ) is a Borel subset of P1(V ) [28, corollary A6], which works as a section with respect to the partition of P1(V ) into G-orbits. 2 Lemma 4.9. Let G be an almost connected real Lie group such that its topological component G0 is simply connected, and for all g 2 G0 all the eigenvalues of the linear operator Ad g are real. Then the action of G on the space LA(G ) of subalgebras of its Lie algebra G via the adjoint representation Ad is type I. Proof. It suces to demonstrate type I-ness on the (invariant) Borel (section 2) subset LAk (G ) of k-dimensional Lie subalgebras of G . Recall that it admits a Borel embedding into the projective space P1(^k G ) given by (linear span of e1 ; : : :; ek ) ! (line containing e1 ^ : : : ^ ek ) [12]. Now the statement of the lemma is due to Proposition 4.8, since under our assumptions on G0 it is exponential. 2 Lemma 4.10. Let G be an almost connected real Lie group such that its topological component G0 is simply connected, and for all g 2 G0 all the eigenvalues of the linear operator Ad g are real. Then the action of G on the space Sac(G) via conjugations is type I. Proof is just the same as that of Lemma 4.4 modulo reference to Lemma 4.9. 2 Theorem 4.11. Let G be an almost connected real Lie group such that its topological component G0 is simply connected, and for all g 2 G0 all the eigenvalues of the linear operator Ad g are real. Let (X; ) be an ergodic G-space with all the stability groups being almost connected. Then all the stability groups over a conull subset of X are conjugate in G. 2
5 The isomorphism problem for stability groups We start with quite an easy observation concerning the nite stability groups. Let us rst consider the properties of convergent sequences of nite subgroups of a l.c.s.c. group G with respect to the Fell topology. If each of subgroups Kn has m elements, and Kn ! K , then it is easy to see that K has at most m elements. For that, it suces to consider the neighborhood of K of the form U(;; F ) with F = fkV : k 2 K g for V a small enough neighborhood of the identity in G. This implies that the set of m-element subgroups Sm is locally closed and hence Borel in S . Moreover, it turns out that the limit subgroup K retains some intrinsic information about the multiplication laws in Kn . In particular, we have Lemma 5.1. Let Kn ; K be the m-element subgroups of a l.c.s.c. group G such that Kn ! K in the Fell topology. If all Kn are isomorphic, then K is isomorphic to each of Kn. T Proof. Let U be a neighborhood of the identity in G small enough that k1 U k2U = ; for any di erent k1; k2 2 K . Then one can choose a neighborhood V of the identity in G such that V k?1 V k U for all k 2 K . Consider now a neighborhood of K of the form U(;; F ) with F = fkV : k 2 K g. It follows from the convergence Kn ! K in the Fell topology that 14
for n large enough and every k 2 K there exists only one element k0 2 Kn such that k0 2 kV , thus providing a one-to-one correspondence between Kn and K . One can easily deduce from the properties of the neighborhoods U and V that the above correspondence is in fact an isomorphism between Kn and K , and hence our statement. 2 Theorem 5.2. Let (X; ) be an ergodic Lebesgue G-space for a l.c.s.c. group G with nite stability groups. Then all the stability groups on a conull subset of X are isomorphic. Proof. Note rst that the number of elements m in the stability group is a.e. constant. This may be proved by an application of an exhaustion argument to the Borel subset f(g; x) : g 2 Gx g G X , using also the obvious constantness of m on each G-orbit and the ergodicity of the G-action on X . Thus the measure on the Fell factor S = S (G) is supported on the subset Sm consisting of m-element subgroups. Let us consider the partition of Sm into the closures of conjugacy classes of subgroups in the Fell topology. To see that this is a well de ned partition, it suces to show that any two such closures are disjoint or coincident. If K is a limit point of a conjugacy class of K1, then clearly K1 is also a limit of a sequence of conjugates of K . Thus if K lies also in the closure of the conjugacy class of some other subgroup K2 , then it follows from the above argument that the both conjugacy classes of K1 and K2 have the same closure. Clearly we obtain a type I equivalence relation on the Fell factor since each equivalence class is closed [22]. On the other hand every equivalence class is G-invariant with respect to the G-action on S . Thus we get a G-invariant Borel map from X into the (standard Borel) quotient space of Sm with respect to the above equivalence relation. By the ergodicity of G this map is constant a.e., that is, the family of stability groups over some conull subset of X is closed. Now the statement of the theorem follows from Lemma 5.1. 2 Lemma 5.3. Let G be a connected matrix Lie group or a closed subgroup of some GL(V ), and (X; ) an ergodic G-space whose stability groups are all almost connected. Then all the stability groups over a conull subset of X are isomorphic. Proof. One may reduce assumptions to the case when G is a closed (Lie) subgroup of GL(V ) for some nite dimensional vector space V [4]. Let (Y; ) be the induced GL(V )-space associated with the above embedding. It may be easily veri ed that the stability groups of this action are all of the form p?1 Gxp, with p 2 GL(V ) and Gx the stability group for the original G-action on X . The induced GL(V )-action is ergodic, and since GL(V ) is algebraic, one may apply Corollary 4.3 to deduce that all the stability groups of this action are conjugate in GL(V ). Now the explicit form of those stability groups already implies our statement. 2 It seems interesting to note that the Mautner group, while being matrix Lie group, is 'wild' in the sense of Kirillov [13]; it was also shown in Example 3.7 that connected stability groups of an ergodic action of Mautner group need not be conjugate. However, Lemma 5.3 implies they should be anyway isomorphic. Lemma 5.4. Let G be a simply connected Lie group, and (X; ) an ergodic G-space whose stability groups are all almost connected. Then all the stability groups over a conull subset of X are isomorphic. Proof. Form as above the Borel maps : X ! Sac (G), and L : Sac (G) ! LA(G ), with (x) being the stability group at x 2 X and L(H ) the Lie subalgebra of G corresponding to a closed subgroup H G. The action of G on LA(G ) via the adjoint representation may be extended to an action of an algebraic group Aut G , which is known to be type I. Moreover, 15
since G is simply connected, every automorphism of G is derived from some automorphism of G, and so we have an action of Aut G on G, and L becomes a G-map. One can see from the proof of Proposition 4.1 that Sac (G) is open in S (G) with respect to the Michael topology, whose Borel structure is the same as that generated by the Fell topology. Let H 2 L(Sac (G)), H = L(H ) for some H 2 Sac(G), then the restriction of the equivalence relation generated by the Aut G -action on Sac(G), to the Borel subset L?1 (H), is given by an action of a Lie group NAut G (H0). Since L?1 (H) admits a Borel identi cation with the collection of nite subgroups of NAut G (H0 )=H0, which is in turn a Borel subset of Sc(NAut G (H0)=H0), we may apply Lemma 3.4 to deduce that the above equivalence relation in L?1 (H) is type I. Now we are in the conditions of lemma 3.2, which imply that the Aut G action on Sac (G) is type I. Clearly is a G-map, and since the Aut G -action on Sac(G) is a type I extension of the G-action via conjugations, (X ) is inside a single Aut G -orbit, and hence our statement. 2
6 The examples of non-isomorphic stability groups We consider some examples of ergodic Lie group actions whose stability groups are not isomorphic a.e. Speci cally, Examples 6.1, 6.2 demonstrate in nite measure preserving actions of that type. Remark 6.3 explains how to produce an action with that property and nite invariant measure, but this action is degenerate (i.e., with non-trivial kernel). All these examples are actions of non-amenable groups, but it turns out that one can restrict the action to an amenable subgroup with the same stabilizers (Remark 6.4). We have no examples of such actions for countable or connected groups. Example 6.1. Consider the real Lie group
80 k > eG = : 00
9 l s a1 > = n t bC CA : k l 2 SL(2; Z); a; b; s; t; u 2 R; > m n 0 1 u ;
0 0 1 together with the family of its closed subgroups 9 80 1 0 s a 1 > > = < C B 0 1 s b CA : a; b; s; u 2 R Le ; = >B @ > ; : 00 00 10 u1
for ; 2 R. The discrete subgroup of Ge 9 80 1 0 0 m 1 > > = B A @ > ; : 00 00 10 01 e ? be the natural projection, and set up is contained in all Le ; . Let p : Ge ! G = G= L ; = Le ; =?. 16
Note that the group Ge is the semidirect product of SL(2; Z) by a nilpotent normal subgroup. Observe also that the intersection of all the subgroups Le ; is exactly the direct product of their common center 9 80 1 0 0 a 1 > > = 2 B = R > ; :@ 00 00 10 01 A
e and an additional copy of R (which corresponds to the parameter u). The quotient group G=C 2 2 is isomorphic to (SL(2; Z) n R ) R. The semidirect product SL(2; Z) n R here is coming from the natural action of SL(2; Z) on R2, and the multiplication law in SL(2; Z) n R2 is as follows: (A; (a; b))(B; (c; d)) = (AB; B ?1 (a; b) + (c; d)). Actually this multiplication law is realized if one identi es the element (A; (a; b)) with the coset 0 a11 a12 a11a + a12b 1 B a21 a22 a21a + a22b C C B @0 0 1 A; 0
0
0
1
in Ge , where A = (aij ) 2 SL(2; Z). Now de ne an action of SL(2; Z) n R2 on R3 by (the following linear representation): (A; (a; b)) (x; y; z ) = (A(x; y ); z ? ya + xb). An easy calculation shows that over a conull subset (with respect to the Lebesgue measure in R3) the stability group of the above action at a point ( s; s; z ) is of the form f(I; ( t; t)) : t 2 Rg, where I is the identity matrix. This action extends to an action of (SL(2; Z) n R2) R with the direct factor R not e be the natural projection map. Composing with , acting at all. Now let : G ! G=C we get an action of G with the torus T2 in the center of G0 (just the kernel of ) not acting at all. It follows from the above observations that the stability groups of this G-action are exactly L ; . Observe that the stability groups L ; have the common center
80 1 > B @ : 00
0 1 0 0
0 0 1 0
9 a1 > = bC C : a; b 2 R 0A 1
> ;
isomorphic to the torus T2, and the commutators of the elements of L ; are all inside the 1-parameter subgroup of T2 9 80 1 0 0 s 1 > > = B : > ; :@ 00 00 01 10 A Thus the isomorphism of L ; and L ; should induce the automorphism of the torus group T2, which intertwines the corresponding 1-parameter subgroups as above. Since any automorphism of T2 is given by some matrix A 2 SL(2; Z), the condition for L ; and L ; 0
0
0
17
0
being isomorphic is ( 0; 0) = A( ; ) for some A 2 SL(2; Z); 2 R. One can easily see that in the above G-action the domains of isomorphism of stability groups are all null sets; in particular, the stability groups are not isomorphic over a conull set. Example 6.2. The previous Example 6.1 may seem somewhat arti cial since that action has a kernel p(C ) R, and the action of G=(p(C ) R) has isomorphic stability groups. One can avoid this unpleasantness via an application of the inducing procedure [27]. Speci cally, for G from Example 6.1, we consider the direct product group G G, together with its automorphism given by (g; h) = (h; g ). Let Gb be the corresponding semidirect b (Gf1g) ! product of Z2 by GG, with G being embedded as a subgroup Gf1g. Let : G= bG be a Borel cross-section of the homogeneous space G= b (G f1g) given by [n (g; h)] = n (1; h). We start from the action of G ' G f1g on R3 from the Example 6.1 and form the induced Gb -action, which can be represented explicitly as follows [27]. The space of this action b f1g) with the natural measure class, and the action itself is given by is R3 (G=G
g(x; h(G f1g)) = ( (g; h(G f1g))x; gh(G f1g)) b (Gf1g) ! Gb being given by (g; h(Gf1g)) = with x 2 R3; g; h 2 Gb , and the cocycle : G= ? 1 (gh(G f1g)) g (h(G f1g)). Observe that the stability groups of this action are the conjugates in Gb of the subgroups L ; f1g, and, just as in Example 6.1, they are non-isomorphic a.e. On the other hand, those conjugates include also subgroups of the form f1g L ; . Hence the intersection of all
those subgroups is trivial, and so the action has trivial kernel. Remark 6.3. The actions in Examples 6.1, 6.2 are in nite measure preserving. If one replaces in Example 6.1 the space R3 of the group action by T3, then a very similar construction allows one to obtain a nite measure preserving action of the same group. In this case, the stability groups become non-connected, but their topological components of the identity are the same as in Example 6.1, hence the non-isomorphism phenomenon. Actually, the only condition for two stability groups being isomorphic in this new example is that they are over the same orbit of the group action; the stability groups over any di erent orbits are non-isomorphic. Remark 6.4. One can easily observe that the actions constructed in the Examples 6.1, 6.2 are amenable [8] actions of non-amenable groups. However, one can arrange an action of an amenable group with non-isomorphic stabilizers via replacing in Examples 6.1, 6.2 SL(2; Z) by (powers of) a single integral matrix A which generates an ergodic group automorphism of the two-dimensional torus.
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