NEW ZEALAND JOURNAL OF MATHEMATICS Volume 26 (1997), 257-262

EPIMORPHISMS OF HAUSDORFF GROUPS BY WAY OF TOPOLOGICAL DYNAMICS

V l a d i m ir P e s t o v (Received November 1995)

Abstract. Recent examples by Uspenskij show that epimorphisms in the cat­ egory of Hausdorff topological groups need not have dense range. We give a criterion for a morphism of Hausdorff groups to be an epimorphism in the language of topological dynamics.

1. Introduction

Let C be a category and H, G € Ob C be two objects. A morphism / : H —► G is an epimorphism in C if every two morphisms g, h from G to any F £ Ob C coincide whenever g o f = h o f . In most commonly used categories of topologo- algebraic structures epimorphisms are either onto or have dense range. This is the case for discrete groups, abelian Hausdorff groups, compact groups, locally compact groups, fc-groups (for the references, cf. [15]), C*-algebras [3] etc. Only recently Uspenskij, answering a long-standing and resistant problem by Hofmann, has made an astonishing discovery: epimorphisms in the category of Hausdorff topological groups need not have dense range [14, 15, 16]. This phenomenon reveals a striking new feature of noncommutativity in the presence of topology and calls for immediate attention because of a large potential significance in the coming era of noncommutative analysis and geometry. In this note we establish a rather transparent and workable criterion for a mor­ phism of topological groups to be an epimorphism in the language of topological dy­ namical systems. Thus, the epimorphism problem for Hausdorff topological groups merges fully into the realm of topological dynamics. In particular, the original Us­ penskij’s example [14, 15] is shown to be closely linked (in fact, equivalent) to an earlier result by Megrelishvili on topological transformation groups [9, 10].

2. Preliminaries

By F(X) we denote the free group on a set X of free generators. Let G be a Hausdorff topological group. A topological G-space X together with a fixed topological group structure on X is called an automorphic G-space [9, 10] if G acts on X by topological group automorphisms. Every topological G-space X admits an (essentially unique) universal morphism i : X —► Fq (X) to an automorphic G-space F g {X), that is, l is a morphism of G-spaces and each morphism of G-spaces / from X to an arbitrary automorphic G-space A admits

1991 AMS Mathematics Subject Classification: 22A05, 54H20, 18A20. Key words and phrases: Epimorphisms, Hausdorff groups, automorphic G-spaces. Research partially supported by the New Zealand Ministry of Research, Science and Technology through the project “Dynamics in Function Spaces” of the International Science Linkages Fund. 258 VLADIMIR PESTOV

a unique factorization / = / o where / : Fq {X) —► A is a morphism between automorphic G-spaces (in an obvious sense). One calls Fq (X) the free topological G-group on X . If the action of G is trivial, then Fg (X) turns into the familiar Markov free topological group on X [6]. It is easy to see that Fq {X) is always algebraically generated by the set t{X ), though it is still an open question whether or not Fg {X) must be algebraically free over t(X ). However, for a given G-space X the free topological G-group Fg {X) is just the Hausdorff replica of the free group F(X) equipped with the finest group topology such that its restriction to X is coarser than the original topology on X and the action G x F (X ) —> F(X) is continuous. The free topological G-group being trivial means that it is isomorphic to Z equipped with the discrete topology and the trivial action of G; in this case, l is constant and maps X to a generator of Z. For a detailed account of free topological G-groups, see [9, 10]. Let F and G be Hausdorff topological groups with a common closed subgroup H . (That is, in each of F and G there is a fixed closed subgroup isomorphic to H , together with a corresponding isomorphism.) The free product of F and G amalgamated over H is a Hausdorff topological group F *h G together with a pair of continuous homomorphisms i : F —> F *h G and j : G —> F *h G such that i\h = j\h (in an obvious sense) and every pair of continuous homomorphisms f ,g from F and G, respectively, to an arbitrary Hausdorff topological group A, satisfying f\H = 9\h, gives rise to a unique continuous homomorphism f * g : F *h G —* A with f = ( f * g) o i and g = ( / * g) o j. The free product with amalgamation exists, is essentially unique, and is algebraically generated by the set i(F) U j(G ) [4]. It is easy to see that in the case of the free product of two copies of a Hausdorff group G with itself, amalgamated over a closed subgroup H C G, both i and j are topological monomorphisms. (As a by-product of his results on epimorphisms, Uspenskij has observed in [15] that even in case where if is a proper closed subgroup, G *h G can coincide with G again; the equivalence (i) (ii) in our main Theorem belongs to him.) The reference for semidirect products of topological groups is [2], 2.6.20.

3. Results

Before stating our criterion, it is convenient to reduce the case of an arbitrary continuous homomorphism between topological groups to that of an embedding of a closed subgroup.

Proposition 1. Let f : H —* G be a continuous homomorphism of Hausdorff topological groups. Then the following are equivalent:

(i) / is an epimorphism of Hausdorff groups.

(ii) The embedding of f ( H ) into G as a closed topological subgroup is an epimor­ phism of Hausdorff groups.

Proof. It is enough to observe that if g, h are continuous homomorphisms from G to any topological group A, then go f = ho f if and only if — ^\j{H)- O EPIMORPHISMS OF HAUSDORFF GROUPS 259

Theorem 2. Let H be a closed subgroup of a Hausdorff topological group G. Denote by X the left topological G-spaceG/H. The following conditions are equiv­ alent. (i) The embedding H <—* G is an epimorphism of Hausdorff groups.

(ii) The free product G *h G of G with itself, amalgamated over H , coin­ cides with G in the sense that both i and j establish an isomorphism G = G *H G.

(iii) Any equivariant morphism of the G-space X to an automorphic G-space is trivial.

(iv) The free topological G-group of the G-space X is trivial.

(v) The only group topology on the free groupF(X), making the natural action of G by left translations continuous, is indiscrete.

Proof. (i) => (ii): Assuming that the free product with amalgamation G *h G is non-trivial, the canonical embeddings i ,j of G into G *# G are different, yet their restrictions to H coincide.

(ii) =$■ (i): Suppose H <—> G is not an epimorphism, and let f ,g : G —► A be such that f ^ g and / \h = 9\h• Fix an x G G with f(x) ± g(x). The diagonal products Id A / and Id A g are topological group monomorphisms from G to the topological group G x A, coinciding with each other on H and such that (Id A f)(x) £ (Id Ag)(G). Now the free product G *h G is clearly different from G because i(x) £ j(G).

(i) =>• (iii): Let F be an automorphic G-space and n : X —> F be a non-trivial morphism of G-spaces. Denote by r the action of G on F. Let A = G x T F be the topological group semidirect product of G with F relative the action r. Put e = 7r(H) £ F C A; the nontriviality of n(X) and transitivity of the action r on 7r(X) imply that for some g+ G G, one has g*£g+~l / e and therefore £<7*£-1 / g*. At the same time £ commutes with all elements of H. Now take for one morphism G —> A the canonical embedding G A, and for the other morphism the conjugate of the above embedding by e, that is, G 3 x t-> £X£~l G A. The two morphisms of G assume different values at g* yet coincide on H.

(iii) =>■ (ii): Suppose that i(G) ^ G *h G. We will construct a non-trivial morphism of G-spaces from X = G/H to an automorphic G-space. The canonical isomorphism between the two copies of G gives rise to an automorphism 0 of G * h G , intertwining the two factors. Since the square of this automorphism is the identity map, it can be interpreted as an action of upon G*hG. Let A — 'L2\x(G*h G) be the corresponding semidirect product. Denote by 1 the generator of Z 2 C A\ the conjugation by 1 in A establishes the canonical isomorphism between i(G) and j(G). Because of non-triviality of the free product, 1 is not central in A, yet 1 commutes with every h G H. Define a continuous action r of G on A by automorphisms by letting for each g G G and a G A

Tg{a) = i(g) > a • i(g)-1. 260 VLADIMIR PESTOV

Now A becomes an automorphic G-space. The map

w : G 3 x (-*• i(x) • 1 • «(x)_1 € A is continuous, nontrivial, its restriction to H is constant, and it commutes with the action r: for all x ,g e G, one has m(gx) = i(gx)-l-i{gx)~ 1 = i(g )i(x)-l •z(x)- 1 = i{g)w{x)i(g)~l = t9 w ( x ). Therefore, w defines a nontrivial morphism of G-spaces a : G /H —► A by the rule

(iii) => (iv): Just apply (iii) to the universal morphism l : X —* F g (X).

(iv) =>. (y): Suppose there is a non-indiscrete topology on F(G/H) making the action r, defined by rg{hH) = (gh)H , g,h e G, continuous. Denote by F the Hausdorff replica of F(G/H) equipped with this topology; let 7r: F(G/H) —► F be the quotient homomorphism. Since the closure of unity in F(G/H) is r-invariant, the action r gives rise to a continuous action, f, of G on F. The subspace G/H of F(G/H ) is non-indiscrete (for otherwise the closure of unity in F(G/H ) would contain both G/H and its group span, F(G/H )) and therefore tt(G/H) is non­ trivial, in contradiction to (ii). (v) =*> (iv): cf. a remark about free topological G-groups made in Section 2.

(iv) => (iii); Since any equivariant morphism of the G-space X to an automor­ phic G-space factors through the free topological G-group and the latter object is assumed to be trivial, the desired conclusion follows. □

Corollary 3. Let X be a transitive G-space such that both the acting group G and the space X are Polish. The following conditions are equivalent:

(i) X admits no non-trivial G-morphisms to automorphic G-spaces;

(ii) The canonical embedding of the stabilizer St^ of any point x € X into G is an epimorphism of Hausdorff topological groups.

Proof. According to Effros’s Microtransitivity Theorem ([1], Th. 2.1), the G-spaces X and G /Stx are isomorphic for every point x G X. Now our Theo­ rem applies. □

4. Final Comments

Remark 4. The first ever example of a nontrivial G-space admitting no non­ trivial G-morphisms to automorphic G-spaces belongs to Megrelishvili [9, 10]. He asked a question on the existence of such a G-space in [8] shortly after having constructed in 1988 [7] a G-space admitting no linearization, and soon afterwards he observed that the same example would do the job. Earlier versions of the pa­ pers [9, 10] were in existence and circulated in a preprint form since at least 1992. The acting group and the phase space in Megrelishvili’s example were both Polish, and in view of our Corollary 3 it contained implicitely an example of a Haus­ dorff group epimorphism without dense range. In fact, Uspenskij has chosen the same setting for his solution to the epimorphism problem [14]: the full group of autohomeomorphisms canonically acting on a compact topological manifold. EPIMORPHISMS OF HAUSDORFF GROUPS 261

(For other remarkable properties of objects of this kind, see Uspenskij’s earlier paper [13].)

Open Questions. From our viewpoint, it is important to understand whether or not epimorphisms have dense range in each of the following categories: (a) (simply) connected Banach-Lie groups and Lie group morphisms; (b) Banach-Lie algebras and their morphisms; (c) (simply) connected regular Frechet-Lie groups and C °° Lie group morphisms as defined in [5]; (d) Banach associative unital algebras and their morphisms; (e) involutive Banach associative unital algebras and their usual morphisms. Note that the questions (a) and (b) are not equivalent in view of the existence of non-enlargeable Banach-Lie algebras [12]. In connection with (e), cf. [3]. We believe that the above questions are intimately linked with infinite-dimensional smooth dynamical systems and representation theory. Free Banach-Lie algebras and their Lie groups [11], as well as free products of Banach-Lie algebras and groups within a suitably restricted category, might come useful in attempts to ob­ tain an analogue of our Theorem for connected Banach-Lie groups and algebras.

Acknowledgements. The author is indebted to Michael Megrelishvili and to Vladimir Uspenskij for stimulating discussions and for providing the manuscripts of papers [9, 10, 14, 15, 16] prior to publication.

References

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10. M. Megrelishvili, Free topological G-groups, New Zeal. J. Math. 25 (1) (1996), 59-72. 11. V. Pestov, Free Banach-Lie algebras, couniversal Banach-Lie groups, and more, Pacific J. Math. 157 (1993), 137-144; corrigendum, ibid., 171 (1995), 585-588. 12. W .T. van Est and T.J. Korthagen, Non-enlargible Lie algebras, Nederl. Akad. Wetensch. Proc. A 26 (1964), 15-31. 13. V.V. Uspenskij, A universal topological group with countable base, Funct. Anal. Appl. 20 (1986), 160-161. 14. V.V. Uspenskij, The solution of the epimorphism problem for Hausdorff topological groups, Seminar Sophus Lie 3 (1993), 69-70. 15. V.V. Uspenskij, The epimorphism problem for Hausdorff topological groups, Topology Appl. 57 (1994), 287-294. 16. V.V. Uspenskij, Epimorphisms of topological groups and Z -sets in the Hilbert cube, in Proc. 2nd Gauss Symposium (Miinchen, 1993), to appear.

Vladimir Pestov School of Mathematical and Computing Sciences Victoria University of Wellington P.O. Box 600 Wellington NEW ZEALAND vladimir .pestov@ vuw.ac. nz