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Vladimir Pestov, Epimorphisms of Hausdorff Groups by Way Of 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 ).
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