On epimorphisms in some categories of infinite-dimensional Lie groups
aa r X i v : . [ m a t h . R T ] M a r Journal of Lie TheoryVolume ?? (??) ??–?? © ?? Heldermann Verlag Version of March 5, 2021 On epimorphisms in some categories of infinite-dimensionalLie groups
Vladimir G. Pestov and Vladimir V. Uspenskij
Abstract.
Let X be a smooth compact connected manifold. Let G = Diff X be the group of diffeomorphisms of X , equipped with the C ∞ -topology, and let H be the stabilizer of some point in X . Then the inclusion H → G , which is amorphism of two regular Fr´echet–Lie groups, is an epimorphism in the category ofsmooth Lie groups modelled on complete locally convex spaces. At the same time,in the latter category, epimorphisms between finite dimensional Lie groups havedense range. We also prove that if G is a Banach–Lie group and H is a properclosed subgroup, the inclusion H → G is not an epimorphism in the category ofHausdorff topological groups. Mathematics Subject Classification 2010:
Key Words and Phrases:
Epimorphism, locally convex Lie group, Fr´echet-Lie group,Banach-Lie group, Hausdorff topological group.
1. Introduction
A morphism f : X → Y in a category K is an epimorphism if for any object Z themapping f ∗ : Mor ( Y, Z ) → Mor (
X, Z ), defined by f ∗ ( g ) = gf , is injective.It is known that epimorphisms in the category of Hausdorff topological groupsneed not have dense range [20, 21, 22]. This result had in its time answered a long-standing open problem, see [16] for a survey of the previous state of knowledge. Itwas already known by that time that in the category of locally compact groupsepimorphisms between finite dimensional Lie groups need not have dense range(Kallman, unpublished, see [16], Remark 2.10), however, epimorphisms betweenlocally compact groups in the wider category of all topological groups must havea dense range ([16], Thm. 2.9).Those results prompt us to investigate intermediate categories of — possiblyinfinite-dimensional — Lie groups, more precisely, group objects in the categoryof C ∞ manifolds modelled on complete locally convex spaces (where we follow theapproach by Neeb [14]). We call such groups locally convex Lie groups and denote thecategory of such groups by LCLG . We show that within this, very wide, categorythere exist epimorphisms even between regular Fr´echet–Lie groups without denserange. A
Fr´echet–Lie group , or an
FL-group , is a locally convex group modelled ona Fr´echet space (= a complete metrizable locally convex space). At the same time,we strengthen Nummela’s result above by observing that already in the category of
Pestov and Uspenskij regular Fr´echet–Lie groups (viewed generally as the best-behaved class after that ofBanach–Lie groups, see [17]) an epimorphism between connected finite dimensionalLie groups must have a dense range.If K is the category of sets, or of groups, or of abelian groups, then epimor-phisms in K are precisely surjective morphisms. If K is the category of abelian Haus-dorff topological groups or of Hausdorff topological vector spaces, then a morphism f : X → Y is an epimorphism if and only if f ( X ) is dense in Y . K.H. Hofmannasked in the 1960s whether the same is true for the category TG of all Hausdorfftopological groups (not necessarily abelian). The answer is negative. For example,let X be a compact connected topological manifold, let G = Homeo X be the groupof all self-homeomorphisms of X , equipped with the compact-open topology, and let H = { f ∈ G : f ( p ) = p } be the stability subgroup at some point p ∈ X . Then theinclusion H → G is an epimorphism in TG [21].Now let X be a compact smooth manifold, and let G = Diff X be the groupof all diffeomorphisms of X . Then G has a natural structure of a regular Fr´echet–Lie group (where regularity implies in particular the existence of an exponentialmap, although we will never use it), see [13, 10]; [17], Ch. VI, §
2; [14], Ex. II.3.14.The corresponding topology on G , the C ∞ -topology, is the topology of uniformconvergence of all derivatives. Let p ∈ X , and let H = { f ∈ G : f ( p ) = p } be thestability subgroup at p . The group H also has a natural regular Fr´echet–Lie groupstructure, see [17], p. 145, Th. 4.5. In view of the result cited above, it is naturalto ask if the inclusion H → G is an epimorphism in the category of the regularFr´echet–Lie groups, or indeed in the wider category of all locally convex Lie groups.Our main result answers this question in the positive. Theorem 1.1.
Let X be a compact connected smooth manifold, let G = Diff X ,and let H ⊂ G be the stability subgroup at some point p ∈ X . Then the inclusion H → G is an epimorphism in the category LCLG of Lie groups modelled on completelocally convex spaces.
Since H is a proper closed subgroup of G , it follows that epimorphisms inthe category of locally convex Lie groups need not have dense range, even betweenregular Fr´echet–Lie groups.At the same time, the inclusion H → G of Theorem 1.1 is not an epimorphismin the category TG (Section 2, Remark 2). In other words, any two Lie groupmorphisms f, g : G → K to a locally convex Lie group K that agree on H mustbe equal, but there exist distinct TG -morphisms f, g : G → K to a Hausdorfftopological group K that agree on H .In Section 2 we prove a criterion for a morphism to be an epimorphism in TG (Theorem 2.1) and observe that Theorem 1.1 follows from the following assertion:( ∗ ) Use the notation of Theorem 1.1. Every G -equivariant smooth map from X to a locally convex Lie G -group is constant .The notion of a locally convex G -group is defined in the beginning of Section 2.It is a locally convex group equipped with a smooth action of G . We prove theassertion ( ∗ ) in Section 3.We also note that if f : G → H is a morphism between finite dimensional estov and Uspenskij G is connected),then f must have dense range in H . Another new result states that any morphism f : G → H between Banach–Lie groups that is an epimorphism in the category TG of Hausdorff topological groups must have dense range. Even more, if H is a properclosed subgroup (not necessarily a Banach–Lie subgroup) of a Banach–Lie group G ,then the inclusion H → G is not an epimorphism of Hausdorff topological groups.Finally, the anonymous referee has kindly provided an example of an embed-ding of connected finite dimensinal Lie groups (the subgroup of upper triangularmatrices in SL ( R )) that is an epimorphism of Banach–Lie groups.A number of unanswered questions are collected at the end of the article.
2. Criterion for an inclusion to be an epimorphism
Let G and F be two topological groups. Let us say that F is a G -group if G actscontinuously on F by automorphisms, that is, a continuous mapping σ : G × F → F is given such that for every g ∈ G the mapping σ g : F → F defined by σ g ( x ) = σ ( g, x ) is an automorphism of F and the mapping g σ g is a group homomorphism.In this situation, the semidirect product G ⋉ σ F can be defined, which is again atopological group. We shall usually write simply gx instead of σ ( g, x ).Similarly, if G and F are locally convex Lie groups (in the sense of [14]), wesay that F is a locally convex G -group if, as above, an action σ : G × F → F is givenwhich is in addition a C ∞ -mapping between manifolds. In this case, the semidirectproduct G ⋉ σ F has a natural locally convex group Lie group structure. If both G and F are Fr´echet–Lie groups, so is the semidirect product.For a subset A of a group G we denote by C G ( A ) the centralizer of A in G ,that is, the subgroup { x ∈ G : xa = ax for every a ∈ A } . Theorem 2.1.
Let i : H → G be a morphism in the category TG of Hausdorfftopological groups. The following are equivalent:1. i is an epimorphism in TG ;2. C K ( f i ( H )) = C K ( f ( G )) for any morphism f : G → K in TG ;3. if F is a topological G -group and x ∈ F is i ( H ) -fixed (that is, hx = x forevery h ∈ i ( H ) ), then x is G -fixed.If H is a closed subgroup of G and i : H → G is the embedding, these three condi-tions are also equivalent to the fourth:(4) for any topological G -group F any G -equivariant continuous mapping f : G/H → F is constant. Proof.
Clearly a morphism i : H → G is an epimorphism in TG if and only ifthe inclusion i ( H ) → G is. Hence without loss of generality we may assume that H is a subgroup of G and i : H → G is the inclusion.We show that not (1) ⇒ not (2). By definition, the inclusion H → G is notan epimorphism if and only if there exist a Hausdorff topological group L and two Pestov and Uspenskij distinct morphisms g, h : G → L which agree on H . Let C = { , a } be a cyclicgroup of order 2. The group C acts on L × L by permutations of coordinates, let K = C ⋉ ( L × L ) be the corresponding semidirect product. Let j : L × L → K be the natural embedding, and let b ∈ K be the image of a ∈ C under the naturalembedding C → K . If x, y ∈ L , then j ( x, y ) commutes with b if and only if x = y .Define a morphism f : G → K by f ( x ) = j ( g ( x ) , h ( x )). Then b commutes with f ( H ) but not with f ( G ). Thus C K ( f ( H )) = C K ( f ( G )).We show that not (2) ⇒ not (1). Suppose f : G → K is a morphism suchthat some b ∈ K commutes with f ( H ) but not with f ( G ). Let σ be the innerautomorphism of K corresponding to b . Then f and σf are distinct morphisms of G into K which agree on H . Thus the inclusion of H into G is not an epimorphism.We show that (2) ⇒ (3). Let F be a G -group, and let x ∈ F be an H -fixed element. Let K be the semidirect product of G and F , and let i : G → K and j : F → K be the canonical embeddings. If the condition (2) holds, then C K ( i ( H )) = C K ( i ( G )). Since x is H -fixed, we have j ( x ) ∈ C K ( i ( H )). It followsthat x ∈ C K ( i ( G )). Hence x is G -fixed.We show that (3) ⇒ (2). Let f : G → K be a continuous homomorphism,and let x ∈ C K ( f ( H )). The group K can be considered as a G -group, equippedwith the G -action defined by ( g, k ) f ( g ) kf ( g ) − . The element x is H -fixed. Ifthe condition (3) holds, then x is also G -fixed, which means that x ∈ C K ( f ( G )).Finally, the equivalence (3) ⇔ (4) is clear, since for any G -group F thereis a natural one-to-one correspondence between H -fixed elements of F and G -equivariant morphisms f : G/H → F , which assigns to every H -fixed element x ∈ F the morphism gH gx of G/H to F . Theorem 2.2.
Let i : H → G be a morphism in the category LCLG of locallyconvex Lie groups. The following are equivalent:1. i is an epimorphism in LCLG ;2. C K ( f i ( H )) = C K ( f ( G )) for any morphism f : G → K in LCLG ;3. if F is a locally convex Lie G -group and x ∈ F is i ( H ) -fixed (that is, hx = x for every h ∈ i ( H ) ), then x is G -fixed. The proof is quite similar to the proof of Theorem 2.1 and hence omitted.The essential point is that semidirect products exist in
LCLG . We did not includethe analogue of condition (4) of Theorem 2.1 in order to avoid possible ambiguitiesof the notion “locally convex Lie subgroup of a locally convex Lie group”. However,if X is a compact smooth manifold, G = Diff X and H is the stabilizer of a point p ∈ X , as in Theorem 1.1, the quotient G/H is well-defined in the category ofmanifolds modelled on (complete) locally convex spaces and can be identified with X . This means that for every manifold M modelled on a (complete) locally convexspace there is a one-to-one correspondence between smooth maps X → M and thosesmooth maps G → M that are constant on cosets gH . Thus we can apply the lastparagraph of the proof of Theorem 1.1 (the equivalence (3) ⇔ (4)) in this situation,and we conclude that the fact that the embedding H → G is an epimorphism in thecategory LCLG is equivalent to the following: estov and Uspenskij (*) If X is a compact connected smooth manifold, G = Diff X , and F is alocally convex Lie G -group, then every G -equivariant smooth mapping j : X → F is constant. Remarks.
1. Let X be a compact connected manifold and G = Homeo ( X ). Let H ⊂ G be the stability subgroup at some point of X . Then the inclusion H → G is an epimorphism in TG [21]. In virtue of Theorem 2.1, this assertion is equivalentto the following: For any topological G -group F any G -equivariant mapping j : X → F isconstant .The latter was proved in [12, Example 3.7] (for the case X is a cube or asphere, but the general case can be proved by the same method, Theorem 3.5 of[12] applies). Thus, as noted in [18, 19], the solution of the epimorphism problem forHausdorff topological groups obtained in [20, 21] can be deduced from Megrelishvili’sresults.We sketch the proof of the fact that j : X → F (as above) must be constant.Let x, y ∈ X . Connect x and y by a smooth arc in X , and pick points a = x, b , a , b , . . . , a n , b n = y going along the arc so that we get a fine partition of thearc. If every b i − is very close to a i , the element h = j ( a ) j ( b ) − j ( a ) j ( b ) − . . . j ( a n ) j ( b n ) − of the group F is very close to j ( x ) j ( y ) − . On the other hand, we can find g ∈ G close to identity such that g ( a i ) is very close to g ( b i ) ( ), and then gh = j ( ga ) j ( gb ) − j ( ga ) j ( gb ) − . . . j ( ga n ) j ( gb n ) − is close to e F , the identity of F .Thus the element c = j ( x ) j ( y ) − of F has the following property: we can find h close to c and g ∈ G close to e G so that gh is close to e F . It follows that c = e F and j ( x ) = j ( y ).2. If G = Diff X and H ⊂ G are as in Theorem 1.1, the inclusion H → G is not an epimorphism in the category TG . Indeed, according to Theorem 2.1, it sufficesto construct a topological vector space V , a jointly continuous linear representationof G on V , and a non-constant G -equivariant map from X to V . We can take for V the space generated by points of X , that is, the space of measures on X with afinite support. Let us describe the topology on V that suits our purposes.Let V be the hyperplane in V consisting of all measures of total mass zero. Itwill be sufficient to describe a locally convex topology on V such that the action of G on V is jointly continuous. Every metric d on X gives rise to the “transportationmetric” (also known as the Kantorovich-Rubinstein metric, and under many othernames, see [23]) ¯ d on V defined by¯ d ( v,
0) = inf nX | c i | d ( x i , y i ) : v = X c i ( x i − y i ) o , c i ∈ R , x i , y i ∈ X and ¯ d ( u, v ) = ¯ d ( u − v, d on X and thetopology on V generated by ¯ d . If g ∈ G is close to identity in the C -topology, A neighborhood of our arc in X can be identified with a Euclidean space so that the arccorresponds to a straight line segment; in that case the existence of g is geometrically obvious. Pestov and Uspenskij g is a C -Lipschitz tranformation of ( X, d ), where
C > G on V easily follows (Proposition 6.2).Comparing this remark with the previous one, one can conclude that if theembedding H → G in the category TG is an epimorphism, then the action of G on G/H is “far from being smooth” in a certain sense.3. Epimorphisms in the the category
AlgGr of linear algebraic groups overa given algebraically closed field need not have a dense range [3] (we are grateful toUgo Bruzzo for pointing out to us this reference as well as [4], and for his help witha copy of the article). For example, if G is a connected linear algebraic group and H is a parabolic subgroup of G , then the embedding H → G is an epimorphismin AlgGr (this readily follows from the fact that every morphism of the projectivevariety
G/H to an affine variety is constant). See [4] for recent advances in thistopic.
3. Idea of the proof of Main Theorem
Let X be a compact connected smooth manifold, G = Diff X , H = { f ∈ G : f ( p ) = p } for some p ∈ X . We want to prove the following: if F is a locally convex Lie G -group and j : X → F is a G -equivariantsmooth mapping, then j is constant. That will suffice, see Theorem 2.2 and the discussion after that.Let f be the Lie algebra of F [13], [10], [14]. Then G acts on f , and the actionmap G × f → f is jointly continuous (in fact, even smooth, see Lemma 4.1). Equipthe dual space f ∗ with the topology of uniform convergence on compact sets. If f is a Fr´echet space, then f ∗ is σ -compact (= the union of countably many compactsets), see Lemma 4.2. In general f ∗ is covered by G -invariant σ -compact subspaces(Corollary 4.7). There is a natural mapping j ∗ : f ∗ → Ω ( X ), where Ω ( X ) isthe space of C ∞ -smooth differential 1-forms on X , equipped with the C ∞ -topology(= the topology of uniform convergence of all derivatives). The construction of j ∗ is explained below. The mapping j ∗ is continuous (Proposition 4.3). It is G -equivariant since j is. We shall show in the next section that Ω ( X ) has no non-zero G -invariant σ -compact subspaces (Proposition 4.5). Since f ∗ is covered by G -invariant σ -compact subspaces, it follows that j ∗ = 0. This means that j isconstant.If M is a locally convex manifold and V is a locally convex vector space,a smooth V -valued differential 1-form on M is a smooth mapping of the tangentbundle of M to V which is linear on every T x M , the tangent space to M at x ∈ M .The space f ∗ can be identified with the space of all smooth left-invariant differential1-forms on F , and the mapping j ∗ : f ∗ → Ω ( X ) from the previous paragraph is justthe restriction of the natural map Ω ( F ) → Ω ( X ).An alternative way to describe j ∗ is the following. Consider the canonical f -valued 1-form θ on F such that θ ( gv ) = v for every g ∈ F and v ∈ f , where gv denotes the tangent vector to F at g obtained from v by the left translation by g .Consider the space Ω ( X, f ) of all smooth f -valued 1-forms on X . Let τ ∈ Ω ( X, f )be the inverse image of θ under j . In other words, if u is a tangent vector to X at x , then τ ( u ) = θ ( j ∗ ( u )), where j ∗ is the differential of j at x . Every h ∈ f ∗ induces estov and Uspenskij
7a natural map h ∗ : Ω ( X, f ) → Ω ( X, R ) = Ω ( X ), and we have j ∗ ( h ) = h ∗ ( τ ).Indeed, both forms have equal values on any vector u tangent to X , namely, thevalue that equals h ( θ ( j ∗ ( u ))).
4. Details of the proof
It remains to prove the statements mentioned in Section 3.
Lemma 4.1. If G and F are locally convex groups and G acts smoothly on F by automorphisms, the the action of G on the Lie algebra of F is jointly continuous(in fact, smooth). Proof.
The action of G on the Lie algebra f on F is exactly the restriction to G × f of the adjoint representation of the semidirect product G ⋉ F , viewed as amap ( G ⋉ F ) × ( g ⋉ f ) → g ⋉ f . But the adjoint representation is a smooth mapping([14], p. 333, Ex. II.3.9). Lemma 4.2. If E is a metrizable LCS, then the dual space E ∗ , equipped with thetopology of uniform convergence on compact sets, is σ -compact. This is of course well known. We decided to include a proof, since this is acrucial point in our arguments.
Proof.
Pick a countable base ( U n ) of neighborhoods of zero in E . Then E ∗ = S U ◦ n , where U ◦ is the polar set, U ◦ = { f ∈ E ∗ : sup {| f ( x ) | : x ∈ U } ≤ } . Every U ◦ n is compact with respect to the w ∗ -topology (the Banach–Alaoglu Theorem).Since U ◦ n is equicontinuous, the w ∗ -topology on U ◦ n is the same as the topology ofuniform convergence on compact sets [5, Ch. 3, page 17, Proposition 5]. Proposition 4.3.
Let X be a compact smooth manifold, F a locally convex Liegroup, f the Lie algebra of F , j : X → F a smooth mapping. The mapping j ∗ : f ∗ → Ω ( X ) considered in Section 3 is continuous. (The topology on f ∗ isthe topology of uniform convergence on compact subsets of f .) Proof.
Pick a Riemannian metric on X . For a smooth vector field v on X denoteby L v the Lie differentiation with respect to v [11, Ch.1, § ( X ) looks as follows: pick smooth vector fields v , . . . , v k on X , andconsider those forms ω ∈ Ω ( X ) for which the form η = L v . . . L v k ω has a smallnorm, that is, | η ( u ) | < ε for all unit tangent vectors u .Suppose vector fields v , . . . , v k and ε > h ∈ f ∗ is small on a certain compact set K and ω = j ∗ ( h ), then | L v . . . L v k ω ( u ) | < ε for all unit tangent vectors u . Consider the space Ω ( X, f )of all smooth f -valued 1-forms on X . If v is a smooth vector field on X , the Liedifferentiation L v is well-defined as a map from Ω ( X, f ) to itself.Consider the forms θ ∈ Ω ( F, f ) and τ ∈ Ω ( X, f ) introduced in the pre-vious section: θ is the canonical f -valued left-invariant 1-form on F , and τ = j ∗ ( θ ). Denote by K the compact subset of f consisting of all vectors of the Pestov and Uspenskij form L v . . . L v k τ ( u ), where u runs over all unit tangent vectors to X . If h is ε -small on K , ω = j ∗ ( h ) = h ∗ ( τ ), and u is a unit tangent vector to X , then L v . . . L v k ω ( u ) = h ∗ ( L v . . . L v k τ )( u ), since h ∗ commutes with every Lie differenti-ation L v . The last expression belongs to h ( K ) and hence is ε -small. Thus j ∗ iscontinuous.A topological space is Polish if it is homeomorphic to a complete separablemetric space. A topological space is Polish if and only if it is ˇCech-complete and hasa countable base (see [9], Th. 4.3.26). If a topological group with a countablebase is locally ˇCech-complete (that is, admits a ˇCech-complete neighborhood ofthe identity), then it is ˇCech-complete ([2], Proposition 4.3.17), hence Polish. Inparticular, every FL-group with a countable base is Polish.Recall that a set in a topological space is meagre if it is contained in theunion of countably many nowhere dense sets. If a space Y is completely metrizable(that is, admits a compatible complete metric), then no meagre subset of Y containsinterior points, in virtue of the Baire category theorem. Lemma 4.4.
Suppose that a topological group G acts continuously on a Hausdorffspace Y . Suppose that for every neighbourhood U of unity in G and for every y ∈ Y the closure of the set U y in Y is not compact. If G is not meagre in itself (inparticular, if G is Polish), then Y contains no non-empty G -invariant σ -compactsubsets. Proof.
Let y ∈ A = S ∞ n =0 K n ⊂ Y , where each K n is compact. We must showthat the orbit Gy is not contained in A . Let F n = { g ∈ G : gy ∈ K n } . Then each F n is closed in G and has no interior points. Since G is not meagre, there exists a g ∈ G \ S n F n , and we have gy / ∈ A . Proposition 4.5. If X is a compact manifold and G = Diff X , the space Ω ( X ) contains no non-zero G -invariant σ -compact subspaces. Proof. ( ) Apply Lemma 4.4 to the group G = Diff X and the space Y =Ω ( X ) \ { } . Since G is Polish, it suffices to check that for any neighbourhood U of the identity in G , any non-zero 1-form ω ∈ Ω = Ω ( X ) and any compactsubset K ⊂ Ω we have U ω K . We’ll confine ourselves to the case when X isa circle. Let ω = f ( θ ) dθ . Assume that U consists of all diffeomorphisms g of X such that the first k derivatives of g are close to those of the identity map. Asthe ( k + 1)-th derivative of g ∈ U can be arbitrary large, the differential forms g ∗ ( ω ) = f ( g ( θ )) g ′ ( θ ) dθ , where g runs over U , cannot all lie in a compact set.If G is a Polish group that acts on an LCS V by linear transformations sothat the action law G × V → V is jointly continuous, we say that V is a G -module. Proposition 4.6. If G is a Polish group and V is a G -module, the topology of V is generated by G -morphisms of V to metrizable G -modules. The referee pointed out that the tools of [8] can be used to prove this proposition. estov and Uspenskij U of zero in V there exist a G -morphism p : V → V ′ to a metrizable G -module V ′ and aneighborhood U ′ of zero in V ′ such that p − ( U ′ ) ⊂ U . Proof.
Let U be a given neighborhood of zero in V . For every g ∈ G find aneighborhood W g of g in G and a neighborhood U g of zero in V so that W g U g ⊂ U .There is a countable collection of W g ’s that cover G ; consider the corresponding U g ’s.With each of these U g do the same as we did with U . We get a larger countablecollection of neighborhoods of zero. Proceed in a similar manner. In countably manysteps we obtain a countable collection B of neighborhoods of zero such that for every O ∈ B and g ∈ G there exist O ′ ∈ B and a neighborhood W of g in G such that W O ′ ⊂ O . We can throw in finite intersections and images under homotheties withcoefficient 1 /
2. We assume that all neighborhoods in B are convex and symmetric.In this way we can achieve that B is a filter base, and the filter F generated by B is the filter of neighborhoods for some pseudometrizable locally convex topology T on V that is coarser that the original topology T . By our construction, the map G × V → V is continuous with respect to T at every point of the form ( g, g, x ) ∈ G × V and O ∈ B . Picka neighborhood W of g and O ′ ∈ B so that W O ′ ⊂ O . Shrinking W if necessary,we may assume that W x ⊂ x + O (we just use the T -continuity of the action).Then W ( x + O ′ ) ⊂ W x + W O ′ ⊂ x + O + O , and the T -continuity of the actionfollows. We take for V ′ the metrizable space associated with the pseudometrizablespace ( V, T ). Corollary 4.7.
Suppose G is a Polish group and V is a G -module. Equip V ∗ with the topology of uniform convergence on compact sets. The space V ∗ is coveredby G -invariant σ -compact subspaces (which may be non-closed). Proof. If p : V → W is a G -morphism to a metrizable G -module W , theimage of p ∗ : W ∗ → V ∗ is σ -compact (Lemma 4.2) and G -invariant. According toProposition 4.6, W ∗ is covered by subspaces of this sort.
5. The case of finite dimensional Lie groupsTheorem 5.1.
Let f : G → H be a Lie group morphism between two finitedimensional Lie groups. Assume that f is an epimorphism in the category of locallyconvex Lie groups (regular Fr´echet–Lie groups, if G has a countable base). Then f has dense range. Proof.
The Lie group H acts smoothly by left translations on the space C ∞ ( H )equipped with the topology of compact convergence with all derivatives ([15], proofof Prop. 4.6). That is, the action mapping λ : H × C ∞ ( H ) → C ∞ ( H ), λ g ( f )( x ) = f ( g − x ), is smooth. The complete locally convex space C ∞ ( H ) is an abelian Liegroup. The morphism f induces a homomorphism between discrete groups f : G/G → H/H , where G and H denote the connected components. This f is in particularan epimorphism in the category of discrete groups (Lie groups of dimension zero),0 Pestov and Uspenskij so is onto by an argument of Kurosh, Livshits and Shul’geifer, see [16], top of p.156 (or Theorem 4.3.1 in [19]). We conclude: the right homogeneous space f ( G ) \ H is connected. Find a smooth function h : f ( G ) \ H → R achieving its maximumexactly on the right coset f ( G ). The composition of h with the right quotientmap H → f ( G ) \ H gives an element h of C ∞ ( H ) whose stabilizer under the action λ is exactly f ( G ). According to Theorem 2.2 (equivalence of (1) and (3)), h is H -invariant, that is, constant. It follows that f ( G ) = H .If G is has a countable base, that is, has at most countably many connectedcomponents, the above argument (using only discrete groups, which are Fr´echet–Lie,indeed zero-dimensional Lie) shows that H has the same property, therefore C ∞ ( H )is a Fr´echet space.Note that in the category of connected finite-dimensional Lie groups epimor-phisms need not have a dense range. The existence of epimorphically embedded com-plex algebraic groups, like the subgroup of upper triangular matrices in GL ( n, C )[3], implies the analogous fact for finite-dimensional complex Lie algebras, and thecase of finite-dimensional real Lie algebras follows by complexification. See the nextsection (Thm. 6.3) for a stronger result: some of those embeddings are epimorphismseven in the category of Banach–Lie groups.
6. The case of Banach–Lie groups
We consider
Banach–Lie groups, or BL-groups, as defined in [6] under the name of(real) Lie groups. We are going to prove the following: if a morphism f : G → H between BL-groups is a an epimorphism in TG , then f has a dense range. Moregenerally: Theorem 6.1.
Let G be a BL group, H a proper closed subgroup. Then theinclusion H → G is not an epimorphism in TG . The argument is the same as in Remark 2, Section 2. We consider a certainmetric d on the space P = G/H and its extension ¯ d over the space V of measureson P with a finite support and total mass zero. We check that the action of G on V is jointly continuous. If V = V ⊕ R is the space of all measures on P with a finite support, there is a natural G -map from P to V , and we invokeTheorem 2.1 (equivalence of (1) and (4)) to conlude that the inclusion H → G is not an epimorphism in TG . Proposition 6.2.
Suppose a topological group G continuously acts on a metricspace ( M, d ) by Lipschitz transformations. Suppose there is a neighborhood U of theunity in G and a constant C > such that for every g ∈ U the g -shift σ g : M → M is C -Lipschitz. Let V be the space of measures on M with a finite support, V thehyperplane of measures of total mass zero. Equip V with the Kantorovich–Rubinsteinmetric ¯ d . Then the action of G on V is jointly continuous. This is essentially a version of Megrelishvili’s result [12, Theorem 4.4]. If G is generated by U (this happens, for example, if G is connected), we can drop the estov and Uspenskij G acts by Lipschitz transformations, as this will follow from thecondition that σ g is Lipschitz for every g ∈ U . Proof.
If ¯ d ( v, < ε , write v = P c i ( x i − y i ) so that P | c i | d ( x i , y i ) < ε . If g ∈ U ,we have ¯ d ( gv, ≤ P | c i | d ( gx i , gy i ) < Cε . It follows that the action G × V → V is jointly continuous at (1 G , σ g impliesthat the action is separately continuous. We conclude that the action is jointlycontinuous at every point ( g , v ): if h is close to 1 G and v ∈ V is small, we have hg ( v + v ) − g v = ( hg v − g v ) + hg v , where both summands are small.A norm on a group G is a function p : G → R + such that: (1) p (1 G ) = 0and p ( x ) > x = 1 G ; (2) p ( x ) = p ( x − ); (3) p ( xy ) ≤ p ( x ) + p ( y ). If G is ametrizable group, its topology is generated by a certain norm p , in the sense thatthe sets of the form { x ∈ G : p ( x ) < ε } constitute a base at 1 G . (Note that notevery norm generates a group topology.) If H is a closed subgroup of G , we candefine a compatible metric d on G/H by d ( a, b ) = inf { p ( g ) : ga = b } . We now prove Theorem 6.1. First we consider the case when G is a connectedBL-group. Denote by g its Lie algebra. We assume that a norm on g is given suchthat g is a Banach space and k [ X, Y ] k ≤ k X k · k Y k . The exponential length normon G is defined by p ( g ) = inf {k X k + . . . k X n k : g = e X . . . e X n } . This norm is compatible with the topology of G [1, Proposition 3.2]. Note that wemay assume that all vectors X i in the definition above are short, that is, belong toa given neighborhood of zero in g : otherwise replace a vector X by k vectors X/k .This will not change the sum of norms or the product e X . . . e X n .Equip G/H with the corresponding metric d defined as above: d ( a, b ) = inf {k X k + . . . k X n k : e X . . . e X n a = b } . Pick a small ε , and let U = { e X : k X k < ε } ⊂ G be the corresponding neighborhoodof 1 G . If k Y k < ε , g = e Y ∈ U and a, b ∈ P , then d ( ga, gb ) = inf {k X k + . . . k X n k : e X . . . e X n ga = gb } (1)= inf {k X k + . . . k X n k : e − Y e X . . . e X n e Y a = b } . Pick short vectors X ′ , . . . , X ′ n so that e X ′ . . . e X ′ n a = b and k X ′ k + · · · + k X ′ n k < d ( a, b ) . (2)Pick short vectors X i so that e X i = e Y e X ′ i e − Y . Then e − Y e X . . . e X n e Y a = b , andhence, according to the formula (1), d ( ga, gb ) ≤ k X k + . . . k X n k . (3)2 Pestov and Uspenskij
We have X i = H ( Y, H ( X ′ i , − Y )) = X ′ i + [ Y, X ′ i ]+ terms of higher degree, where H is the Hausdorff series [6, Ch.2, § ε is small enough, we have k X i k ≤ k X ′ i k ,therefore, from Eq. (2) and (3), d ( ga, gb ) < d ( a, b ).By Proposition 6.2, the action of G on V is jointly continuous. This provesTheorem 6.1 in the case G is connected.If we drop the assumption that G is connected, the argument with connectedcomponents used in the previous section shows that we can assume that H meetsall connected components of G . In that case G/H = G / ( G ∩ H ), and wehave seen above that there is metric on the manifold G/H such that G acts onthis manifold by Lipschitz transformations. Actually, the whole group G acts byLipschitz transformations. This can be deduced from the following observation:every automorphism of the connected Lie group G is Lipschitz, if G is equippedwith the exponential length metric as above. To see that the latter statement istrue, note that the tangent automorphism σ of the Lie algebra g is such that C − k X k ≤ k σX k ≤ C k X k for some constant C > f : G → H in the category of BL-groups have a dense range, in par-ticular if G and H are (connected) finite-dimensional Lie groups? The anonymousreferee provided the following counterexample to all three questions. Theorem 6.3.
Let B be the subgroup of upper-triangular matrices in SL ( R ) .Then the embedding B → SL ( R ) is an epimorphism in the category of BL-groups. Proof.
It suffices to prove the analogous fact for Lie algebras which reduces tothe following. Let g be a Banach–Lie algebra. Consider sl -triples ( h, e, f ) and( h, e, f ) in g , that is,[ h, e ] = 2 e, [ h, f j ] = − f j , [ e, f j ] = h. We must prove that f = f . Assume the contrary: f = f − f = 0. Put s = span { e, h, f } , v n = (ad f ) n f , and V = span { v , v , . . . } . Since [ e, f ] = 0and [ h, f ] = − f , it is easy to see that V is an s -module such that each v n is anad h -eigenvector with the eigenvalue − − n . As ad h is a bounded operator, thevectors v n are zero for n large enough. We therefore obtain a finite-dimensionalrepresentation of sl in which h has negative spectrum. That is impossible: thespectrum of h in a finite-dimensional representation of sl is symmetric with respectto zero [7, Ch.8, §
7. Open questions
Let f : G → H be two groups from a category marking a row, and suppose f isan epimorphism in a category marking a column. Does f necessarily have a denserange? Here is a summary of what we know. estov and Uspenskij × × × X X X f.-d. Lie × × ? X X
Banach-Lie × ? ? X Fr´echet-Lie × × ?loc. conv. Lie × ?top. groups × As we mentioned in Section 6, the three crosses appearing in the BL-columnare due to the anonymous referee. We cordially thank the referee for this valuablecontribution.We have seen that whether or not the inclusion i : H → G of a proper closedsubgroup is an epimorphism in a suitable category depends on the dynamical system( G, G/H ). For example, if it is smooth (or, more generally, Lipschitz, see Proposition6.2), the inclusion i is not an epimorphism of Hausdorff groups. In order for i to bean epimorphism in the category TG , the action of G on G/H must be “sufficientlymixing”. Formalizing this criterion could be an interesting task.
References [1] H. Ando, M. Doucha, Y. Matsuzawa,
Large scale geometry of Banach–Liegroups, arXiv:2011.10376v2 [math.OA], version of Dec. 11, 2020.[2] A. Arhangel’skii and M. Tkachenko,
Topological groups and related structures,
Atlantis Studies in Mathematics, , Atlantis Press, Paris; World ScientificPublishing, Hackensack, NJ, 2008.[3] F. Bien, A. Borel, Sous-groupes ´epimorphiques de groupes alg´ebriques lin´eairesI,II,
C. R. Acad. Sci. Paris S´er. I, (1992), 649–653, 1341–1346.[4] M. Brion,
Epimorphic subgroups of algebraic groups,
Math. Res. Lett. (2017),no. 6, 1649–1665.[5] N. Bourbaki, Espaces vectoriels topologiques,
Springer, 2007.[6] N. Bourbaki,
Lie groups and Lie algebras. Chapters 1–3 , Springer, 1989.[7] N. Bourbaki,
Lie groups and Lie algebras. Chapters 7-9 , Springer, 2005.[8] H. Gl¨ockner,
Solutions to open problems in Neeb’s recent survey on infinite-dimensional Lie groups,
Geom. Dedicata (2008), 71-86.[9] R. Engelking,
General topology,
Second edition. Sigma Series in Pure Mathe-matics, , Heldermann Verlag, Berlin, 1989.[10] R.S. Hamilton, The inverse function theorem of Nash and Moser,
Bull. Amer.Math. Soc (1982), No. 1, 65–222.4 Pestov and Uspenskij [11] S. Kobayashi, K. Nomizu,
Foundations of Differential Geometry. Vol. 1 , Inter-science Publishers, 1963.[12] M.G. Megrelishvili,
Free topological G-groups,
New Zealand J. Math. (1996),59–72.[13] J. Milnor, Remarks on infinite-dimensional Lie groups, in book: Relativit´e,groupes et topologie II. North-Holland, 1984. Pp. 1007–1057.[14] K.-H. Neeb,
Towards a Lie theory of locally convex groups,
Jpn. J. Math. (2006), 291–468.[15] K.-H. Neeb, On differentiable vectors for representations of infinite dimensionalLie groups,
J. Funct. Anal. (2010), 2814–2855.[16] E.C. Nummela,
On epimorphisms of topological groups,
General Topology Appl. (1978), 155–167.[17] H. Omori, Infinite-Dimensional Lie Groups,
Translations of MathematicalMonographs, , American Mathematical Society, 1997.[18] V. Pestov,
Epimorphisms of Hausdorff groups by way of topological dynamics,
New Zealand J. Math. (1997), 257–262.[19] V. Pestov, Topological groups: Where to from here? , Topology Proceedings (1999), 421–502; arXiv:math/9910144.[20] V. Uspenskij, The solution of the epimorphism problem for Hausdorff topologicalgroups,
Seminar Sophus Lie (1993), 69–70.[21] V. Uspenskij, The epimorphism problem for Hausdorff topological groups,
Topol-ogy Appl. (1994), 287–294.[22] V. Uspenskij, Epimorphisms of topological groups and Z -sets in the Hilbert cube, Symposia Gaussiana. Proceedings of the 2nd Gauss Symposium (Munich, 1993).Conference A: Mathematics and Theoretical Physics (edited by M. Behara, R.Fritsch, R.G. Lintz). Walter de Gruyter & Co., Berlin – New York, 1995. Pp.733-738.[23] C. Villani,
Optimal transport. Old and new,
Grundlehren der MathematischenWissenschaften , Springer-Verlag, Berlin, 2009.
Vladimir G. PestovDepartamento de Matem´aticaUniversidade Federal de Santa CatarinaTrindadeFlorian´opolis, SC, 88.040-900Brazil andand