Embeddings of Automorphism Groups of Free Groups into Automorphism Groups of Affine Algebraic Varieties

For every integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>0$$\end{document}, we construct a new infinite series of rational affine algebraic varieties such that their automorphism groups contain the automorphism group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Aut}(F_n)$$\end{document} of the free group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_n$$\end{document} of rank \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} and the braid group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_n$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} strands. The automorphism groups of such varieties are nonlinear for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geq 3$$\end{document} and are nonamenable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\geq 2$$\end{document}. As an application, we prove that every Cremona group of rank \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\geq}\,3n-1$$\end{document} contains the groups \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Aut}(F_n)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_n$$\end{document}. This bound is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1$$\end{document} better than the bound published earlier by the author; with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_n$$\end{document}, the order of its growth rate is one less than that of the bound following from a paper by D. Krammer. The construction is based on triples \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G,R,n)$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} is a connected semisimple algebraic group and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R$$\end{document} is a closed subgroup of its maximal torus.


Introduction
The trend of the last decade has been the study of abstract-algebraic, topological, algebro-geometric, and dynamical properties of automorphism groups of algebraic varieties.This paper is related to this topic and continues the research started in author's paper [11].
In [11], an infinite series of irreducible algebraic varieties is constructed in whose automorphism group embeds the automorphism group of a free group F n of rank n.This has applications to the problems of linearity and amenability of automorphism groups of algebraic varieties and that of the embeddability of various groups into Cremona groups.To formulate the results obtained in this paper, we recall the construction introduced in [11].
Consider a connected algebraic group G. Denote X is the group variety of the algebraic group We fix in F n a free system of generators f 1 , . . ., f n .For any w ∈ F n and x = (g 1 , . . ., g n ) ∈ X, g j ∈ G for all j, denote by w(x) the element of G obtained from the word w in f 1 , . . ., f n by replacing f j with g j for each j.For each σ ∈ Aut(F n ), the mapping σ X : X → X, x → (σ(f 1 )(x), . . ., σ(f n )(x)). ( is an automorphism of the algebraic variety X (but not, in general, of the group G n ).The mapping σ → (σ −1 ) X is a group homomorphism Aut(F n ) → Aut(X).It defines an action of the group Aut(F n ) by automorphisms of the variety X commuting with the diagonal action G on X by conjugation.Let us assume that for the restriction of this action to a closed subgroup R of the group G, there is a categorical quotient π X/ /R : X → X/ /R (for example, this property holds if R is finite, see [15,Prop. 19; p. 50, Expl. 2)], or if G is affine and R is reductive, see [12, 4.4]).Then it follows from the definition of categorical quotient (see [12,Def. 4.5]) that σ X descends to a uniquely defined automorphism σ X/ /R of X/ /R, which has the property In this case, there arises a group homomorphism defining the action of the group Aut(F n ) by automorphisms of the variety X/ /R.For some (but not all) G and R, homomorphism ( 5) is an embedding.Namely, in [11] it is proved that (a) in the following cases, homomorphism ( 5) is an embedding: • G is nonsolvable and R is finite; • G is reductive, R = G, n = 1 and G contains a connected simple normal subgroup of one of the following types: (b) in the following cases, homomorphism (5) is not an embedding: • G is solvable, R is finite and n 3, • G is reductive, R = G and either n 2, or n = 1 and G does not contain a connected simple normal subgroup of either of types (6).This leads to the following general question: Question.Is it possible to classify the triples (G, R, n), where G is a connected reductive algebraic group, R is its closed subgroup, and n is a positive integer, for which homomorphism ( 5) is an embedding?
The main result of the present paper is Theorem 1.1, in which the next step after [11] is taken towards answering the question posed: we add one more class to the triples of the specified type found in [11]: Theorem 1.1.Let G be a connected semisimple algebraic group and let R be a closed subgroup of its maximal torus.Then homomorphism (5) is an embedding.
As an application we obtain the following Theorem 1.2, in which B n denotes the braid group on n strands.Theorem 1.2.We keep the assumptions of Theorem 1.1.Then X/ /R is an affine algebraic variety such that We also explore the rationality problem: Theorem 1.3.The variety X/ /R from Theorems 1.1, 1.2 is rational.
As an application we strengthen by 1 the bounds obtained in [11, Cor. 9 and Rem.10]: Theorem 1.4.For any integer n 1, the Cremona group of rank 3n−1 contains the group Aut(F n ) and, for n 3, the braid group B n .
Remark.As is proved in [4] by D. Krammer, the braid group B n embeds into GL n(n−1)/2 .Hence B n embeds into the Cremona group of rank n(n − 1)/2.The order of the growth rate of this bound for the minimal rank of the Cremona group containing B n is one bigger than that of the bound from Theorem 1.4.
The proofs of Theorems 1.1-1.4 are given in Section 6.

Conventions and notation
In what follows, algebraic varieties are considered over an algebraically closed field k.We use the results of paper [6] and statement [12,Prop. 3.4] obtained under the condition char(k) = 0. Therefore, we also assume that this condition holds.With respect to algebraic geometry and algebraic groups we follow [1].
The identity element of a group considered in multiplicative notation is denoted by e (it will be clear from the context which group is meant).
The statement that a group A contains a group B means the existence of a group monomorphism B ֒→ A, by which B is identified with its image.
C (A) is the center of a group A. A • m and A m are respectively, the orbit and the stabilizer of a point m with respect to a considered action of a group A (it will be clear from the context which action is meant).
The kernel of an action α : A × M → M of a group A on a set M is the following normal subgroup of A: By homomorphisms of algebraic groups we mean algebraic homomorphisms, and by their actions on algebraic varieties we mean algebraic actions.In particular, for an algebraic group A, we denote by Aut(A) the group of its algebraic automorphisms.
The multiplicative group k × of the field k is considered as the algebraic group G m , and its additive group is considered as G a .

Terminology and some general results
Recall (see [12]) the terminology and results used below, which concern an action of α of an algebraic group H on an irreducible algebraic variety Y .
(a) An action α is said to be stable if there is a nonempty open set in Y , the H-orbits of whose points are closed in Y .
(b) A subgroup H * of H is called the stabilizer in general posisiton (s.g.p.) of the action α if there is a nonempty open set in Y such that for any its point y, the subgroups H y and H * are conjugate in H.
(c) If H is reductive and Y is smooth and affine, then Let β be an action of the group H on an algebraic variety Z such that (k) ker(α) = ker(β), and let ϕ : Z → Y be an H-equivariant morphism such that ϕ −1 (H•y) = ∅.Then for every point z ∈ ϕ −1 (H •y) the following properties hold: Proof.In view of (y 1 ), the nonempty H-invariant subset ϕ −1 (H • y) is closed in Z and hence contains the closure H •z of the orbit H •z.
Assume that (z 1 ) fails, i.e., H The restriction of the morphism ϕ to the orbit H•v is an H-equivariant and therefore a surjective morphism , which together with (7) gives On the other hand, since H z ⊇ ker(β), from (y 2 ) and (k) it follows that dim(H •y) dim(H •z).This contradicts (8) and proves (z 1 ).The restriction of the morphism ϕ to the orbit H •z is an H-equivariant, and therefore a surjective morphism of H•z → H•y.Hence, there exists h ∈ H for which ϕ(h • z) = y.Therefore, ker(α) (e 1 ) For every H-equivariant mapping ϕ : M → M there is an element h ∈ H such that (e 2 ) For every element h ∈ H, the map ϕ : M → M defined by formula ( 9) is H-equivariant.
Proof.(e 1 ) Fix a point m 0 ∈ M. Since the action is transitive, for any point m ∈ M there is an element This follows directly from (9) in view of the commutativity of H.

Reduction
The proof of Theorem 1.1 is based on the following geometric description of the kernel of homomorphism (5): Proof.In this case, the variety X is affine, which implies (see [9, §2 and Append.1B]) that the morphism π is surjective, its fibers are Rinvariant, and for each point b ∈ X/ /R, the fiber π −1 (b) contains a unique closed R-orbit O b .It follows from (4) that the restriction of the morphism σ X to the fiber π −1 (b) is its R-equivariant isomorphism with the fiber π −1 (σ X/ /R (b)).In view of the uniqueness of closed orbits in the fibers, this means that σ Under the conditions of Lemma 4.1, the algebra k[X] R of all Rinvariant elements of the algebra k[X] of regular functions on X is finitely generated, X/ /R is the affine algebraic variety with the algebra of regular functions k[X/ /R] = k[X] R , and the comorphism corresponding to morphism (3) is the identity embedding k[X] R ֒→ k[X].This implies that for any reductive closed subgroup S of G containing R, the identity embedding k[X] S ֒→ k[X] R determines a dominant morphism X/ /R → X/ /S.This morphism is Aut(F n )-equivariant.Therefore, the kernel of the action of the group Aut(F n ) on X/ /R lies in the kernel of its action on X/ /S.
In the situation considered in Theorem 1.1, this gives the following.By the assumption, in it, R is a subgroup of some maximal torus T of the group G. Therefore, it follows from what has been said that it suffices to prove Theorem 1.1 for In what follows, we assume that the group G satisfies the conditions of Theorem 1.1, i.e., is connected and semisimple.Note that the kernel of the action of the group T on X is C (G), since C (G) ⊂ T (see [1,13.17,Cor.2(d)]).

Principal Luna stratum for action of T on X
The variety X is smooth, and the group T is reductive.Therefore, the diagonal action of the torus T on X by conjugation determines the Luna stratifications of the varieties X and X/ /T .In what follows, X pr denotes the principal stratum of this stratification of the variety X. • It suffices for us to prove that (i) the action of the torus T on X under consideration is stable; (ii) C (G) is its stabilizer in general position, or, in other words, that there is a nonempty open subset of X, for all points x of which property (b 2 ) holds.Indeed, suppose this subset exists.Due to its openness, its intersection with the open set X pr is nonempty.Let x be a point of this intersection.Since C (G) is the kernel of the action of the group T on X, from condition (b 2 ) it follows that the normal bundle of the orbit T • x is equivariantly isomorphic to V. From this and from the definition of the Luna strata it follows that a closed T -orbit from X lies in X pr if and only if its normal bundle is equivariantly isomorphic to V. In particular, the dimension of this orbit is dim(T ).It remains to note that the T -orbit of any point y ∈ X pr is closed.Indeed, if this were not the case, then the unique closed T -orbit in the fiber π −1 X/ /T (π X/ /T (y)) ⊆ X pr lying in its closure had dimension strictly less than dim(T •y) dim(T ), which contradicts the dim(T )dimensionality of this closed orbit.
• Let us now prove that properties (i) and (ii) indeed hold.It suffices to prove them for n = 1.Indeed, suppose that for n = 1 they are proved, i.e., there is a nonempty open subset of G such that T -orbit of every its point x is closed in G and T x = C (G).Then, as explained above, G pr is the set of all such points x.Let π i : X = G n → G be the natural projection onto the ith factor.Applying Lemma 3.1 to it, we infer that property (b 2 ) holds for each point x of a nonempty set π −1 i (G pr ), which means that properties (i) and (ii) hold.
• It remains to prove that (i) and (ii) hold for n = 1.In [16, 6.11], it is proved that the action of G on itself by conjugation is stable and its s.g.p. is T .From [5,Thm. and Sect. 3] and the reductivity of T , it follows that the natural action of T on G/T is stable.These two facts imply, according to [10,Prop. 6], that (i) holds for n = 1.
• Let Φ be the root system of the group G with respect to the torus T in which subsystems of positive and negative roots with respect to some base in Φ are fixed.For any α ∈ Φ, there is an embedding of algebraic groups ε α : G a ֒→ G, such that (see [3, 26.3.Thm.], [16, 2.1]).Consider in G the "big cell" Θ (see [3, 28.5 Prop.]), i.e., the set of all elements of the form where the factors in the products are taken with respect to some fixed orders on the sets of positive and negative roots.The set Θ is open in G and each of its elements can be uniquely written as (12) (see [1, 14.5.Prop.(2), 14.14.Cor.], [16, 2.2, 2.3]).In view of (11), it is Tinvariant.The set Θ 0 of all elements of the form (12) with x α = 0 for each α ∈ Φ has the same properties.Let a ∈ Θ 0 and c ∈ T .It follows from (11) and the indicated uniqueness that the condition c ∈ T a is equivalent to the condition c ∈ ker(α) for all α ∈ Φ.
In turn, it follows from ( 11), (12) and the openness of Θ 0 that ( 13) is equivalent to the property that c belongs to the kernel of the action of T on G, i.e., ( 13) is equivalent to the inclusion c ∈ C (G).This proves that T a = C (G).Hence, (ii) holds for n = 1.This completes the proof of (b).
(c) This follows from (b), since each fiber of the canonical morphism of any Luna stratum in X contains a unique orbit closed in X.
(d) As is explained in the proof of statement (b), the set X pr contains the set n i=1 π −1 i (G pr ), from where we get From ( 14) it follows that dim(X \ X pr ) n(dim(G) − 1) = dim(X) − n.This proves (d).
6. Proofs of Theorems 1.1-1.4 Proof of Theorem 1.1.As is explained in Section 4, we can (and shall) assume that equality (10) holds.Arguing by contradiction, suppose that the kernel of homomorphism ( 5) contains an element σ ∈ Aut(F n ), σ = e.The cases n = 1 and n 2 will be considered separately: in each of them the proof is based on the properties that do not hold in the other.
Case n = 1.The order of Aut(F 1 ) is 2 and σ(f For any element t ∈ T we have T • t = t.In view of Lemma 4.1, this implies that σ X (t) = t.Together with (15) this shows that t 2 = e for any t ∈ T .This conclusion contradicts the fact that the set of orders of elements of the torsion subgroup of any torus of positive dimension is not upper bounded (see [1, 8.9.Prop.]).
• Since the kernel of the considered action of the torus T on X is C (G) (see Theorem 5.1(a)), this action defines a faithful (that is, with trivial kernel) action on X of the torus The orbits of this action of the torus S, and hence the categorical quotient and the Luna stratifications are the same as those of the action of the torus T .Below, instead of the original action of the torus T , we consider the indicated action of the torus S.
• Theorem 5.1(b) and Lemmas 4.1, 3.2(e 1 ) imply the existence of a set-theoretic mapping ψ : (X/ /T ) pr → S such that σ X (x) = ψ(π X/ /T (x)) • x for each point x ∈ X pr . ( Let us prove that the set-theoretic mapping is a morphism of algebraic varieties.According to Theorem 5.1(b) and what was said in part (iii) of Section 3, the canonical morphism X pr → (X/ /T ) pr is an étale trivial bundle with fiber S. Since algebraic tori are special groups in the sense of Serre (see [14,Prop. 14]), this bundle is locally trivial in the Zariski topology.Hence, X pr is covered by Sinvariant open sets for which there are S-equivariant isomorphisms of them with varieties of the form U × S, where U is an open subset of (X/ /T ) pr , and the torus S acts through translations of the second factor.If we identify them by these isomorphisms, then the restriction of mapping (18) to any of these open sets has the form The issue therefore boils down to proving that α is a morphism of algebraic varieties.To this end, note that since σ X is a morphism, then is also a morphism in view of (17).Hence is a morphism as well.Moreover, is a morphism too.It remains to note that α = γ • β.
• Thus, there exists a rational mapping θ : X S, which is defined everywhere on the open set X pr and coincides on it with morphism (18).Since n 2, it follows from Theorem 5.1 The torus S can be identified with the product of several copies of the group k × .Then θ is given by a set of rational functions θ i : X k, which are compositions of the mapping θ with projections of this product onto the factors.Each θ i is regular and does not vanish on X pr .Since X is smooth, it follows from this and (20) that the divisor of θ i on X is zero, that is, θ i is regular and does not vanish on the whole of X.Thus, we have a morphism θ i : X → k × .Since X is the group variety of the connected algebraic group G n , it follows from this and from [13,Thm. 3] that θ i is the product of a character of this group and a constant.But due to semisimplicity, G n has no nontrivial characters.Hence θ i is a constant.This means that there is an element s ∈ S for which θ(X) = s.
• Fix an element t ∈ T that maps to s under the natural surjection T → S (see ( 16)).We have proven that σ Since σ = e, it follows from [11, Thm.2(b 1 )] that σ X = id X .In view of (21) and Theorem 5.1(a), this gives It follows from ( 21), (1), and (2) that for each i ∈ {1, . . ., n} the following group identity holds In particular, for each g ∈ G the equality obtained by substituting g 1 = . . .= g n = g into (23) holds.Since σ(f i ) is a noncommutative Laurent monomial in f 1 , . . .f n , this means that there exists an integer d such that the following group identity holds: Notice that d = 1 and d = −1.
Indeed, in view of (24), if d = 1, then t ∈ C (G) contrary to (22).If d = −1, then for any g, h ∈ G the following equality holds: which means that the group G is commutative and contradicts its semisimplicity.Further, if r is a positive integer, then the following group identity holds: t r gt −r = g d r for each g ∈ G.
is also a T -equivariant, and therefore, an R-equivariant birational morphism.
Since R and V n are respectively a diagonalizable group and an Rmodule, the field k(V ) R is rational over k (see [12,Sect 2.9]).Since Θ n is open in X, this implies that the field k(X) R is also rational over k.But the action of T on X is stable, and C (G) is the s.g.p. for it by Theorem 5.1(b).Since R is reductive, from [5,Thm. and Sect. 3] it follows that the natural action of R on T /C (G) is stable.Hence, according to [10,Prop. 6], the action of R on X is stable.In view of [12,Prop. 3.4], this implies that k(X) R is the field of fractions of the algebra k[X] R = k[X/ /R].This is what the rationality of the variety X/ /R means.
Proof of Theorem 1.4.Let G = SL 2 , so that dim(G) = 3 and dim(T ) = 1.It follows from here and from Theorem 5.1(c) that dim(X/ /T ) = 3n − 1.Hence, in view of the rationality of the variety X/ /T (Theorem 1.3), the group Aut(X/ /T ) embeds into the Cremona group of rank 3n − 1.The claim of the theorem now follows from Theorem 1.2 and the fact that every Cremona group embeds into any Cremona group of a higher rank.

Lemma 3 . 2 .
Let a commutative group H act transitively on a set M.

Lemma 4 . 1 .
Let G be a connected affine algebraic group and let R be its closed reductive subgroup.The following properties of an element σ ∈ Aut(F n ) are equivalent: (a) σ lies in the kernel of homomorphism (5); (b) σ X (O) = O for every closed R-orbit O in X.

Theorem 5 . 1 .
Let G be a connected semisimple algebraic group with a maximal torus T acting diagonally by conjugation on the group variety X of the group G n .(a) The kernel of the specified action is C (G).(b) The following properties of a point x ∈ X are equivalent: (b 1 ) x ∈ X pr ; (b 2 ) the orbit T •x is closed in X, and T x = C (G). (c) Each fiber of the canonical morphism of the Luna stratum X pr is a T -orbit equivariantly isomorphic to T /C (G).(d) codim X (X \ X pr ) n. Proof.Statement (a) is obvious.(b) Denote by V the trivial codim X (T )-dimensional vector bundle over T /C (G).Note that dim(T /C (G)) = dim(T ) since the group C (G) is finite (see [1, 14.2.Cor.(a)]).
are endowed with the Luna stratifications defined as follows.The fact that the points a, b ∈ Y / /H belong to the same Luna stratum means that the normal vector bundles to the unique H-orbits closed in the fibers π −1 In view of (i) and (ii), there are (unique) open Luna strata in Y / /H and Y .They are called the principal strata and denoted by (Y / /H) pr and Y pr respectively.Lemma 3.1.We keep the previous notation H, Y , α.Let y ∈ Y be a point such that • The s.g.p. exists.• The varieties Y and Y / /H Y / /H (a) and π −1 Y / /H (b) orbits are Hequivariantly isomorphic.The Luna strata in Y are the sets of the form π −1 Y / /H (L), where L is a Luna stratum in Y / /H.The Luna stratifications have the following properties: (i) the set of all Luna strata is finite; (ii) all Luna strata in the varieties Y / /H and Y are smooth locally closed subvarieties of these varieties; (iii) for any Luna stratum L in Y / /H there exists an affine variety F endowed with an action of H such that the restriction of the morphism π Y / /H to the stratum π −1 Y / /H (L) (called the canonical morphism of the Luna stratum π −1 Y / /H (L)) is an étale trivial bundle π −1 Y / /H (L) → L with fiber F .