JORDAN PROPERTY FOR LIE GROUPS AND AUTOMORPHISM GROUPS OF COMPLEX SPACES

VLADIMIR L. POPOV^{∗}

Abstract. We prove that all connected finite-dimensional real Lie groups are Jordan. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifolds are Jordan.

1. Introduction. We recall the definition introduced in [Po 2011, Def. 2.1]:

Definition 1. Given a group G, put

J_{G} := sup_{F} min_{A} [F : A],

where F runs over all finite subgroups of G and A runs over all normal abelian subgroups of F. If J_{G} ≠ ∞, then G is called a Jordan group and J_{G} is called the Jordan constant of G. In this case, we also say that G enjoys the Jordan property.

Informally, the Jordan property of G means that all finite subgroups of G are “almost abelian” in the sense that they are extensions of abelian groups by groups taken from only a finite list. Definition 1 is inspired by the classical theorem of Jordan [Jo 1878] claiming that J_{GL_{n}(ℓ)} ≠ ∞ holds for every n and every field ℓ of characteristic zero. If ℓ is algebraically closed, then, for every fixed n, the constant J_{GL_{n}(ℓ)} is independent of ℓ, so we denote it simply by J(n). It has been computed in [Co2007]; in particular,

J(n) = (n + 1)! for all n ⩾ 71 and n = 63, 65, 67, 69.

For more examples of Jordan groups see [Po 2014_{1}].

Below variety means algebraic variety over a fixed algebraically closed field k of characteristic zero; in particular, any algebraic group is defined over k. If G is either an algebraic group or a topological group, G^{0} denotes the identity component of G.

After being posed seven years ago in [Po 2011, Sect. 2] (see also [Po 2014_{1}, Sect. 2]), the following problem was explored by a number of researchers (see the most recent brief survey and references in [PS 2017, Sect. 1]):

Problem. Describe varieties X for which the group Aut(X) is Jordan.

∗ This work is supported by the RSF under a grant 14-50-00005.

At present (April 2018) it is still unknown whether there are varieties X such that the group Aut(X) is non-Jordan (note that complex manifolds whose automorphism groups are non-Jordan do exist, see below Remark 2). On the other hand, by now for many types of varieties X it is shown that the group Aut(X) is Jordan. In particular, S. Meng and D.-Q. Zhang recently proved the following

Theorem 1 ([MZ 2015, Thm. 1.6]). For every projective variety X, the group Aut(X) is Jordan.

Given a variety X, we denote by Aut(X)^{0} the identity component of Aut(X) in the sense of [Ra 1964]; see also [Po 2014_{2}]. By [Ra 1964, Cor. 1], if X is complete, then Aut(X)^{0} is a connected (not necessarily affine) algebraic group. Jordan’s theorem cited above implies the claim that every affine algebraic group is Jordan; see [Po 2014_{1}, Thm. 2]. The key ingredient of the proof of Theorem 1 given in [MZ 2015] is the proof that the extension of this claim to all (i.e., not necessarily affine) algebraic groups holds true. The latter proof is rather involved.

In the present note we obtain, with a very short proof, a general result, from which the above-mentioned extension immediately follows (see Theorem 4 below). Namely, we prove that every finite-dimensional connected real Lie group is Jordan (the more precise and general statements are formulated in Theorems 2, 3, and Corollary 3 below). Then in Sections 5–7 we apply this to showing that some transformation groups of complex spaces and Riemannian manifolds are Jordan (see Theorems 5, 7, 8, 9 below).

The question of whether the Lie groups are Jordan was posed to me by A. M. Vershik (see [Po 2014_{3}, 95:20]) whom I thank. I am grateful to Yu. G. Zarhin for the valuable comments.

2. Lie groups. We now explore the Jordan property for finite-dimensional real Lie groups G. Note that non-Jordan groups of this type do exist, because every discrete group is a 0-dimensional real Lie group and there are non-Jordan discrete groups (see [Po 2014_{1}, 1.2.5]). Therefore, the Jordan property of G can be expected only under some constraint on the component group G/G^{0}.

To formulate this restriction we recall the following definition introduced in [Po 2011, Def. 2.9]:

Definition 2. Given a group H, put

b_{H} := sup_{F} |F |,

where F runs over all finite subgroups of H. If b_{H} ≠ ∞, then the group H is called bounded.

In particular, every finite group H is bounded and b_{H} = |H|.

In Theorems 2, 3 and Corollary 4 below, we consider the class of finitedimensional real Lie groups G whose component group G/G^{0} is bounded.

Note that every compact Lie group K belongs to this class, because K/K^{0} is finite.

Theorem 2. Let G be a finite-dimensional real Lie group whose component group G/G^{0} is bounded. Then G is Jordan.

Proof. By [Po 2011, Lem. 2.11] (or [Po 2014_{1}, Thm. 5]), we may (and shall) assume that G is connected. This assumption implies the existence of a compact Lie subgroup K of G such that every compact subgroup of G is conjugate to that of K (see, e.g., [Ho 1965, Chap. XV, Thm. 3.1(iii)]). In particular, every finite subgroup of G is conjugate to that of K. This and Definition 1 show that G is Jordan if and only if K is, and if they are, then

J_{G} = J_{K}.  (1)

Being compact, the group K admits a faithful finite-dimensional representation, i.e., is isomorphic to a subgroup of GL_{m}(ℝ) for some m (see, e.g., [OV 1990, Chap. 5, §2, Thm. 10]). Since the latter group is Jordan, K is Jordan as well (see [Po 2014_{1}, Thm. 3(i)]. This completes the proof. □

Corollary 1. For every finite-dimensional real Lie group G whose component group G/G^{0} is bounded, the set of isomorphism classes of all finite simple subgroups of G is finite.

We now dwell on estimating the Jordan constants of Lie groups whose component group is finite, with a view of proving that the class of such groups enjoys a property stronger than that of all its members to be Jordan (see Corollary 3 below). Seeking only this goal, we did not seek to improve the estimates obtained.

Lemma 1. Let S be a simply connected simple affine algebraic group. Then the minimum rdim S of dimensions of faithful linear algebraic representations of S is given by the following table:

type of S A_{ℓ}_{ℓ⩾1} B_{ℓ}_{ℓ⩾2} C_{ℓ}_{ℓ⩾2} D_{ℓ}_{ℓ⩾3, ℓ odd} D_{ℓ}_{ℓ⩾4, ℓ even} E_{6} E_{7} E_{8} F_{4} G_{2} ; rdim S ℓ + 1 2^{ℓ} 2ℓ  2^{ℓ−1} 2ℓ + 2^{ℓ−1} 27 56 248 26 7

Remark 1. In the proof of Lemma 1 below, a faithful representation of S of dimension rdim S is explicitly specified for each type of S.

Proof of Lemma 1. By Lefschetz’s principle (see, e.g., [Si 1968, VI.6]), we may (and shall) assume that k is ℂ. We fix a maximal torus T of S. Let α_{1}, …, α_{ℓ} ∈ (Lie T)^{∗}, ϖ_{1}, …, ϖ_{ℓ} ∈ (Lie T)^{∗}, and α_{1}^{∨}, …, α_{ℓ}^{∨} ∈ Lie T be respectively the system of simple roots, fundamental weights, and simple coroots of Lie T with respect to a fixed Borel subalgebra of Lie S containing Lie T; we number them as in [OV 1990].

The center Z of S is a finite subgroup of T. Fix a subset Z̃ of Lie T whose image under the exponential map Lie T → T is the set of all nonidentity elements of Z.

For every dominant weight λ ∈ (Lie T)^{∗}, let R(λ) be an irreducible representation of Lie S with the highest weight λ. The dimension of R(ϖ_{i}) for every i is specified in [OV 1990, Ref. Chap., §2, Table 5, pp. 299–305]. Note that Weyl’s dimension formula implies

dim R( ∑_{i=1}^{ℓ} λ_{i}ϖ_{i}) ⩾ dim R( ∑_{i=1}^{ℓ} μ_{i}ϖ_{i}) if λ_{i} ⩾ μ_{i} for every i. (2)

Since S is simply connected, R(λ) is the differential of a finite-dimensional linear algebraic representation R(λ) of S. Since S is simple, for every finite set D of nonzero dominant weights and R(D) := ⨁_{λ∈D} R(λ), we have ker R(D) ⊆ Z. Hence

R(D) is faithful ⟺ for every x ∈ Z̃ there is λ ∈ D with λ(x) ∉ ℤ. (3)

As is well known, dim R(ϖ_{1}) is the minimum of dimensions of nonzero finite-dimensional algebraic representations of S (see [OV 1990, pp. 299– 305]).

If S is of type E_{8}, F_{4}, or G_{2}, then Z is trivial; hence in this case R(ϖ_{1}) is faithful and therefore we have the equality

rdim S = dim R(ϖ_{1}),  (4)

which proves the claim of Lemma 1 for these types.

If S is of type A_{ℓ} or C_{ℓ}, then S is respectively SL_{ℓ+1} and Sp_{2ℓ}. Since for these groups R(ϖ_{1}) is the tautological faithful representation, in this case (4) holds as well, which proves the claim of Lemma 1 for these types.

For the other types, we apply (3) to the set Z̃ taken from [OV 1990, Ref. Chap., §2, Table 3, p. 298]. Below is used that for any λ_{i}, μ_{i} ∈ k,

the value of ∑_{i=1}^{ℓ} λ_{i}ϖ_{i} ∈ (Lie T)^{∗} in ∑_{i=1}^{ℓ} μ_{i}α_{i}^{∨} ∈ Lie T is ∑_{i=1}^{ℓ} λ_{i}μ_{i}. (5)

If S is of type E_{7}, then Z̃ consists of only one element ζ := (α_{1}^{∨} + α_{3}^{∨} + α_{7}^{∨})/2. By (5), we have ϖ_{1}(ζ) = 1/2 ∉ ℤ, so R(ϖ_{1}) is faithful. Therefore, in this case again (4) holds, which proves the claim of Lemma 1 for this type.

If S is of type E_{6}, then Z̃ consists of two elements ζ := (α_{1}^{∨} − α_{2}^{∨} + α_{4}^{∨} − α_{5}^{∨})/3 and 2ζ. Since ϖ_{1}(z) = 1/3 ∉ ℤ, ϖ_{1}(2z) = 2/3 ∉ ℤ, in this case again R(ϖ_{1}) is faithful; whence (4) holds. This proves the claim of Lemma 1 for this type.

If S is of type B_{ℓ}, then Z̃ consists of only one element α_{ℓ}^{∨}/2. This and (3),

(5) imply that R(D) is faithful if and only if D contains ∑_{i=1}^{ℓ} λ_{i}ϖ_{i} with odd λ_{ℓ}. Using (2), from this we infer that R(ϖ_{ℓ}) is the faithful representation of minimal dimension. Hence rdim S = dim R(ϖ_{ℓ}). This proves the claim of Lemma 1 for this type.

If S is of type D_{ℓ}, ℓ ⩾ 3, ℓ odd, then Z̃ consists of three elements

ζ := (α_{1}^{∨} + α_{3}^{∨} + ⋯ + α_{ℓ−2}^{∨})/2 + (α_{ℓ−1}^{∨} − α_{ℓ}^{∨})/4, 2ζ, 3ζ.  (6)

¿From (3), (5), (6) we infer that R(D) is faithful if and only if D contains ∑_{i=1}^{ℓ} λ_{i}ϖ_{i} such that 4 is coprime to either λ_{ℓ−1} or λ_{ℓ}. This and (2) show that R(ϖ_{ℓ}) is the faithful representation of minimal dimension. Hence rdim S = dim R(ϖ_{ℓ}), proving the claim of Lemma 1 for this type.

If S is of type D_{ℓ}, ℓ ⩾ 4, ℓ even, then Z̃ consists of three elements

ζ_{1} := (α_{1}^{∨} + α_{3}^{∨} + ⋯ + α_{ℓ−1}^{∨})/2, ζ_{2} := (α_{ℓ−1}^{∨} + α_{ℓ}^{∨})/2, ζ_{1} + ζ_{2}. (7)

Hence if R(D) is faithful, then D contains ∑_{ı=1}^{ℓ} λ_{i}ϖ_{i} with odd λ_{ℓ} or λ_{ℓ−1} and ∑_{ı=1}^{ℓ} μ_{i}ϖ_{i} with odd μ_{i} for some odd i ≠ ℓ−1. On the other hand, since in this case Z is not cyclic, Schur’s lemma implies that |D| ⩾ 2. ¿From this it is not difficult to deduce that R(ϖ_{1})⊕R(ϖ_{ℓ}) is the faithful representation of minimal dimension. Hence rdim S = dim R(ϖ_{1}) + dim R(ϖ_{ℓ}) = 2ℓ + 2^{ℓ−1}. This completes the proof of Lemma 1. □

Corollary 2. Every simply connected simple affine algebraic group of rank ℓ admits a faithful linear algebraic representation of dimension at most 2^{ℓ}+10.

Proof. Clearly if an algebraic group admits a faithful linear algebraic representation, then it admits a faithful linear algebraic representation of any bigger dimension. In view of this, the claim follows from the inequality rdim S ⩽ 2^{ℓ} + 10, which, in turn, follows from Lemma 1: indeed, the latter shows that rdim S ⩽ 2^{ℓ} if the type of S differs from F_{4} and G_{2}, and that rdim S = 2^{ℓ} + 10 and 2^{ℓ} + 3 respectively for the types F_{4} and G_{2}. □

Theorem 3. Let G be an n-dimensional real Lie group whose component group G/G^{0} is bounded. Then

J_{G} ⩽ b_{G/G^{0}}J(n(2^{n} + 10))^{b_{G/G^{0}}} .  (8)

Proof. By [Po 2011, Lem. 2.11] (or [Po 2014_{1}, Thm. 5]), we may (and shall) assume that G is connected; in particular,

b_{G/G^{0}} = 1.  (9)

We use the notation of the proof of Theorem 2. Since G is connected, K is connected, too; see [Ho 1965, Chap. XV, Thm. 3.1(ii)]. Hence (see [Bo1982, §1, Prop. 4]) there are

(i) the compact simply connected simple Lie groups K_{1}, …, K_{d}; (ii) a compact torus S; (iii) a group epimorphism with finite kernel

π: K̃ := K_{1} × ⋯ × K_{d} × S → K.  (10)

By [Po 2014_{1}, Thm. 3(ii)], from (iii) we infer that

J_{K} ⩽ JK̃.  (11)

Every K_{i} is a real form of the corresponding simply connected simple complex affine algebraic group. The rank ℓ_{i} of the latter is equal to that of K_{i}. By Corollary 2 we then conclude that K_{i} admits an embedding in GL_{2^{ℓ_{i}}+10}(ℂ). Since ℓ_{i} ⩽ dim K̃ = dim K ⩽ n, this in turn implies that K_{i} admits an embedding in GL_{2^{n}+10}(ℂ). Clearly, S admits an embedding in GL_{dim S}(ℂ), and therefore, in view of dim S ⩽ dim K̃, also in GL_{2^{n}+10}(ℂ).

This and the definition of K̃ (see (10)) show that K̃ admits an embedding in the direct product of d+1 copies of GL_{2^{n}+10}(ℂ), hence in GL_{(d+1)(2^{n}+10)}(ℂ). In turn, since, in view of (10), we have d + 1 ⩽ dim K̃, from this we infer that K̃ admits an embedding in GL_{n(2^{n}+10)}(ℂ); whence,

JK̃ ⩽ J(n(2^{n} + 10)).  (12)

Now, putting (1), (11), (12), (9) together, we complete the proof. □

Recall from [MZ 2015] the following

Definition 3. A family F of groups is called uniformly Jordan if every group in F is Jordan and there is an integer J_{F} such that J_{G} ⩽ J_{F} for every G ∈ F.

Corollary 3. Fix an integer n ⩾ 0. Let L_{n} be the family of all connected n-dimensional real Lie groups. Then

(i) the family L_{n} is uniformly Jordan;

(ii) one can take J_{L_{n}} = J(n(2^{n} + 10)).

Proof. This follows from (8) because b_{G/G^{0}} = 1 for every G ∈ L_{n}. □

Corollary 4. For every integer n ⩾ 0, the set of isomorphism classes of finite simple groups embeddable in n-dimensional connected real Lie groups is finite.

3. Algebraic groups. We now consider several applications of Theorems 2 and 3. First, we apply them to algebraic groups, answering Question 1.2 in [MZ 2015]:

Theorem 4. Every (not necessarily affine) n-dimensional algebraic group G over an algebraically closed field k of characteristic 0 is Jordan. Moreover,

J_{G} ⩽ [G : G^{0}]J(n(2^{2n+1} + 20))^{[G:G^{0}]}.  (13)

Proof. In this case, G/G^{0} is finite. By Lefschetz’s principle, we may (and shall) assume that k is ℂ. Then G has a structure of 2n-dimensional real Lie group whose identity component is G^{0}. The claim then follows from Theorem 3. □

Statement (i) of the next corollary is one of the main results of [MZ 2015]:

Corollary 5. Fix an integer n ⩾ 0. Let A_{n} be the family of all (not necessarily affine) connected n-dimensional algebraic groups over an algebraically closed field k of characteristic 0. Then

(i) ([MZ 2015, Thm. 1.3]) the family A_{n} is uniformly Jordan; (ii) one can take J_{A_{n}} = J(n(2^{2n+1} + 20)).

Proof. This follows from (13). □

4. Automorphism groups of complex spaces. The next application is to automorphism groups of complex spaces.

Let C be a (not necessarily reduced) complex space. There exists a topology on Aut(C) with respect to which Aut(C) is a topological group (see [Ak 1995, 2.1]).

Theorem 5. For every compact complex space C, the group Aut(C)^{0} is Jordan.

Proof. By [Ka 1965], the compactness of C implies that Aut(C) is a complex Lie group. The claim then follows from Theorem 2. □

We do not know whether the statement of Theorem 5 remains true if Aut(C)^{0} is replaced by Aut(C). By [PS 2017, Thm. 1.5], the answer is affirmative if C is a connected compact two-dimensional complex manifold. By Theorem 1, it is also affirmative if C is a projective variety. On the other hand, we recall that by [CPS 2014] there are connected smooth compact real manifolds whose diffeomorphism groups are non-Jordan (this disproves Ghys’ conjecture).

Remark 2. There are connected noncompact complex manifolds, whose automorphism groups are non-Jordan. Indeed, by [Wi 2002], for any countable group Γ, there is a noncompact Riemann surface M such that Aut(M) is isomorphic to Γ; whence the claim because of the existence of countable non-Jordan groups (see [Po 2014_{1}, Sect. 1.2.5]).

In actual fact, using the idea exploited earlier in [Po 2015], one can prove more than is said in Remark 2, showing the existence of connected complex manifolds with monstrous automorphism groups, namely:

Theorem 6. There is a 3-dimensional simply connected noncompact complex manifold M such that

(i) the group Aut(M) contains an isomorphic copy of every finitely presentable (in particular, every finite) group;

(ii) every such copy is a discrete transformation group of M acting freely.

Proof. It follows (see, e.g., [Ro 1995, Thm. 12.29]) from Higman’s embedding theorem [Hi 1961] that there is a universal finitely presented group, i.e., a finitely presented group U containing as a subgroup an isomorphic copy of every finitely presented group. In turn, by [ABCKT 1996, Cor. 1.66] the finite presentability of U implies the existence of a connected 3-dimensional compact complex manifold B whose fundamental group is isomorphic to U. Consider the universal cover π: B̃ → B. Then B̃ is a simply connected noncompact 3-dimensional complex manifold and the deck transformation group of π is a subgroup of Aut B̃ isomorphic to U, which acts on B̃ freely. Hence one can take M = B̃. □

Remark 3. For M from Theorem 6, the group Aut(M) is non-Jordan, because for every integer n, there is a finite simple group of order > n (cf. [Po 2014_{1}, Example 4].

5. Automorphism groups of hyperbolic complex manifolds. The next application is to complex manifolds hyperbolic in the sense of Kobayashi (in particular, to bounded domains in ℂ^{n}).

Theorem 7. Fix an integer n ⩾ 0. Let H_{n} be the family of groups Aut(M)^{0}, where M runs over all connected complex manifolds hyperbolic in the sense of Kobayashi and of complex dimension n. Then

(i) the family H_{n} is uniformly Jordan; (ii) one can take J_{H_{n}} = J((2n + n^{2})(2^{2n+n^{2}} + 10)); (iii) for every point x ∈ M, the Aut(M)-stablizer Aut(M)_{x} of x is Jordan

and J_{Aut(M)_{x}} ⩽ J(n).

Proof. Let M be a connected complex manifolds hyperbolic in the sense of Kobayashi and of complex dimension n. By [Ko 2005, Thms. 2.1, 2.6], Aut(M) is a real Lie group of dimension ⩽ 2n + n^{2}; whence (i) and (ii) by Theorems 2 and 3. By [Ko 2005, Thm. 2.6], the isotropy representation of Aut(M)_{x} is faithful and its image is isomorphic to a subgroup of the unitary group U(n); whence (iii). □

Remark 4. The group Aut(M)^{0} in the formulation of Theorem 7 cannot be replaced by Aut(M). Indeed, it follows from the construction in [Wi 2002] that the Riemann surface M in Remark 2 is hyperbolic in the sense of Kobayashi. Therefore there are connected hyperbolic complex manifolds M such that the group Aut(M) is not Jordan.

However, as the next theorem shows, for complex hyperbolic manifolds M of a special type, the Jordan property holds for the whole Aut(M) rather than only for Aut(M)^{0}.

Theorem 8. For every strongly pseudoconvex bounded domain M with smooth boundary in ℂ^{n}, the group Aut(M) of all biholomorphic transformations of M is Jordan.

Proof. If the Lie group Aut(M) is compact, then the claim follows from Theorem 2. If the group Aut(M) is non-compact, then, by the Rosey–Wong theorem [Ro 1979], [Wo 1977], the domain M is biholomorphic to the unit ball B_{n} in ℂ^{n}. Since Aut(B_{n}) is PU(n, 1) (see [Ak 1995, Sect. 2.7, Prop. 3]), and the latter Lie group is connected (see [He 1962, Chap. IX, Lem. 4.4]), the claim then follows from Theorem 2. □

Corollary 6. For every strongly pseudoconvex bounded domain M with smooth boundary in ℂ^{n}, the set of isomorphism classes of all finite simple groups of biholomorphic transformations of M is finite.

6. Isometry groups of Riemannian manifolds. The last application is to isometry groups Iso(M) of Riemannian manifolds M. They are topological groups with respect to the compact-open topology [Ko 1995].

Theorem 9. Fix an integer n ⩾ 0. Let R_{n} be the family of groups Iso(M)^{0}, where M runs over all connected n-dimensional Riemannian manifolds. Then

(i) the family R_{n} is uniformly Jordan; (ii) one can take J_{R_{n}} = J((n^{2} + n)(2^{(n^{2}+n−2)/2} + 5)); (iii) for every point x ∈ M, the Iso(M)-stabilizer Iso(M)_{x} of x is Jordan; (iv) if the manifold M is compact, then the group Iso(M) is Jordan.

Proof. It is known (see, e.g., [Ko 1995, Chap. II, Thms. 1.2 and 3.1]) that Iso(M) is a real Lie group of dimension at most n(n + 1)/2, the group Iso(M)_{x} is compact for every x, and the group Iso(M) is compact if the manifold M is compact. The claims then follows from combining these facts with Theorems 2 and 3. □

Remark 5. The group Aut(M)^{0} in the formulation of Theorem 9 cannot be replaced by Aut(M). Indeed, it follows from the construction in [Wi 2002] that the Riemann surface M in Remark 2 is a two-dimensional Riemannian manifold and Aut(M) = Iso(M). Therefore there are connected Riemannian manifolds M such that the group Iso(M) is not Jordan.

7. Concluding remarks. In view of (1), computing the Jordan constants of connected real Lie groups is reduced to that of compact such groups. For instance, the results of [Co2007] may be interpreted as computing the Jordan constants of all unitary groups:

J_{U_{n}} = J(n) for every n.

This leads to the following natural

Problem. Compute the Jordan constants of all simple compact connected real Lie groups.

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Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow, 119991, Russia

E-mail address: popovvl@mi.ras.ru