The length and depth of compact Lie groups

Let $G$ be a connected Lie group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0>G_1>\cdots>G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. In this paper, we introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups $G$. We obtain best possible bounds on the length of $G$ in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on $\dim G'$ in terms of the chain difference of $G$, which is its length minus its depth.


Introduction
where each G i is a maximal subgroup of G i−1 . The depth of finite solvable groups was studied by Kohler [14], and more generally by Shareshian and Woodroofe [19] in relation to lattice theory. We refer the reader to [3,5] for recent work on the length and depth of finite groups and finite simple groups.
In [4], we extended these notions to algebraic groups. Let G be a connected algebraic group defined over an algebraically closed field of characteristic p 0. The length and depth of G, denoted by l(G) and λ(G), are defined to be the maximal and minimal length of a chain of subgroups as in (1), respectively, where each G i is a maximal connected subgroup of G i−1 . The length of G can be computed precisely. Indeed, if B is a Borel subgroup of G = G/R u (G) and r is the semisimple rank ofḠ, then [4,Theorem 1] states that For simple algebraic groups, this generalizes a theorem of Solomon and Turull [21, Theorem A*] on finite quasisimple groups. Several results on the depth of algebraic groups are also established in [4]. For example, if p = 0 we can calculate the exact depth of every simple algebraic group G, obtaining λ(G) 6, with equality if and only if G is of type A 6 (see [4,Theorem 4]). We also show that the depth of an algebraic group behaves rather differently in positive characteristic. For example, [4,Theorem 5(iii)] states that the depth of a simple classical type algebraic group tends to infinity with the rank of the group. In this paper we initiate the study of length and depth for connected Lie groups. Let g be a finite dimensional semisimple Lie algebra over C with compact real form g 0 (so g 0 is a compact Lie algebra over R with g 0 ⊗ R C ∼ = g). Under the Lie correspondence, g 0 is the Lie algebra of a compact semisimple (real) Lie group G. We write G(C) for the corresponding complex Lie group, which can be viewed as a semisimple algebraic group over C. For example, if g = sl n (C), then G = SU n and G(C) = SL n (C). Up to isomorphism and isogenies, the compact simple Lie groups are as follows: SU n (n 2), Sp n (n 4 even), SO n (n 7), G 2 , F 4 , E 6 , E 7 , E 8 . ( More generally, it is well known that any compact connected Lie group G is of the form G = G Z (G) 0 , where G is a commuting product of compact simple Lie groups and Z (G) 0 is a torus (a direct product of k copies of T ∼ = SO 2 , the circle group). We will write Cl n to denote any one of the classical type groups SU n , Sp n and SO n . We will also write T k for a k-dimensional torus. It is natural to transfer the definition of length and depth from algebraic groups to Lie groups. So we define the length l(G) of a connected Lie group G to be the maximal length of a chain of (closed) subgroups as in (1), where each G i is a maximal connected subgroup of G i−1 (a sequence of subgroups with this property is called an unrefinable chain). Similarly, the depth λ(G) is the minimal length of such a chain. Note that these parameters are independent of any choice of isogeny type. In particular, we may (and will) always assume that if G is a nonabelian compact connected Lie group, then G = i S i is a commuting product of simple groups given in (3).
We are now ready to state our main results. For a classical type group G = Cl n , define Our first result determines the exact length of every compact simple Lie group.

Theorem 1 Let G be a compact simple Lie group.
(i) If G = Cl n is of classical type, then l(G) = f G (n).
(ii) If G is of exceptional type, then l(G) is as follows: 5 11 13 17 20 As an immediate corollary, we deduce that 2r l(G) < 3r for all compact simple Lie groups G of rank r . In view of (2), we also note that In fact, it is easy to see that l(G) 2r , independently of Theorem 1. Indeed, let T be a maximal torus of G, let {α 1 , . . . , α r } be a corresponding set of simple roots and consider a chain of Levi subgroups where {α 1 , . . . , α k } is the set of simple roots in the root system of L k . Then we get We can use Theorem 1 to calculate the length of an arbitrary compact connected Lie group G. Recall that G = G Z (G) 0 , where the commutator subgroup G is semisimple (or trivial) and Z (G) 0 is a torus (see [13,Theorem 4.29], for example).

Theorem 2 Let G be a compact connected Lie group and write z
In particular, z + 2r l(G) z + 3r − t.
We now discuss the relationship between the length of a compact Lie group G and its dimension. We trivially have l(G) dim G. The next result shows that l(G) is close to dim G if and only if G is almost abelian.

Theorem 3 Let G be a compact connected Lie group.
For a collection C of compact connected Lie groups, the set is bounded if and only if the set Note that part (ii) above implies parts (i) and (iii). The method of proof of Theorem 3 also enables us to characterize algebraic groups whose length is close to their dimension; see Proposition 4.4 below.
As for lower bounds on l(G) in terms of dim G, we show the following.

Theorem 4 Let G be a compact connected Lie group and let
In particular, l(G) This may be compared with Theorem 3 of [4], showing that l(G) > 1 2 dim G for algebraic groups G over algebraically closed fields.
The proofs of Theorems 3 and 4 rely on the classification of compact simple Lie groups and on Theorems 1 and 2 above. The first lower bound in Theorem 4 is best possible; indeed it is attained for G = E 8 . Similarly, the second lower bound is asymptotically best possible, since In addition, we characterize compact connected Lie groups G of small length as follows.

Theorem 5 Let c be a positive integer and let G be a compact connected Lie group satisfying
Then either dim G is c-bounded, or G has a normal simple subgroup of type SO n of c-bounded codimension.
Here c-bounded means bounded above by some function of c only. Next we turn to the depth of compact Lie groups. For simple groups, we have the following result (see [4,Theorem 4] for the analogous statement for simple algebraic groups over C). 7 3 in all other cases.
We do not obtain a precise formula for the depth of an arbitrary compact Lie group, but we give bounds, as follows. (ii) Let G be a compact connected Lie group and set z = dim Z (G) 0 and G = m i=1 S k i i , where the S i are pairwise non-isomorphic simple groups. Then By Theorem 6, the upper bound on λ(G) in (ii) is at most z + 1 + i (k i + 3). Our next result concerns compact Lie groups G satisfying the condition l(G) = λ(G). Finite groups with this property were characterized by Iwasawa [10] -they are precisely the supersolvable groups. For algebraic groups over algebraically closed fields, a partial result was proved in [4, Theorem 6], but not a full characterization. For compact Lie groups, we prove the following result. More generally, the chain difference of G is defined by cd(G) = l(G) − λ(G). This invariant has been studied for finite groups and finite simple groups (see [2,5], for example), with particular interest in groups with chain difference one. Here we determine the compact Lie groups with this property. Combining Theorem 10 with part (iii) of Theorem 3, we immediately obtain the following somewhat surprising consequence.

Corollary 11
The following are equivalent for a collection C of compact connected Lie groups.
Indeed, conditions (i)-(iii) are all equivalent to the set {dim G − dim Z (G) 0 : G ∈ C} being bounded.
We refer the reader to [18] for results on the length and depth of non-compact Lie groups. The layout of the paper is as follows. In Sect. 2 we prove some results on the subgroup structure of compact Lie groups, in particular determining their maximal connected subgroups. Section 3 contains some further preliminary results on lengths and depths of compact groups which are needed for the proofs of the main results. These proofs are given in Sect. 4.

Subgroups of compact Lie groups
The following result provides a close link between the connected subgroups of a compact Lie group G and the connected reductive subgroups of the corresponding complex Lie group. This result is surely well-known, but we have been unable to find a proof in the literature. Proof Let X denote the set of connected subgroups of G, and Y the set of connected reductive subgroups in G(C). For X ∈ X , let X C ∈ Y be the complexification of X . Define to be the map that sends the class X G to the class (X C ) G(C) . We shall show that φ is a bijection.
To see that φ is surjective, let Y ∈ Y, and let Y 0 be a maximal compact subgroup of Y .
It remains to show that φ is injective. Let X 1 , X 2 ∈ X , and suppose X C . Now consider the Cartan decomposition of G(C): Now x 1 k, kx 2 ∈ G and p, p x 2 ∈ P, so by uniqueness in the Cartan decomposition, x 1 k = kx 2 . It follows that X k 1 = X 2 , so X 1 and X 2 are conjugate in G. This proves the injectivity of φ, as required.
Since the complexification map X → X C is inclusion-preserving, the following is an immediate consequence of Lemma 2.1.

Corollary 2.2 There is a bijective correspondence between conjugacy classes of maximal connected subgroups in G and conjugacy classes of maximal connected reductive subgroups in G(C).
Here, by maximal connected reductive subgroups in G(C), we mean subgroups that are maximal among connected reductive subgroups of G(C).
In the next two results we use Corollary 2.2, together with known results on the subgroup structure of G(C), to describe the maximal connected subgroups of compact simple Lie groups. For subgroups of maximal rank, this was done by Borel and de Siebenthal [1].

Proposition 2.3
Let G = Cl n be a compact simple Lie group of classical type with natural module V of dimension n over C, and let M be a maximal connected subgroup of G. Then one of the following holds: Table 1 The maximal connected reducible and tensor product subgroups of compact simple Lie groups of classical type G Reducible subgroups Tensor product subgroups (n = ab)  Table 1;

is a reducible or tensor product subgroup of G, as listed in
(ii) G = SU n and M = Sp n (with n 4 even) or SO n ; (iii) M is a compact simple Lie group acting irreducibly on V , and M is not isomorphic to a classical group on V .  Table 2.
Proof The results of Dynkin [7] show that the maximal connected subgroups of the complex simple group G(C) are parabolic subgroups, together with subgroups as in Table 2, but excluding the subgroups D 5 T 1 < E 6 and E 6 T 1 < E 7 (these are of course Levi factors of parabolics). By [1], these two Levi subgroups are the only ones that are maximal among connected reductive subgroups of G(C). Hence the maximal connected reductive subgroups of G(C) are precisely those in the table, and the result now follows via Corollary 2.2.

Length and depth: preliminary results
We start by recording some elementary properties of the length and depth of connected Lie groups, which we will use repeatedly throughout the paper. Note that the proof of [6, Lemma 2.1] goes through to give part (i).
Lemma 3.1 Let G be a connected Lie group with a connected normal subgroup N .

Lemma 3.2 If G is a connected compact Lie group, then
Proof We proceed by induction on the dimension of G. The conclusion is clear if G is a torus (every maximal connected subgroup has codimension 1), so assume Proof Part (i) is obvious, so let us consider (ii). If G = T 2 then λ(G) = 2 by Lemma 3.2; and if G = SU 2 , then G has a maximal connected subgroup T 1 by Corollary 2.2, so λ(G) = 2 again. Conversely, suppose G is a connected compact Lie group with λ(G) = 2. By Lemma 3.2, either G = T 2 or G = G . In the latter case, G has a maximal connected subgroup T 1 by part (i), and it follows that G = SU 2 . This completes the proof.

Lemma 3.4 Let G(C) be a complex semisimple Lie group, with compact form G. Then l(G) < l(G(C)).
Proof It follows from Lemma 2.1 that l(G) l(G(C)). To see that the inequality is strict, we apply [4, Corollary 2], which states that every unrefinable chain of G(C) of maximum length includes a maximal parabolic subgroup.
In the statement of the next result, we refer to the function defined in (4).  N (H , k) can be read off from [15], and they are listed in Table 3. By definition of the function f G , for G = Cl N we have f G (N ) 5     Proof For G = SU n , we have the following unrefinable chain of connected subgroups of length f G (n) = 2n − 2: Similarly, if G = Sp n with n = 2k, there is an unrefinable chain of length k − 1. Since Sp 2 ∼ = SU 2 has length 2, it follows that l((Sp 2 ) k ) = 2k and thus Finally, consider G = SO n . Suppose first that n is not divisible by 4, and write n = 4k + s with 1 s 3. There is an unrefinable chain SO n > SO 4 × SO n−4 > · · · > (SO 4 ) k × SO s of length k. Since SO 4 ∼ = (SU 2 × SU 2 )/Z 2 , SO 3 ∼ = SU 2 /Z 2 and SO 2 ∼ = T 1 (where Z 2 is a cyclic group of order 2), we see that l((SO 4 ) k × SO s ) = 4k + s − 1 and the result follows. Similarly, if n = 4k, then the above chain SO n > · · · > (SO 4 ) k has length k − 1 and thus l(G) 5k − 1 = f G (n).

Proof of Theorem 1
First assume G = Cl n is a compact simple Lie group of classical type. We prove that l(G) f G (n) by induction on n. In view of Lemma 3.7, this will complete the proof of part as required. A similar argument applies when G is a symplectic or orthogonal group. For example, if G = Sp n and M = SU n/2 T 1 , then Similarly, if G = SO n and M = SO k × SO n−k , then induction gives Next suppose that M is a tensor product subgroup, as in  Table 2. By applying part (i), it is an easy exercise, starting with the case G = G 2 and working down the rows of the table, to compute l(M) in every case. In this way, we obtain the values for l(G) recorded in part (ii) of the theorem. For example, if G = F 4 then by taking M = B 4 we can construct a chain 123 of length 11. Similarly, in the other cases we take to build chains of length 5, 13, 17 and 20, respectively.

Proof of Theorem 2
As in the statement of the theorem, let G be a compact connected Lie group and write G = G Z (G) 0 , where G = t i=1 S i is a commuting product of simple groups. Then Lemma 3.1(i) implies that l(G) = z + i l(S i ), where z = dim Z (G) 0 . Let r i = rank(S i ) and r = i r i = rank(G ). Then 2r i l(S i ) 3r i − 1 by Theorem 1, hence

Proof of Theorem 3
Let G = G Z (G) 0 be a compact connected Lie group.

Lemma 4.1 If G is simple, then l(G) 2 3 dim G, with equality if and only if G = SU 2 .
Proof This follows easily from Theorem 1.
Denote (G) = dim G − l(G). The additivity of dim and l implies the additivity of , namely (G) = (G/N ) + (N ), where N is a connected normal subgroup of G.

Lemma 4.2 If G is semisimple, then (G) dim G 3 (G).
Proof The first inequality is trivial for any compact connected Lie group G, and the second reduces to the case where G is simple, by additivity. For G simple we have 3l(G) 2 dim G by Lemma 4.1, which yields dim G 3 (G) as required.

Lemma 4.3 We have (G) = (G ).
Proof This is clear since We can now prove Theorem 3. It suffices to prove part (ii), namely In view of Lemma 4.3, this reduces to (G ) dim G 3 (G ), which is Lemma 4.2 (since G is semisimple). This completes the proof of the theorem.
A similar method enables us to characterize connected algebraic groups G, over algebraically closed fields, of large length. We clearly have l(G) dim G, and by [4,Theorem 3] equality holds if and only if G = R(G).A t 1 for some t 0, where R(G) is the radical of G (and the extension is not necessarily split). Here we extend this by showing that dim G −l(G) is bounded if and only if the codimension of R(G).A t 1 is bounded, where t is the multiplicity of A 1 in the semisimple group G/R(G). More precisely, we prove the following.

Proposition 4.4 Let G be a connected algebraic group over an algebraically closed field. Set
(G) = dim G − l(G) and let t 0 be as above. Then Proof If G is simple and not isomorphic to A 1 , then by applying [4, Corollary 2] we deduce that with equality if and only if G = A 2 , and this implies For an arbitrary connected algebraic group G, write G/R(G) = A t 1 S 1 · · · S k , where t, k 0 and each S i is a simple algebraic group that is not isomorphic to A 1 . Since is additive and (R(G)) = (A 1 ) = 0, we conclude that (G) dim(G/R(G).A t 1 ) 8 (G), as required.

Proof of Theorem 4
We first express the length of a simple classical compact Lie group in terms of its dimension.

Lemma 4.5 Let S = Cl n be a simple classical compact Lie group and let d = dim S. Then
where n ≡ k(mod 4) and 0 k 3.
Proof This follows from Theorem 1 and the well known formulae for the dimensions of the relevant groups. Lemma 4.5.

Corollary 4.6 Let S and d be as in
The next result also deals with exceptional groups.

Lemma 4.7 Let S be a compact simple Lie group and set
Then Proof If S is classical, this follows from Corollary 4.6 (since 5 · 2 −3/2 = 1.7677 . . . > 9/8). For S exceptional, the lower bound follows from Theorem 1.
Note that the inequality in Lemma 4.7 is essentially best possible. For example, and and define α as in (5). We will need the following elementary inequalities.

Lemma 4.8 Let x, y be real numbers.
Proof This is easily reduced to quadratic inequalities in one variable. For example, let us prove part (ii). By squaring both sides of the required inequality we reduce it to √ x y √ x + y + 1 2 , namely to x y x + y + √ x + y + 1 4 . Since (x − 3)(y − 3) 0 we obtain x y 3(x + y) − 9 so it suffices to show that 2(x + y) √ x + y + 9.25. Let w = √ x + y. Then we have to show that 2w 2 − w − 9.25 0, which follows from the fact that w √ 6 > (1 + √ 75)/4.
Let G be a compact connected Lie group. We prove Theorem 4 by induction on dim G, the base case dim G = 1 being trivial. Suppose dim G > 1 and let N = 1 be a connected normal subgroup of G of minimal dimension. If Z (G) 0 = 1 then N is a 1-dimensional torus. Set x = dim G/N . By the induction hypothesis, l(G/N ) β( √ x − α) and by applying part (i) of Lemma 4.8 we obtain as required. We therefore may assume Z (G) 0 = 1, so G = t i=1 S i is semisimple. If t = 1 then G is simple and the result follows from Lemma 4.7. So suppose t 2. We may assume N = S 1 , and set x = dim G/N and y = dim N . Then x y 3. Suppose first that N = E 6 , E 7 , E 8 . Then the induction hypothesis, Lemma 4.7 and part (ii) of Lemma 4.8 yield It remains to deal with the case where N = S 1 is E 6 , E 7 or E 8 . Then x y 78. Combining part (iii) of Lemma 4.8 with the induction hypothesis we obtain This completes the proof.

Proof of Theorem 5
Let the constants α and β be as defined above in (5) and (6). Let G be a compact connected Lie group with Let N be a connected normal subgroup of G.
by Theorem 4. This is a contradiction, so In particular, if z = dim Z (G) 0 then z < β( √ z + c + α) and thus z is c-bounded. Therefore, we may assume that G is semisimple.
Suppose G has t factors of type E 8 , so there is a connected normal subgroup N with G/N = (E 8 ) t . By (7), we have and we deduce that t is c-bounded. In the same way, we see that the number of exceptional factors of G is c-bounded so we may assume that G is a product of classical groups. If G/N = (SU n ) t for some n and t, then (7) implies that and thus n and t are c-bounded. The same conclusion holds if G/N = (Sp n ) t . Therefore, to complete the proof we may assume that G = k i=1 SO n i , where k 2 and n 1 n i 7 for all i. By Theorem 1, we have Claim. If (n 1 , . . . , n k ) is a k-tuple of integers with k 2 and n 1 n i 7 for all i, then either (n 1 , k) = (7, 2), or , the claim implies that k i=2 n i (βc) 2 and thus the normal subgroup SO n 1 has c-bounded codimension. Therefore, it suffices to prove the number-theoretic claim.
First assume the n i are all equal. We need to show that for (n 1 , k) = (7, 2). If k is fixed, the expression on the left hand side is increasing in n 1 and it is routine to verify the desired bound.
Now assume that at least one n i is less than n 1 , say n 2 < n 1 , and set These bounds imply that f (n 2 + 1, n 3 , . . . , n k ) f (n 2 , . . . , n k ), so f is minimal when n i = 7 for all 2 i k. Finally, we note that f (7, . . . , 7) = 5 4 is an increasing function in both n 1 and k, and by setting (n 1 , k) = (8, 2) we see that f (7, . . . , 7) > 0. This justifies the bound in (8) and the proof of Theorem 5 is complete.
In the other direction, if G has dimension d, or has a normal subgroup isomorphic to SO n of codimension d, then l(G) β( √ dim G + c), where c is d-bounded; indeed, this follows easily from Lemma 4.5.

Proof of Theorem 6
This is very similar to the proof of the analogous result for complex simple Lie groups (see [4,Theorem 4]). Let G be a compact simple Lie group of rank r . It will be convenient to adopt the Lie notation for classical groups, so that A r = SU r +1 and so on. By Lemma 3.3, we have λ(G) 2, with equality if and only if G = SU 2 , so we may assume r 2. Note that λ(G) 3, with equality if and only if G has a maximal A 1 subgroup.
If G = C r , then by applying [8] and Corollary 2.2, we see that G has a maximal A 1 subgroup and thus λ(G) = 3. Next assume G = B r , with r 3. If r 4 then λ(G) = 3. However, if r = 3 then G does not have a maximal A 1 subgroup, so λ(G) 4. In this case, equality holds since is an unrefinable chain. Similarly, by arguing as in the proof of [4,Theorem 4], we see that λ(G) = 4 if G = D 2r , or A r with r 3 and r = 6. Since A 2 has a maximal A 1 subgroup, we have λ(A 2 ) = 3, so to complete the proof of Theorem 6 for classical groups, we may assume that G = A 6 . Here λ(G) 5 since B 3 is a maximal subgroup and λ(B 3 ) = 4 as above. In addition, λ(G) 4 since G does not have a maximal A 1 subgroup. Let M be a maximal connected subgroup of G. By inspecting [8], we deduce that either M = B 3 , or M = A 5 T 1 , A 4 A 1 T 1 or A 3 A 2 T 1 is the Levi factor of a maximal parabolic subgroup of G. If M is a Levi factor, then λ(M) λ(A k ) for some k ∈ {3, 4, 5} and we conclude that λ(M) 4 for each connected maximal subgroup M of G. Therefore λ(G) = 5.
Finally, if G is an exceptional group, then we can repeat the argument in the proof of [4,Theorem 4]. We omit the details.

Proof of Theorem 7
We first prove part (i). Let G = S k , where S is a compact simple Lie group. We proceed by induction on k, noting that the case k = 1 is obvious. Assume Now consider part (ii), where z = dim Z (G) 0 and G = m i=1 S k i i . The upper bound for λ(G) follows from part (i) and Lemma 3.1(ii). We now prove the lower bound by induction on i k i . We have λ(G) = λ(G ) + z by Lemma 3.2, so we may assume that G = G . The case i k i = 1 is trivial, so assume i k i 2. Let M be a maximal connected subgroup of G such that λ(M) = λ(G) − 1. As above, without loss of generality, one of the following holds: (a) k 1

Proof of Theorem 8
Let G be a compact connected Lie group and write G = G Z (G) 0 and z = dim Z (G) 0 . By Lemmas 3.1(i) and 3.2, we have l(G) = l(G ) + z and λ(G) = λ(G ) + z. In particular, the result is trivial if G is a torus, so assume that G = 1.
Conversely, suppose that l(G) = λ(G). Write G = t i=1 S i , a commuting product of simple groups S i . Then l(G) = z + i l(S i ) and λ(G) z + i λ(S i ). Hence l(S i ) = λ(S i ) for all i, so G = (SU 2 ) t and l(G) = z + 2t. It remains to show that t = 1. To see this, suppose t 2 and note that there is an unrefinable chain , a contradiction. This completes the proof.

Proof of Theorem 9
By Theorems 1 and 6, we see that SU 3 is the only compact simple Lie group with chain difference one. It follows easily that the compact semisimple Lie groups with chain difference one are SU 3 , (SU 2 ) 2 and SU 3 SU 2 . The rest of the argument is very similar to the proof of Theorem 8 above.

Proof of Theorem 10
We start with some preparations. Proof It suffices to show that l(S) 2λ(S) − a, which is easily deduced from Theorems 1 and 6.
Next, we deal with homogeneous semisimple groups.  We can now prove Theorem 10. Lemma 4.11 enables us to reduce to the case where G is semisimple. Write G = m i=1 S k i i where m 5, k i 0 and the S i are pairwise non-isomorphic simple groups as in (3), labelled so that S 1 , . . . , S 5 are SU 2 , SU 3 , SU 4 , Sp 4 , SO 7 , respectively. Set Note that G = G 1 × G 2 , l(G) = l(G 1 )+l(G 2 ) and cd(G) cd(G 1 )+cd(G 2 ), so it suffices to show that l(G) 2(cd(G 1 ) + cd(G 2 )) + 2. Therefore, to complete the proof of the main statement of Theorem 10, it remains to show that l(G 1 ) 2cd(G 1 ) + 2.
To prove the second statement, recall that, by Theorem 4 we have where the constants α, β are defined as in Sect. 4.4 (see (5) and (6)). Combining this with the first assertion of Theorem 10 we obtain This completes the proof.
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