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Which Nilpotent Groups are sel-similar?
Olivier Mathieu
To cite this version:
Which Nilpotent Groups are self-similar?
Olivier Mathieu
† †Institut Camille Jordan du CNRS, Universit´e de Lyon,
F-69622 Villeurbanne Cedex, mathieu@math.univ-lyon1.fr
January 21, 2021
Abstract
Let Γ be a finitely generated torsion free nilpotent group. A cri-terion for the existence of a faithfull transitive self-similar action of Γ on a rooted tree A∗ is established. Then a formula is given for the minimal cardinal of an alphabet A such that some subgroup Γ0 ⊂ Γ of finite index admits a faithfull and transitive self-similar action. The formula has some numer theoretical content.
As an application, it is shown that the fundamental group of a nilmanifold endowed with an Anosov diffeomorphism is transitive self-similar. It is also shown that the fundamental group of the non-affine nilmanifold found in [3] is not self-similar.
Contents
1 Self-similar actions and self-similar data 6
2 Rational points of a torus 8
3 Special Gradings 11
5 The Campbell-Hausdorff lattices 23
6 Relative complexity of multiplicative lattices 28
7 Proof of Theorems 2 and 3 30
8 Examples of not self-similar FGTF nilpotent groups 34
9 Absolute Complexities 36
10 Some Nilpotent Lie groups of infinite complexity. 40
Introduction
Let A be a finite alphabet and let A∗ be the set of all words over A. An action of a group Γ on the rooted tree A∗ is called self-similar iff for any γ ∈ Γ and a ∈ A there exists γa∈ Γ and b ∈ A such that
γ(aw) = bγa(w) for any w ∈ A∗.
Moreover, the action of Γ on A∗ is called transitive (respectively level transi-tive) if Γ acts transitively on A = A1 (respectively on each level An).
The group Γ is called self-similar (transitive self-similar, level transitive self-similar) if Γ admits a faithfull self-similar action (respectively a faith-full transitive similar action, respectively a faithfaith-full level transitive self-similar action) on A∗ for some finite alphabet A.
Self-similar groups appeared in the early eighties, e.g. in the works of R. Grigorchuk [10] [11] and in the joint works of N. Gupta and S. Sidki [13] [14]. See also the monography [24] for an extensive bibliography before 2005 and [25] [2] [9] [16] [12] for more recent works. A general question is to determine which groups Γ are self-similar. In this paper, we will investigate the case of finitely generated torsion-free nilpotent groups (called FGTF nilpotent groups in the sequel). For them, the question has been explicitely raised in some talks of S. Sidki. The systematic study of self-similar representations of nilpotents groups started with [4].
To explain the main result, a few definitions are required. A grading of a Lie algebra m is a decomposition m = ⊕n∈Zmn such that [mn, mm] ⊂ mn+m
for all n, m ∈ Z. It is called special if m0∩ z = 0, where z is the center of m.
[18][26], Γ is a cocompact lattice in a unique connected, simply connected (or CSC in what follows) nilpotent Lie group N . Let nR be the Lie algebra
of N and set nC= C ⊗ RnR.
The main result is the following
Theorem 2. The following assertions are equivalent (i) The group Γ is transitive self-similar,
(ii) the group Γ is level transitive self-similar, and (iii) the Lie algebra nC admits a special grading.
As a consequence, let’s mention
Corollary 1. Let M be a nilmanifold endowed with an Anosov diffeomor-phism. Then π1(M ) is level transitive self-similar.
Let N be a CSC nilpotent Lie group, with Lie algebra nR. Let’s assume
that nC has a special grading and that N contains at least one cocompact
lattice. Theorem 2 suggests the following question
Given a cocompact lattice Γ ⊂ N , how to find an explicit faithfull transitive (or level transitive) self-similar action on some rooted tree A∗?
In order to give a partial answer to this question we will investigate the degree of such actions, where the degree is the cardinal of A.
To explain our next result, more notions are now defined. The complexity of a cocompact lattice Γ of N , denoted by cp Γ, will be the smallest degree of a faithfull transitive self-similar action of Γ on some A∗.
Recall the the commensurable class ξ of a cocompact lattice Γ0 in N is
the set of all cocompact lattices of N which share with Γ0 a subgroup of
finite index. Its complexity cp ξ is defined by cp ξ = MinΓ∈ξcp Γ,
and a cocompact lattice Γ ∈ ξ with cp Γ = cp ξ is called minimal. Indeed a faithfull transitive self-similar action of minimal degree cp ξ is automatically level transitive, so restricting the definition to level transitive actions provides the same complexity for ξ.
By Malcev’s Theory, the commensurable class ξ determines a canonical Q-form n(ξ) of the Lie algebra nR. Let S(n(ξ)) be the set of all automorphisms
f of n(ξ) such that Spec f |zC contains no algebraic integer.
Let λ 6= 0 be an algebraic number, let O(λ) be the ring of integers of Q(λ). Set πλ = {x ∈ O(λ)|xλ ∈ O(λ)} and d(λ) = Card O(λ)/πλ. For any
ht h =Q
λ∈Spec h/Gal(Q) d(λ) mλ,
where Spec h/Gal(Q) is the list of eigenvalues of h up to conjugacy by Gal(Q) and where mλ is the multiplicity of the eigenvalue λ.
Theorem 4. We have
cp ξ = Minh∈S(n(ξ))ht h.
Moreover for any Γ ∈ ξ, there are minimal cocompact lattices Γ1, Γ2 ∈ ξ with
Γ1 ⊂ Γ ⊂ Γ2.
Of course, our definition of complexity of is a bit naive. A really concrete notion of complexity should be the size of the input needed to implement an automaton describing a transitive self-similar action on A∗, and A is only part of it, but we did not investigate this direction.
Theorem 4 is formula for cp ξ, however finding cp Γ for an individual group is an ugly process. With the notation of the previous theorem it follows easilyly that cp Γ ≤ cp ξ Min([Γ : Γ1], [Γ2 : Γ]), see Corollary 3. There
exists also a faithfull level transitive self-similar action of Γ, but the proof provides a very large bound for its degree.
The framework of nonabelian Galois cohomology shows another concrete aspect of Theorem 4. Indeed the commensurable classes (up to conjugacy) in N are classified by the Q-forms of some classical objects with a prescribed R-form, see Corollary 3 of ch.9. Roughly speaking, the commensurable classes are arithmetic objects, and their complexity is an invariant of some arithmetic group.
As an illustration of the previous vague sentence, we investigate a class N of Lie groups N . The commensurable classes in N are classified, up to conjugacy, by the positive definite quadratic forms q on Q2. For such a q, let ξ(q) be the corresponding commensurable class in N . Then Theorem 5 states that
cp ξ(q) = F (d)e(N )
where e(N ) is an explicit invariant of the special grading of nC, where −d is
the discriminant of q, and F (d) is the minimal n > 0 which is solution of the diophantine equation a2+ db2 = 4n2 with ab 6= 0 (except that F (d) = 7 for d = 3).
Section 8 involves a different type of corrolary of Theorem 2. Among FGTF nilpotent groups, some of them are similar but not transitive self-similar. Some of them are not even self-similar, since Theorem 2 implies the next
Corollary 2. Let M be one of the non-affine nilmanifolds appearing in [3]. Then π1(M ) is not self-similar.
About the proofs. The proofs of the papers are based on different ideas. Theorem 1 is a statement about rational points of algebraic tori. Its proof is based on standard results of number theory, including the Cebotarev’s Theorem. It is connected with the density of rational points for connected groups proved by A. Borel [6], see also [27].
In addition of Theorem 1 and the standard theory of algebraic groups, Theorems 2 and 4 are based on the new notion of minimal Campbell-Haus-dorff lattices, relative to some commutative reductive group H.
The notion of Campbell-Hausdorff lattices is a refinement of Malcev idea, that some cocompact lattices of nilpotent Lie groups have a structure of Lie ring. Campbell-Hausdorff lattices are Lie rings which are stable by higher operations, coming from the Campbell-Hausdorff formula. Equivalently, it means that they are rings over a certain Z-form of the Lie operad.
Given a semi-simple isomorphism h of a Q-vector space V , let O(h) be the subring of F ∈ Q[h] ⊂ End(V ) such that all eigenvalues of F are alge-braic numbers. A lattice Λ of V is called minimal relative to a commutative reductive group H ⊂ GL(V ), if it is an O(h)-module for any h ∈ H. The existence of minimal Campbell-Hausdorff lattices relative to a given group of automorphisms H, proved in Lemma 23, is a key point in the proof of Theo-rems 2 and 4. It should be noted that the Lie bracket, and higher operations, are not O(h)-linear.
Also, the proof of Corollary 1 is based on a paper of A. Manning [20] about Anosov diffeomorphisms.
The existence of non-self-similar FGTF nilpotent group is based on very difficult computations, which, fortunately, were entirely done in [3].
1
Self-similar actions and self-similar data
This section, based on [24], explains the correspondence between the faithfull transitive similar actions and some virtual endomorphisms, called self-similar data.
1.1 Self-similar actions and self-similar groups
Let A be a finite alphabet. The set A∗ = ∪n≥0An of all words over the
alphabet A is a rooted tree. Its root is the empty word ∅ and the edges are the pairs {w, wx} whith w ∈ A∗ and x ∈ A. An action of a group Γ on the tree A∗ means an action on the set A∗ which preserves the root and the adjacency of the vertices. An action is called self-similar iff for any γ ∈ Γ and a ∈ A there exists γa ∈ Γ and b ∈ A such that
γ(aw) = bγa(w) for any w ∈ A∗.
Note that any action of Γ on A∗ preserves each level set An, thus the action cannot be transitive for the usual meaning. Instead, an action of Γ on A∗ is called transitive (level transitive) if Γ acts transitively on A (respectively transitively on each level An). The degree of an action is the cardinal of the alphabet A.
The group Γ is called self-similar (transitive self-similar, level transitive self-similar) if Γ admits a faithfull self-similar action (respectively a faith-full transitive similar action, respectively a faithfaith-full level transitive self-similar action) on A∗ for some finite alphabet A.
1.2 Core and f -core
Let Γ be a group and Γ0 be a subgroup. The core of Γ0 is the biggest normal subgroup K / G with K ⊂ Γ0. Equivalently the core is the kernel of the left action of Γ on Γ/Γ0.
Now let f : Γ0 ! Γ be a group morphism. By defintion the f-core is the biggest normal subgroup K / G with K ⊂ Γ0 and f (K) ⊂ K.
1.3 Self-similar data
Let Γ be a group. A virtual endomorphism of Γ is a pair (Γ0, f ), where Γ0 is a subgroup of finite index and f : Γ0 ! Γ is a group morphism. A self-similar datum is a virtual endomorphism (Γ0, f ) with a trivial f -core.
Assume given a faithfull transitive self-similar action of Γ on a tree A∗. Let a ∈ A, and let Γ0 be the stabilizer of a. By definition, for each γ ∈ Γ0 there is a unique γa∈ Γ such that
for any w ∈ A∗. Let f : Γ0 ! Γ be the map γ 7! γa. Since the action
is faithfull, γa is uniquely determined and f is a group morphism. Also it
follows from Proposition 2.7.4 and 2.7.5 of [24] that the f -core of Γ0 is the kernel of the action, therefore it is trivial (without using the terminology of, the f -core is described explicitely in [24], Proposition 2.7.5). Therefore (Γ0, f ) is a self-similar datum.
Conversely, a sef-similar datum (Γ0, f ) determines a faithfull transitive self-similar action of Γ on a tree A∗, where A = Γ/Γ0, see ch 2 of [24] for details, especially subsection 2.5.5 of [24]). In conclusion, we have
Lemma 1. Let Γ be a group. There is a correspondence between the self-similar data (Γ0, f ) and the faithfull transitive self-similar actions of Γ on the rooted tree A∗, where A ' Γ/Γ0.
In particular, the degree of the self-similar action associated with (Γ0, f ) is [Γ : Γ0].
The correspondence is indeed a bijection modulo some conjugacy and a more precise statement can be found in [24].
1.4 A technical lemma
Let Γ be a finitely generated group.
Lemma 2. Let A, B be two subgroups of finite index. We have [A : A ∩ B] ≤ [Γ : B].
Moreover if Γ is nilpotent, then
[A : A ∩ B] divides [Γ : B].
Proof. Since A/(A ∩ B) is isomorphic to the A-orbit of 1 in Γ/B, we have [A : A ∩ B] ≤ [Γ : B].
Let’s now assume that Γ is nilpotent. Let K be the core of A ∩ B and set Γ = Γ/K, A = A/K and B = B/K. Since [A : A ∩ B] = [A : A ∩ B] and [Γ : B] = [Γ : B], we can assume that Γ = Γ, what implies that Γ is finite. Since Γ, A, B and A ∩ B are the direct product of their p-Sylow subgroups, we can also assume that Γ is a finite p-group, for some prime integer p. Since it is already proved that [A : A ∩ B] ≤ [Γ : B], we have
[A : A ∩ B] | [Γ : B].
Let Γ be a group, and let (Γ0, f ) be a virtual endomorphism. Let Γn be the
subgroups of Γ inductively defined by Γ0 = Γ, Γ1 = Γ0 and for n ≥ 2
Γn = {γ ∈ Γn−1| f (γ) ∈ Γn−1}
Lemma 3. The sequence n 7! [Γn : Γn+1] is not increasing.
Proof. For n > 0, the map f induces an injection of the set Γn/Γn+1 into
Γn−1/Γn, thus we have [Γn: Γn+1] ≤ [Γn−1 : Γn].
The virtual endomorphism (respectively a self-similar datum) (Γ0, f ) is called good if [Γn: Γn+1] = [Γ/Γ0] for all n.
Let (Γ0, f ) be a self-similar datum, and let A∗ be the corresponding tree on which Γ acts. If a is the distinguished point in A ' Γ/Γ0, then Γn is
the stabilizer of an. If the self-similar datum (Γ0, f ) is good, then [Γ : Γ n] =
Card An and therefore Γ acts transitively on An. Exactly as before, we have
Lemma 4. Let Γ be a group. There is a correspondence between the good self-similar data (Γ0, f ) and the faithfull level transitive self-similar actions of Γ on A∗, where A ' Γ/Γ0.
1.6 Fractal self-similar data
Let Γ be a group. A self-similar datum (Γ0, f ) is called fractal (or recurrent) if f (Γ0) = Γ. An action of Γ on a rooted tree A∗ is called fractal if it is a transitive self-similar action and the corresponding self-similar datum is fractal. Indeed, the corresponding self-similar datum is only defined up to conjugacy, but it is clear that the notion of a fractal action is well-defined, see [24] section 2.8. Obviously a fractal action is level transitive.
The group Γ is called fractal if Γ admits a faithfull fractal action on some rooted tree A∗.
2
Rational points of a torus
on X(H), or, equivalently such that H is L-isomorphic to Gr
m, where Gm
denotes the multiplicative group. Moreover, we have χ(h) ∈ L∗
for any χ ∈ X(H) and any h ∈ H(Q).
Let OLbe the ring of integers of L and let P be the set of its prime ideals.
Recall that a fractional ideal is a non-zero finitely generated OL-submodule
of K. A fractional ideal I is called integral if I ⊂ OL. If the fractional ideal
I is integral and I 6= OL, then I is an ordinary ideal of OL.
Let I be the set of all fractional ideals and I+be the subset of all integral ideals. Given I and J in I, their product is the OL-module generated by all
products ab where a ∈ I and b ∈ J . Since OL is a Dedekind ring, we have
I ' ⊕π∈PZ [π] I+' ⊕
π∈PZ≥0[π],
where P is the set of maximal ideals of O. Indeed the additive notation is used for for the group I and the monoid I+: view as an element of I the fractional ideal πm1
1 . . . πnmn will be denoted as m1[π1] + · · · + mn[πn].
Since Gal(L/Q) acts by permutation on P, I is a ZGal(L/Q)-module. For S ⊂ P, set
IS = ⊕π∈P\SZ [π].
Lemma 5. Let S ⊂ P be a finite subset and let r > 0 be an integer.
The Gal(L/Q)-module IL contains a free ZGal(L/Q)-module M (r) of
rank r such that
(i) M (r) ∩ IL+= {0}, and (ii) M (r) ⊂ IS.
Proof. Let S0 be the set of all prime numbers which are divisible by some π ∈ S. Let Σ be the set of prime numbers p that are completely split in K, i.e. such that O/pO ' F[L:Q]p . For p ∈ Σ, let π ∈ P be a prime ideal over p.
When σ runs over Gal(L/Q) the ideals πσ are all distinct, and therefore [π]
generates a free ZGal(L/Q)-module of rank one.
By Cebotarev theorem, the set Σ is infinite. Choose r + 1 distinct prime numbers p0, . . . pr in Σ \ S0, and let π0, . . . , πr ∈ P such that O/πi = Fpi.
For 1 ≤ i ≤ r, set τi = [πi] − [π0] and let M (r) be the ZGal(L/Q)-module
generated by τ1, . . . , τr.
Obviously, the Gal(L/Q)-module M (r) is free of rank r and M(r) ⊂ IS.
It remains to prove that M (r) ∩ IL+= {0}. Let A = P
1≤i≤r,σ∈Gal(L/Q)
be an element of M (r) ∩ IL+. We have A = B − C, where B = P 1≤i≤r,σ∈Gal(L/Q) mi,σ[πσi], and C = P σ∈Gal(L/Q) ( P 1≤i≤r mσ i)[π0σ].
Thus the condition A ∈ IL+ implies that mσ
i ≥ 0, for any 1 ≤ i ≤ k and σ ∈ Gal(L/Q), and
P
1≤i≤r
mσi ≤ 0, for any σ ∈ Gal(L/Q). Thus all the integers mσ
i vanish. Therefore M (r) intersects I+ trivially.
For π ∈ P, let vπ : L∗ ! Z be the corresponding valuation.
Lemma 6. Let S ⊂ P be a finite Gal(L/Q)-invariant subset and let r > 0 be an integer.
The Gal(L/Q)-module L∗ contains a free ZGal(L/Q)-module N(r) of rank r such that
(i) N (r) ∩ OL= {1}, and
(ii) vπ(x) = 0 for any x ∈ N (r) and any π ∈ S.
Proof. Set L∗S = {x ∈ L∗|vπ(x) = 0, ∀π ∈ S} and let θ : L∗S ! IS be the
map x7! Pπ∈P vπ(x) [π].
By Lemma 5, IS contains a free ZGal(L/Q)-module M (r) of rank r such
that M (r) ∩ I+ = {0}. Note that Coker θ is a subgroup of the class group Cl(L) of L. Since, by Dirichelet theorem, Cl(L) is finite, there is a positive integer d such that d.M (r) lies in the image of θ. Since M (r) is free, there is a free ZGal(K/Q)-module N(r) ⊂ L∗S of rank r which is a lifting of M (r),
i.e. such that θ induces an isomorphism N (r) ' d.M (r). Since θ(OL\ 0) lies
in I+, we have θ(N (r) ∩ O
L) = {0}. It follows that N (r) ∩ OL = {1}.
The second assertion follows from the fact that N (r) lies in L∗S.
For π ∈ P, let Oπ and Lπ be the π-adic completions of OL and L. Let
x, y ∈ L and let n > 0 be an integer. In what follows, the congruence x ≡ y modulo nOπ
means xπ ≡ yπmod nOπ, where xπ and yπ are the images of x and y in Lπ.
Theorem 1. Let H be an algebraic torus defined over Q, and let L be its splitting field. Let n > 0 be an integer and let S ⊂ P be the set of prime divisors of n.
There exists h ∈ H(Q) such that
(i) χ(h) is not an algebraic integer, for any non-trivial χ ∈ X(H), and (ii) χ(h) ≡ 1 mod nOπ for any χ ∈ X(H) and any π ∈ S.
Proof. Step 1. First an element h0 ∈ H(Q) satisfying Assertion (i) and (iii) vπ(χ(h0)) = 0, for any π ∈ S and any χ ∈ X(H)
is found.
As a Z-module, X(H) is free of rank r where r = dim H. The co-multiplication ∆ : X(H) ! X(H) ⊗ ZGal(L/Q) provides an embedding of X(H) into a free ZGal(L/Q)-module of rank r. By lemma 6, there a free ZGal(L/Q)-module N (r) ⊂ L∗S of rank r with N (r) ∩ OL= {1}. Let
µ : X(H) ⊗ ZGal(L/Q) ! N (r),
be an isomorphism of ZGal(L/Q)-modules, and set h0 = µ ◦ ∆.
Since H(Q) = HomGal(L/Q)(X(H), L∗), the embedding h0 is indeed an
element of H(Q). Viewed as a map from X(H) to L∗, h0 is the morphism χ ∈ X(H) 7! χ(h).
Since Im h0∩ OL = 1 and h is injective, χ(h0) is not an algebraic integer
if χ is a non-trivial character. Since Im h0 ⊂ L∗
S, we have vπ(χ(h0)) = 0 for
any χ ∈ X(H). Therefore h0 satisfies Assertions (i) and (iii).
Step 2. Let χ1, . . . , χr be a basis of X(H). Since vπ(χi(h0)) = 0 for any
π ∈ S, the element χi(h0) mod nOπ is an inversible element in the finite ring
Oπ/nOπ. Therefore there is a positive integer mi,π such that
χi(h0)mi,π ≡ 1 mod nOπ.
Set m = lcm(mi,π) and set h = h0m. Obviously h satisfies Assertion (i).
Moreover we have χi(h) ≡ 1 mod nOπ, for all π ∈ S and all 1 ≤ i ≤ r, and
therefore h satisfies Assertion (ii) as well.
3
Special Gradings
Let n be a finite dimensional Lie algebra defined over Q and let z be its center. The relations between the gradings of C ⊗ n and the automorphisms of n are investigated now.
the algebraic group of automorphisms of n. By definition, G is defined over Q, and we have G(K) = Aut nK for any field K of characteristic zero. The algebraic group G = Aut n cannot be entirely recovered from the abstract group G(Q) = Aut nQ, since the later is not always Zariski dense. The
notation n underlines that n is viewed as the functor in Lie algebras K 7! nK.
These subtilities are really relevant in chapter 9 and 10.
Let H ⊂ G be a maximal torus defined over Q, whose existence was proved by C. Chevalley in [7], see also [6] Theorem 18.2.
By definition, a K-grading of n is is a decomposition of nK nK = ⊕
n∈ZnKn
such that [nK
n, nKm] ⊂ nKn+m for all n, m ∈ Z. Such a grading is called special
(respectively very special) if zK ∩ nK
0 = 0 (respectively if nK0 = 0). Such a
grading is called non-negative if nK
n = 0 for n < 0.
For any field K of characteristic zero, a K-grading of n can be identified with an algebraic group morphism
ρ : Gm ! G
defined over K, where Gm denotes the multiplicative group.
Consider the following two hypotheses HK and HK+:
(HK) The Lie algebra n admits a special K-grading.
(H+K) The Lie algebra n admits a non-negative special K-grading. Lemma 7. (i) The hypotheses HC and HQ are equivalent.
(ii) The hypotheses H+
C and H +
Q are equivalent.
Some result similar to Assertion (ii) is proved in [8].
Proof. Note that the implications H
Q ⇒ HCand H + Q ⇒ H
+
C are tautological.
Step 1. We claim that any grading of nC is G(C)-conjugate to a grading
defined over Q. Let
nC= ⊕ n∈ZnCn
be a grading of nC and let ρ : G
m ! G be the corresponding algebraic group
morphism defined over C. Since any maximal torus of G is G(C)-conjugate to H, see [6], it can be assumed that ρ(Gm) ⊂ H.
Let X(H) be the character group of H. The group morphism ρ is de-termined by the dual morphism L : X(H) ! Z = X(Gm). Thus ρ is
automaticaly defined over Q. Therefore the claim is proved.
Step 2: Proof of Assertion (i). It follows from the previous step HC ⇒ H
Q,
Step 3: Proof of Assertion (ii). It follows from the first step that H+ C ⇒ H + Q. So we can assume H+ Q. Let nQ = ⊕ k∈Z≥0n Q k
be a non-negative special grading G of nQ. Let ρ : G
m ! G be the
cor-responding algebraic group morphism defined over Q. As before, it can be assumed that ρ(Gm) ⊂ H and let L : X(H) ! Z be the dual morphism of ρ.
The action of H over nQ provides a weight decomposition
nQ = ⊕
χ∈X(H)nQχ
of nQ, relative to which we have
nQ
k = ⊕L(χ)=knQχ
for any k ≥ 0.
Let K be the splitting field of H. Since Gal(K) acts trivially over X(H), the group X(H) is a Gal(K, Q)-module. So we can define a morphism L0 : X(H) ! Z by
L0(χ) =P
σ∈Gal(K,Q) L(χ σ).
for any χ ∈ X(H). For any k ∈ Z, set mQ k = L L0(χ)=k nQ λ.
Therefore the decomposition nQ = ⊕
k∈ZmQk is a new grading Gnew of nQ.
Since L0 is Gal(Q)-invariant, Gnewis defined over Q. It remains to prove that
Gnew is non-negative and special.
Let Ω be the space of weights of nQ. Since the given grading G is
non-negative, we have L(χ) ≥ 0 for any χ ∈ Ω. Since Ω is stable by Gal(K, Q), we have L0(χ) ≥ L(χ) ≥ 0 for any χ ∈ Ω and therefore Gnew is non-negative.
Similarly, let Ω0 be the set of weights of zQ. We have L0(χ) ≥ L(χ) > 0
for any for any χ ∈ Ω0. Thus Gnew is also special.
Lemma 8. Let Λ be a finitely generated abelian group and let S ⊂ Λ be a finite subset containing no element of finite order. Then there exists a morphism L : Λ! Z such that
L(λ) 6= 0 for any λ ∈ S.
Proof. The group structure on Λ will be denoted additively. Let F be the subgroup of finite order element in Λ. Using Λ/F instead of Λ, it can be assumed that Λ = Zd for some d an 0 /∈ S. Let’s choose a positive integer N
L(a1, . . . , ad) =
P
1≤i≤daiN i−1.
For any λ = (a1, . . . , ad) ∈ S, there is a smallest index i with ai 6= 0. We have
L(λ) = aiNi−1 modulo Ni. Since |ai| < N , it follows that L(λ) 6= 0 mod Ni
and therefore L(λ) 6= 0.
For f ∈ G(Q), let Spec f be the set of all eigenvalues of f in Q.
Lemma 9. Let f ∈ G(Q). There is a f -invariant Z-grading of nQ such that
all eigenvalues of f on nQ
0 are roots of unity.
In particular, if Spec f contains no root of unity, then nQ admits a very
special grading.
Proof. Let Λ ⊂ Q∗ be the subgroup generated by the Spec f . For any λ ∈ Λ denote by E(λ) ⊂ nQ the corresponding generalized eigenspace of f . Let R
be the set of all roots of unity in Spec f and set S = Spec f \ R.
By Lemma 8, there is a morphism L : Λ! Z such that L(λ) 6= 0 for any λ ∈ S. Let G be the decomposition
nQ = ⊕ k∈ZnQk
of nQ defined by nQ
k = ⊕L(λ)=kE(λ). Since [E(λ), E(µ)] ⊂ E(λµ) and L(λµ) =
L(λ) + L(µ) for any λ, µ ∈ Λ, it follows that G is a grading of the Lie algebra nQ. Moreover we have
nQ
0 = ⊕λ∈RE(λ),
from which the lemma follows.
Let S(n) be the set of all f ∈ G(Q) such that Spec f |z contains no
alge-braic integers, where Spec f |z is the set of all eigenvalues of f |z in Q.
Lemma 10. The following statements are equivalent (i) the Lie algebra nC admits a special grading, and
(ii) the set S(n) is not empty.
Proof. Consider the following assertion (A) H0(H(Q), zQ) = 0.
The proof of the lemma is based on the following ”cycle” of implications (i) =⇒ (A) =⇒ (ii) =⇒ (i).
Step 1: proof that (i) implies (A). By hypothesis and Lemma 7, there exists a special grading of nQ. Let ρ : G
m ! G be the corresponding group
H0(H(Q), zQ) ⊂ H0(ρ(Q∗), zQ) = 0.
Thus Assertion A is proved.
Step 2: proof that (A) implies (ii). Assume Assertion A. By Theorem 1, there exists f ∈ H(Q) such that χ(f ) is not an algebraic integer for any non-trivial character χ ∈ X(H). It follows that Spec f |z contains no algebraic
integers and therefore S(n) is not empty.
Step 3: proof that (ii) implies (i). Assume that S(n) is not empty and let f ∈ S(n). Since Spec f |z contains no roots of unity, it follows from Lemma 9
that the Lie algebra nQ, and therefore nC, admits a special grading. Assertion
(i) is proved.
Lemma 11. The following are equivalent:
(i) the Lie algebra nQ admits a nonnegative special grading, and
(ii) there exists g ∈ G(Q) such that all eigenvalues of g are algebraic integers but Spec g−1|z contains no algebraic integers.
Proof. First prove that (i) =⇒ (ii). Let nQ = ⊕
k≥0nQk be a non-negative
special grading of nQ. Let g : n ! n be the automorphism defined by
g(v) = 2kv if v ∈ n
k. So all eigenvalues of g are integers. Moreover z∩n0 = 0,
so Spec g−1|z contains no algebraic integers.
Next prove that (ii) =⇒ (i). Let g : nQ ! nQ be an automorphism
satisfying Assertion (ii). Let K ⊂ Q be the subfield generated by Spec g and let L : K∗ ! Z be the map defined by
L(x) =P
p vp(NK/Q(x))
where the sum runs over all prime numbers p and where NK/Q : K∗ ! Q∗ denotes the norm map.
For any integer k, set
nQ k =
L
L(x)=k
E(x)
where E(x)⊂ nQdenotes the generalized eigenspace associated to x ∈ Spec g.
We have [E(x), E(y)] ⊂ E(xy) and L(xy) = L(x) + L(y), for any x, y ∈ K.
Therefore the decomposition
nK = ⊕ k∈ZnQk
is a grading G of the Lie algebra nQ. Since the function L is Gal(Q)-invariant,
the grading G is indeed defined over Q. It remains to prove that G is non-negative and special.
is not an algebraic integer, we have NK/Q(x) 6= ±1 and L(x) > 0. Thus the grading is special, so Assertion (i) holds.
4
Height and relative complexity
In this section, we define the notion of the height of an isomorphism h of a finite dimensional vector space V over Q and the notion of minimality of a lattice. At the end of the section, V is a nilpotent algebra over some operad, and we the existence of lattices which are both rings an minimal relative to a family of automorphisms.
4.1 Height, complexity and minimality
Let V be a finite dimensional vector space over Q and let h ∈ GL(V ). Recall that a lattice of V is a finitely generated subgroup Λ which contains a basis of V . Let S(h) be the set of all couple of lattices (Λ, E) of V such that E ⊂ Λ and h(E) ⊂ Λ. By definition, the height of h, is the integer
ht(h) := Min(Λ,E)∈S(h)[Λ : E].
Let Smin(h) be the set of all couples (Λ, E) ∈ S(h) such that [Λ : E] = ht(h).
Similarly, for a lattice Λ of V , the h-complexity of Λ is the integer cph(Λ) := Min(Λ,E)∈S(h)[Λ : E].
It is clear that cph(Λ) = [Λ : E], where E = Λ ∩ h−1(Λ). The lattice Λ is called minimal relative to h if cph(Λ) = ht(h).
For the proofs, a technical notion of relative height is needed. Let C(h) ⊂ End V be the commutant of h and let A ⊂ C(h) be a subring. Let SA(h) be
the set of all couple of A-modules (Λ, E) in S(h). The A-height of h is the integer
htA(h) := Min(Λ,E)∈SA(h)[Λ : E].
(with the convention that htA(h) = ∞ if SA(h) is empty). Obviously, we
have htA(h) ≥ ht(h) = htZ(h). Let S A
min(h)) be the set of all couples (Λ, E) ∈
SA(h)) such that [Λ : E] = ht A(h).
4.2 Height and filtrations
Lemma 12. Let 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V be a fitration of V , where
each vector space Vi is a A[h]-submodule. For i = 1 to n, set hi = hVi/Vi−1.
Then we have
htA(h) ≥
Q
1≤i≤n htA(hi).
Moreover if V ' ⊕Vi/Vi−1 as a A[h]-module, then we have
htA(h) =Q1≤i≤n htA(hi).
Proof. Clearly it is enough to prove the lemma for n = 2. Let (Λ, E) ∈ SA
min(h). Set Λ1 = Λ ∩ V1, E1 = E ∩ V1, Λ2 = Λ/Λ1 and E2 = E/E1. We
have
[Λ : E] = [Λ1 : E1][Λ2 : E2].
Since (Λ1, E1) ∈ SA(h1) and (Λ2, E2) ∈ SA(h2), we have
htA(h) ≥ htA(h1) htA(h2),
what proves the first assertion.
Next, we assume that V ' V1 ⊕ V2 as a A[h]-module. Let (Λ1, E1) ∈
SA
min(h1), (Λ2, E2) ∈ SminA (h2) and set Λ = Λ1 ⊕ Λ2 and E = E1⊕ E2. We
have
[Λ : E] = [Λ1 : E1][Λ2 : E2] = htA(h1) htA(h2).
Therefore htA(h) ≤ htA(h1) htA(h2). Hence htA(h) = htA(h1) htA(h2).
4.3 Computation of the height
For any algebraic number λ ∈ Q, let O(λ) be the ring of integers of the number field Q(λ). It should be noted that λ /∈ O(λ), except if λ is an algebraic integer. Set πλ = {x ∈ O(λ)| xλ ∈ O(λ)}. Then πλ is an integral
ideal and the integer
d(λ) := Card O(λ)/πλ
is its norm.
Let h ∈ GL(V ) as before. For each λ ∈ Spec h, let mλ be its multiplicity.
Note that the functions λ7! mλ and λ7! d(λ) are Gal(Q)-invariant, so they
can be viewed as functions defined over Spec h/Gal(Q). Lemma 13. Assume that h is semi-simple. We have
ht(h) =Q d(λ)mλ,
where the product runs over Spec h/Gal(Q).
eigenvalues λ1, . . . , λn of h are conjugate by Gal(Q), and the
correspond-ing Gal(Q)-orbit is denoted by λ. Under these simplifycorrespond-ing hypotheses, the formula to be proved is
ht(h) = d(λ).
Step 1: Proof that ht(h) ≥ d(λ). Set K = Q(λ1, . . . , λn), let U = K ⊗ V , let
˜
h = 1 ⊗ h be the extension of h to U and let {v1, . . . , vn} be a K basis of U
such that ˜h.vi = λivi. We have U = ⊕1≤i≤nUi, where Ui = K vi.
Let O be the ring of integers of K. For each 1 ≤ i ≤ n, set ˜hi = h|Ui.
Since each Ui is a O[˜h]-module, Lemma 12 shows that
htO(˜h) =Q1≤i≤n htO(˜hi).
Next, the integers htO(˜hi) are computed. Let ( ˜Λ, ˜E) ∈ SminO (˜hi). Since
Ui = Kvi, we have ˜Λ = πvi for some invertible ideal π ⊂ K. By minimality,
it is clear that ˜E = {v ∈ ˜Λ| ˜hiv ∈ Λ}. Therefore we have ˜E = π πλivi. It
follows that [ ˜Λ : ˜E] = [π : ππλi] = [O : Oπλi] = [O(λi) : πλi] [K:Q(λi)] = d(λ)[K:Q(λi)].
Since [Q(λi : Q] = n, we have htO(˜hi) = d(λ)[K:Q]/n, and therefore
htO(˜h) =QihtO(˜hi) = d(λ)[K:Q].
Now let (Λ, E) ∈ Smin(h). Set ˜Λ = O ⊗ Λ and ˜E = O ⊗ E. We have
[ ˜Λ : ˜E] = [Λ : E][K:Q], and therefore
ht(h)[K:Q] = [ ˜Λ : ˜E] ≥ htO(˜h) = d(λ)[K:Q].
Thus the inequality ht(h) ≥ d(λ) is proved.
Step 2: Proof that ht(h) ≤ d(λ). Set L = Q[h] ⊂ End V . Since it is assumed that V is a simple Q[h]-module, L is field extension of Q isomorphic Q(λ1).
Since dimKV is a one dimension, V can be identified with L, and h can
be be identified with the mutiplication by λ1. Under these identifications,
(O(λ1), πλ1) belongs to S(h). Thus we have
d(λ) = [O(λ1) : πλ1] ≥ ht(h).
Remark In number theory, the Weil height of an algebraic number λ is H(λ) = d(λ)1/nθ, where θ involves the norms at infinite places. Therefore ht(h) is essentially the Weil’s height of h, up to the factor at infinite places.
uniquely defined by the following three conditions: hs and hu commutes, hs
is semi-simple and hu is unipotent.
Lemma 14. We have
ht(h) = ht(hs).
Proof. Lemma 12 shows that ht(h) ≥ ht(hs), so it is enough to prove the
opposite inequality. Using the same lemma, it can be assume that the Q[h]-module V is indecomposable. Set L = Q[hs] ⊂ EndV , let λ be the
multipli-cation by hs in L and let n be the lenght of the Q[h]-module V . Then L is
a number field, and there is an isomorphism V ' L ⊗ Q[t]/(tn),
relative to which hs can be identified with λ⊗ and hu with 1 + 1 ⊗ t and
where n = dimL V . Set Λ = O(λ), E = πλ. We have [Λ : E] = d(λ), so by
Lemma 13, the couple (Λ, E) belongs to Smin(λ). Set ˜Λ = Λ ⊗ Z[t]/(tn) and
˜
E = E ⊗ Z[t]/(tn). Then we have
( ˜Λ, ˜E) ∈ Smin(hs) ∩ S(h)
from which it follows that ht(h) ≤ ht(hs).
4.4 A simple criterion of minimality
Assume that h is semi-simple. Let P (t) be its minimal polynomial, let P = P1. . . Pk be the factorization of P into irreducible factors, where k is the
number of factors. For 1 ≤ i ≤ k, set Ki = Q[t]/(Pi(t)). Since Ki is a
number field, its ring of integers Oi is well defined. Set O(h) = ⊕1≤i≤kOi.
As before, we have h /∈ O(h), except if all eigenvalues of h are algebraic integers.
Lemma 15. Let Λ be a lattice of V which is an O(h)-module. Then Λ is minimal relative to h.
Proof. For each 1 ≤ i ≤ k, let eibe the unit of Oiand set Λi = eiΛ, Vi = eiV .
We have Λ = ⊕1≤i≤kΛi. Thus by Lemma 12, it is enough to show that each
lattice Λi of Vi is minimal relative to h|Vi.
Therefore, it can be assumed that k = 1. Let λ be an eigenvalue of K and let mλ be its multiplicity. Then K = Q[h] is a subfield of End V isomorphic
Set E = πλΛ. Since Λ is a O(h)-module, we have E ⊂ Λ and h.E ⊂ Λ,
so (Λ, E) belongs to S(h). Since the rank of the O(h)-module Λ is mλ, we
have [Λ : E] = d(λ)mλ. It follows from Lemma 13 that [Λ : E] = ht(h), and
therefore Λ is minimal relative to h.
4.5 O(h)-lattices
As before, let V be a finite dimensional vector space over Q and let h ∈ GL(V ) be semi-simple. Let K ⊃ Spec h be a Galois extension of Q, let O be the ring of integers of K and set U = K ⊗ V . Let
U = ⊕λ∈Spec hUλ
be the eigenspace decomposition of U .
Lemma 16. Let ˜Λ ⊂ U be a lattice. Assume that (i) ˜Λ is an O-module,
(ii) ˜Λ is Gal(K/Q)-module,
(iii) ˜Λ = ⊕λ∈Spec hΛ˜λ, where ˜Λλ = ˜Λ ∩ Uλ.
Then H0(Gal(K/Q), ˜Λ) is a O(h)-lattice of V .
Proof. Since H0(Gal(K/Q), K ⊗ V ) = V , it follows that H0(Gal(K/Q), ˜Λ)
is a lattice of V . One only needs to prove that it is an O(h)-module.
For each µ ∈ Spec h/Gal(K, Q), set ˜Λ(µ) = ⊕λ∈µΛ˜λ. It is enough to
prove the lemma for each component ˜Λ(µ) of ˜Λ. Equivalently, it can be assumed that Spec h consists of only one Gal(K, Q)-orbit.
Set Spec h = {λ1, . . . , λn}, where n = Card Spec h. Let ei : ˜Λ! ˜Λ be the
projection on the component ˜Λλi relative to the decomposition
˜
Λ = ⊕λ∈µΛ˜λi.
Set ˜O = ⊕1≤i≤nOei. The stabilizer λ1 is the subgroup Gal(K/Q(λ1)) in
Gal(K/Q). Since Gal(K/Q) permutes the component of ˜O, we have H0(Gal(K/Q), ˜O) = H0(Gal(K/Q(λ
1), ˜Oe1) = O(λ1) = O(h).
Since ˜Λ is a ˜O-module, it follows that H0
(Gal(K/Q), ˜Λ) is a O(h)-module. Let H ⊂ GL(V ) be a commutative reductive algebraic subgroup defined over Q. Let X(H) be its character group, let K be a its splitting field, let O be the ring of integers of K and set U = K ⊗ V . Let
U = ⊕χ∈X(H)Uχ
Lemma 17. Let ˜Λ ⊂ U be a lattice. Assume that (i) ˜Λ is an O-module,
(ii) ˜Λ is Gal(K/Q)-module,
(iii) ˜Λ = ⊕χ∈X(H)Λ˜χ, where ˜Λχ = ˜Λ ∩ Uχ.
Then H0(Gal(K/Q), ˜Λ) is a O(h)-lattice of V , for any h ∈ H.
Proof. Since K is the spitting field of H, we have Spec h ⊂ K. Let U = ⊕λ∈Spec hEλ be the eigenspace decomposition of U relative to h. For each
λ ∈ Spec h, we have
Eλ = ⊕χ(h)λUχ.
Therefore ˜Λ = ⊕λ∈Spec hΛ∩E˜ λ. It follows from Lemma 16 that H0(Gal(K/Q), ˜Λ)
is a O(h)-lattice.
4.6 Minimal lattices with a ring structure
As before, let V be a finite dimensional vector space over Q and let h ∈ GL(V ). Assume moreover that V is endowed with an algebra structure, rel-ative to which h is an algebra isomorphism. In this subsection we investigate how to find lattices which are both minimal relative to h and subrings. The difficulty comes from the fact that minimal lattices are not stable by tensor products, i.e. if Λi is minimal relative to hi for i = 1 or 2, usually Λ1⊗ Λ2
is not minimal relative to h1⊗ h2. That’s why the convoluted Lemma 16 is
used.
More precisely, the question investigated in this section involves two re-finements. First, we are looking at ring structures over an operad instead of ordinary ring structures. Second, we are interested in lattices which are minimal relative to a family of ring automorphisms instead of a single auto-morphism, and the refined Lemma 17 is used.
After this introduction, some precise definitions and hypotheses are now provided. Let E = (E (n))n≥1 be a linear operad over Q, see e.g. [17] for the
definition of an operad. We assume that dim E (n) < ∞ for any n and that E(1) = Q1 has dimension 1, where 1 is the identity. The main axiom of the operad is the existence of linear maps
Θk,m1,...,mk : E (k) ⊗ E (m1) ⊗ · · · ⊗ E (mk)! E(m1+ · · · + mk)
satisfying some axioms, see [17] for more details. Let V be a vector space over Q. A vector space V endowed with a family of linear maps satisfying the obvious compatibility conditions is called an E -algebra.
Θk,m1,...,mk(EZ(k) ⊗ EZ(m1) ⊗ · · · ⊗ EZ(mk)) ⊂ EZ(m1+ · · · + mk),
for all (k, m1, . . . , mk). The notion of EZ-ring is defined as before, except that
the structure is defined on a abelian group Λ instead of a vector space V . A finite dimensional E -algebra V is called nipotent if the maps E (n) ! Hom(V⊗n, V ) are zero whenever n 0. Assume now that V is nilpotent. It is clear that V contains a lattice Λ which is a EZ-ring. Let G = Aut V be the group of automorphism of the E -algebra V , and let H ⊂ G be a reductive commutative algebraic subgroup defined over Q.
Lemma 18. Let S ⊂ V be a finite set. Then there exists a lattice Λ of V such that
(i) Λ contains S, (ii) Λ is a EZ-ring, and
(iii) Λ is a O(h)-module, for any h ∈ H(Q).
Proof. First some notations are introduced. Let K be the splitting field of H, let O be its ring of integer and let X(H) be the character group of H. Set U = K ⊗ V , let
U = ⊕χ∈XUχ
be the weight space decomposition of U . For each χ ∈ X, let eχ : U ! Uχ
be the corresponding projection. Let L be any lattice of V containing S. Set ˜
L = ⊕χ∈XL˜χ,
where ˜Lχ = eχ(O ⊗ L), for any χ ∈ X. Note that U is a finite dimensional
nilpotent E -algebra and let ˜Λ be the EZ-subring generated by ˜L.
Next, we claim that ˜Λ satisfies the hypotheses of Lemma 17. For any µ ∈ E (n), let µ : U⊗n ! U be the corresponding operation. Since h is an isomorphism of E -algebra, we have
µ(Uχ1 ⊗ · · · ⊗ Uχn) ⊂ Uχ1...χn,
for any χ1, . . . , χn ∈ X and where the group structure on X is noted
multi-plicatively. Since we have ˜ Λ =P n≥1 P µ∈EZ(n) µ( ˜L n) =P n≥1 P χ1,...,χn P µ∈EZ(n) µ( ˜Lχ1 ⊗ · · · ⊗ ˜Lχn),
it follows that ˜Λ = ⊕χ∈XΛ˜χ, where ˜Λχ = ˜Λ ∩ Uχ, i.e. the hypothesis (iii) of
Lemma 17 is satisfied.
Since it is obvious that it is a Gal(K, Q)-stable O-module, ˜Λ satisfies all hypotheses of Lemma 17. Set Λ = H0(Gal(K/Q), ˜Λ). Then it is clear that
5
The Campbell-Hausdorff lattices
In this chapter, we recall Malcev’s Theorem. Then we collect some related results, which are due to Malcev or viewed as folklore results. Then it is easy to characterize the self-similar data for FGTF nilpotent groups. The end of the section is devoted to a more subtle point, namely the existence of Campbell-Hausdorff lattices which are minimal relative to a commutative reductive group H of automorphisms.
5.1 The Campbell-Hausdorff formula
Let L(2) be the free Lie algebra over Q generated by the Lie variables X1, X2.
It admits a grading
L(2) = ⊕r,sLr,s(2)
where Lr,s(2) is the space of Lie polynomials which are homogenous of degree
r in X1 and s in X2. Let
L(2) = Q
r,s Lr,s(2)
be the completion, for some suitable topology, of the Lie algebra L(2). Indeed L(2) is endowed with a group structure, where the product X.Y of any two elements X, Y ∈ L(2) is defined by the Campbell-Hausdorff formula
X.Y = X + Y +P
r,s≥1 Hr,s(X, Y )
where Hr,s(X1, X2) belongs to Lr,s(2). For example, we have H1,1(X, Y ) =
1/2[X, Y ] and H2,1(X, Y ) = 1/12[X[X, Y ]]. There is a very large literature
on this subject, for example see [5] for some explicit formula for the Lie polynomials Hr,s.
The Campbell-Hausdorff formula has a meaning in different contexts, see [5]. For a nilpotent Lie algebra n, the series is finite and it provides an algebraic group structure on n.
5.2 Three types of lattices
Let n be a nilpotent Lie algebra over Q. So n admits two group structures, the addition and the the Campbell-Hausdorff product. To avoid confusion, the Campbell-Hausdorff product is called the multiplication and it is denoted accordingly.
A finitely generated multiplicative subgroup Γ is called a multiplicative lattice if Γ mod [n, n] generates the Q-vector space n/[n, n], or, equivalently, if Γ generates the Lie algebra n. Let N be the CSC nilpotent Lie group with Lie algebra nR = R ⊗ n. A discrete subroup Γ of N is called a cocompact
lattice if N/Γ is compact.
It should be noted that three distinct notions of lattices will be used in the sequel: the additive lattices, the multiplicative lattices and the cocompact lattices. When it is used alone, a lattice is always an additive lattice. This very commoun terminology could be confusing, because the multiplicative lattices or a cocompact lattices are not necessarily lattices: the reader should read ”multiplicative lattice” or ”cocompact lattice” as single words.
5.3 Malcev’s Theorem
Obviously, any multiplicative lattice Γ of a finite dimensional nilpotent Lie algebra over Q is a FGTF nilpotent group. Conversely, Malcev proved in [18]
Malcev’s Theorem. Let Γ be a FGTF nilpotent group.
1. There exists a unique nilpotent Lie algebra n over Q wich contains Γ as a multiplicative lattice.
2. There exists a unique CSC nilpotent Lie group N which contains Γ as a cocompact lattice.
3. The Lie algebra of N is R ⊗ n.
The Lie algebra n of the previous theorem will be called the Malcev Lie algebra of Γ. When Γ = Z, then n = Q and N = R. In general, it is convenient to denote n as Q ⊗ Γ.
5.4 The coset index
From now on, let n will be a finite dimensional nilpotent Lie algebra. The coset index, which is defined now, generalizes the notions of index for additive lattices and for multiplicative lattices.
Recall that the nilpotency index of n is the smallest integer n such that Cn+1n = 0, where (Cnn)
n≥0 is its descending central series. The following
lemma is folklore, and its proof is done by induction on the nilpotency index of n.
(i) Γ ⊂ Λ, and
(ii) Γ is a finite union of Λ0-cosets.
A subset X ⊂ n is called a coset if it is a Λ-coset for some additive lattice Λ of n. Let F be the set of all finite union of cosets in n. Given X ⊃ Y ∈ F , there is a lattice Λ such that X and Y are both finite union of Λ-coset. The coset index of Y in X is the number
[X : Y ]coset = Card X/ΛCard Y /Λ
The numerator and denomanitor of the previous expression depends on the choice of Λ, but [X : Y ]coset is well defined. In general, the coset index is not
an integer. Obviously if Λ ⊃ Λ0 are additive lattices in n, we have [Λ : Λ0]coset = [Λ : Λ0].
Similarly, for multiplicative lattices there is
Lemma 20. Let Γ ⊃ Γ0 be multiplicative lattices in n, we have [Γ : Γ0]coset = [Γ : Γ0].
The proof, done by induction on the nipotency index of n, is skipped. As a consequence, if Γ ⊃ Γ0 are both additive and multiplicative lattices, their index [Γ : Γ0] is unambiguously defined.
5.5 Morphims of FGTF nilpotent groups
Lemma 21. Let Γ, Γ0 ⊂ n be multiplicative lattices in n and let f : Γ0 ! Γ
be a group morphism. Then f extends uniquely to a Lie algebra morphism ˜
f : n! n. Moreover ˜f is an isomorphism if f is injective.
When f is an isomorphism, the result is due to Malcev, see [18], Theorem 5. Obviously Malcev’s proof holds when f is injective. In general, the lemma is a folklore result. It is implicitely used in Homotopy Theory, see e.g. [1] but we did not found a precise reference. So a proof, essentially based on Hall’s collecting formula (see Theorem 12.3.1 in [15]), is now provided.
Proof. Step 1. Let x ∈ n. We have xn = nx for any n ∈ Z. Since Γ contains
an additive lattice by Lemma 19, we have mZx ⊂ Γ for some m > 0. Thus there is a unique map ˜f : n ! n extending f such that ˜f (nx) = n ˜f (x) for any x ∈ n and n ∈ Z. It remains to prove that
˜
Step 2. Let n be the nilpotency index of n. Set L(2, n) = L(2)/Cn+1L(2),
where L(2) denotes the free Lie algebra over Q generated by X1, X2. Let
Γ(2, n) ⊂ L(2, n) be the multiplicative subgroup generated by X1 and X2.
As before, m(X1+ X2) and m[X1, X2] belongs to Γ(2, n) for some m > 0.
Thus there are two elements w1, w2 of the free group over two generators,
such that
w1(X1, X2) = m(X1+ X2) and w2(X1, X2) = m[X1, X2].
Since L(2, n) is a free in the category of nilpotent Lie algebras of nilpo-tency index ≤ n, we have
w1(x, y) = m(x + y) and w2(x, y) = m[x, y]
for any x, y ∈ n. From this it follows easily that ˜f is a Lie algebra morphism.
5.6 Self-similar data for FGTF nilpotent groups
Let Γ ⊃ Γ0 be multiplicative lattices of n, let f : Γ0 ! Γ be a group morphism and let ˜f : n! n be its extension.
Lemma 22. The virtual endomorphism (Γ0, f ) is a self-similar datum iff ˜f belongs to S(n). In particular ˜f is an isomorphism.
Moreover, if (Γ0, f ) is a fractal datum, then all x ∈ Spec ˜f−1 are algebraic integers.
Proof. Let z be the center of n and let Z(Γ0) be the center of Γ0. Since Γ0 contains a set of generators of n, we have Z(Γ0) = Γ0∩z. Moreover by Lemma 19, Z(Γ0) is an additive lattice of z.
Let (Γ0, f ) be a virtual endomorphism and let K be the f -core. Since any non-trivial normal subgroup of Γ0 intersects Z(Γ0), it follows that
First let’s assume that the virtual endomorphism (Γ0, f ) is self-similar, i.e. that K is trivial. Since Z(Γ0) contains no normal subgroup invariant by f , we Kerf ∩Z(Γ0) = 1 and f is injective. By Lemma 21, ˜f is an isomorphism and therefore ˜f (z) ⊂ z.
Let’s assume by contradiction that ˜f /∈ S. Then there is a nonzero finitely generated subgroup E ⊂ z such that ˜f (E) ⊂ E. There is m > 0 such that mE ⊂ Z(Γ0). Then mE is a f -invariant normal subgroup in Γ0, which contradisct that K is trivial.
Moreover let’s assume that (Γ0, f ) is a fractal datum. Let Λ be the addi-tive lattice generated by Λ. Since
˜
f−1(Λ) ⊂ Λ, all x ∈ Spec ˜f−1 are algebraic integers.
5.7 The Campbell-Hausdorff operad CH ⊂ Lie
Let X1, X2, . . . be an infinite set of Lie variables. For n ≥ 1, let (Lie(n))
be the space of multilinear Lie polynomials over X1, . . . , Xn over the field
Q. The family Lie of vector spaces (Lie(n))n∈Z is the Lie operad over Q.
The Lie operad has been investagated by many authors, see e.g. [17] for an account. As an operad, it is generated by Lie(2) = Q [X1, X2]. Let LieZ be
the Z-suboperad generated by [X1, X2]. It is a Z-form of the operad Lie.
The Campbell-Hausdorff operad CH, which is defined below, is another Z-form of the operad Lie. For any integers r, s ≥ 1, let Br,s ∈ Lie(r + s) be
uniquely defined by the following two conditions
(i) Br,s is symmetric into the first r variables and the last s variables, and
(ii) Br,s(X1, . . . , X1, X2, . . . , X2) = Hr,s(X1, X2),
where, in the previous expression, the variable X1 is repeated r times and X2
is repeated s times. For example, B2,1(X1, X2, X3) = 1/24([X1, [X2, X3]] +
[X2, [X1, X3]]). By definition, the Campbell-Hausdorff operad CH is the
Z-suboperad of Lie generated by all elements Br,s. We have
LieZ ⊂ CH ⊂ Lie,
Since it is clear that CH(n) is finitely generated as an abelian group, CH(n) is a lattice of the Q-vector space Lie(n) for any n ≥ 1. Finding an explicit basis of the Z-module CH(n) looks difficult, and of no interest here.
A additive lattice Λ ⊂ n is called a Campbell-Hausdorff lattice if Λ is a CH-subring (i.e. stable by all operations Br,s). Thus a Campbell-Hausdorff
lattice is both an additive lattice and a multiplicative lattice.
5.5 Minimal Campbell-Hausdorff lattices
Set G = Aut n and let H ⊂ G be a reductive commutative subgroup defined over Q.
Lemma 23. Let Γ be a multiplicative lattice of n. Then there exists a Campbell-Hausdorff lattice Λ and an integer n > 0 such that
In particular, n contains some Campbell-Hausdorff lattices which are O(h)-module, for any h ∈ H(Q)).
Remark: In general, the Lie bracket and other higher products in Λ are not O(h)-linear.
Proof. Let S be a finite set of generators of Γ. By Lemma 18, there exists a Campbell-Hausdorff lattice Λ and an integer n > 0 such that
(i) Λ ⊃ S, and
(ii) Λ is an O(h)-module, for any h ∈ H(Q)).
Since Λ is a multiplicative group, we have Λ ⊃ Γ. Moreover, by Lemma 19, Γ is a finite union of cosets, therefore it is a finite union of nΛ-cosets for some n > 0.
6
Relative complexity of multiplicative
lat-tices
This chapter is the mutiplicative analogue of ch.4. The main result is the refined criterion of minimality. Together with Theorem 1, they are the main ingredients of the proof of Theorem 2.
Throughout the whole chapter, n is finite dimensional nilpotent Lie alge-bra over Q, and z is its center.
6.1 Complexity of multiplicative lattices
Let f ∈ Aut n and let Γ be a multiplicative lattice of n. The complexity of Γ relative to f is the integer
cpf(Γ) = [Γ : Γ0],
where Γ0 = Γ ∩ f−1(Γ). The multiplicative lattice Γ is called minimal relative to f if cpf(Γ) = ht(f ). When Γ is both an additice and multiplicative lattice, it follows from Lemma 20 that the complexity of Γ as an additive lattice or as an multiplicative lattice is the same. Therefore the notation cpf(Γ) is unambiguous.
Lemma 24. Let Γ be multiplicative lattices of n. Then we have cpf(Γ) ≥ ht(f ).
Proof. Let 0 = z0 ⊂ z1 ⊂ . . . zk = n be the ascending central series of n, i.e.
set Γi = Γ ∩ zi, Γ0i = Γ 0∩ z
i. Also for 0 < i ≤ k, set zi = zi/zi−1, Γi = Γi/Γi−1,
and Γ0i = Γ0i/Γ0i−1. It is clear that f (zi) ⊂ zi. Thus let fi : zi ! zi be the
induced map.
It is clear that Γ0i ⊂ Γi are additive lattices of zi and fi(Γ 0 i) ⊂ Γi. By definition, we have [Γi : Γ 0 i] ≥ ht(fi) Note that [Γ : Γ0] =Q 1≤i≤k[Γi : Γ 0 i],
and by Lemmas 13 and 14, we have ht(f ) =Q
1≤i≤k ht(fi),
and therefore [Γ : Γ0] ≥ ht(f ).
Remark: when f is semi-simple, the Lemma 23 shows that there exists a multiplicative lattice Λ with cpfΛ = ht(f ). Therefore a lattice of minimal complexity is the same that a minimal lattice relative to f .
6.2 A property of the minimal multiplicative lattices Let f ∈ Aut n and let Γ be a multiplicative lattice of n.
Lemma 25. Assume that Γ is minimal relative to f and let Γ0 = Γ ∩ f−1(Γ). Then the virtual morphism (Γ0, f ) is good.
Proof. Let Γ0, Γ1, . . . be the multiplicative lattices inductively defined by
Γ0 = Γ and Γn+1 = Γn∩ f−1(Γn). By hypothesis we have
[Γ0 : Γ1] = ht(f ).
By Lemma 3, the sequence [Γn : Γn+1] is not decreasing. By Lemma 24,
we have [Γn : Γn+1] ≥ ht(f ). Therefore [Γn : Γn+1] = ht(f ) for all n. Thus
the virtual endomorphism (Γ0, f ) is good.
Remark: indeed the converse is also true. If (Γ0, f ) is good, then Γ is minimal relative to f . Since we will not use it, the proof is skipped.
6.3 A refined criterion of minimality
A refined version of Lemma 15 is now provided. Let Γ be a multiplicative lattice in n and let h ∈ Aut n be semi-simple. Let L be the field generated by Spec h, let O be its ring of integers and let P be the set of prime ideals of O.
Let Λ be a lattice which is an O(h)-module. Assume moreover that Λ ⊃ Γ and Γ is an union of nΛ-cosets
Lemma 26. Let S be the set of divisors of n in P. Assume that λ ≡ 1 mod nOπ,
for any λ ∈ Spec h and any π ∈ S. Then Γ is minimal relative to h.
Proof. Step 1. Since Spec h lies in Oπ for all π ∈ S, there exists a positive
integer d, which is prime to n, such that dλ ∈ O for all λ ∈ O. Moreover we can assume that d ≡ 1 mod n.
Let λ ∈ Spec h. We have dλ ≡ 1 mod nOπ for all π ∈ S. Therefore we
have
dλ ∈ 1 + nO,
for all λ ∈ Spec h. Set H = dh. Since Spec dH and Spec (H − 1)/n lie in O, it follows that
H ∈ O(h) and H ∈ 1 + nO(h).
Step 2. Set Λ0 = Λ ∩ h−1Λ. Since all eigenvalues of h are units in Oπ
whenever π divides n, the height of h is prime to n. By Lemma 15, we have [Λ : Λ0] = ht(h). Therefore we get Λ = Λ0+ nΛ. It follows that Γ = ` 1≤i≤k gi+ nΛ
for some g1, ..., gk ∈ Λ0, where k = [Γ : nΛ] and where` is the symbol of the
disjoint union. Since H(gi) ≡ gimod nΛ, we get that h(gi) ∈ gi + nΛ ⊂ Γ.
Therefore we have
Γ0 ⊃ `
1≤i≤k
gi+ nΛ0,
Therefore we have [Γ0 : nΛ0] ≥ k = [Γ : nΛ]. It follows that [Γ : Γ0] ≤ [nΛ : nΛ0] = ht(h).
Thus by Lemma 24 we have [Γ : Γ0] = ht(h) and Γ is minimal relative to h.
7
Proof of Theorems 2 and 3
7.1 Proof of Theorem 2. Let n be a finite dimensional nilpotent Lie algebra defined over Q and let z be its center. Set G = Aut n be the algebraic group of automorphisms of n. As before S(n) is the set of all f ∈ G(Q) such that Spec f |z contains no algebraic integers.
Theorem 2. The following assertions are equivalent (i) The group Γ is transitive self-similar,
(ii) the group Γ is level transitive self-similar, and (iii) the Lie algebra nC admits a special grading.
Proof. Let consider the following assertion (A) S(n) 6= ∅
The implication (ii) ⇒ (i) is tautological. Together with the Lemmas 7(i) and 10, the following implications are already proved
(ii) ⇒ (i) ⇒ (A) ⇔ (iii). Therefore, it is enough to prove that A ⇒ (ii).
Step 1. Definition of some h ∈ G(Q). Let’s assume that S(n) 6= ∅, and let f ∈ S(n). Since the semi-simple part of f is also in G, it can be assumed that f is semi-simple. Let K ⊂ G be the Zariski closure of the subgroup generated by f and set H = K0.
There is a lattice Λ which is a O(h)-module for any h ∈ H(Q) and an integer n such that Λ ⊃ Γ and Γ is an union of nΛ-coset. Indeed the statement is an easy case of Lemma 23.
Let X(H) be the group of characters of H, let K be the splitting field of H, let O be the ring of integers of K, let P be the set of prime ideals of O and let S be set set of all π ∈ P dividing n.
By Theorem 1, there exists h ∈ H(Q) such that, for any non-trivial χ ∈ X we have
(i) χ(h) is not an algebraic integer, and (ii) χ(h) ≡ 1 mod nOπ for any π ∈ S.
Step 2. Let Γ0 = Γ ∩ h−1(Γ). We claim that the virtual morphism (Γ0, h) is a good self-similar datum.
It follows from Lemma 26 that the virtual endomorphism (Γ0, h) is good. Moreover, let Ω0 be the set of weights of H over zQ. There is an integer
l such that fl ∈ K0 = H. The spectrum of fl on zQ are the numbers χ(fl)
when χ runs over Ω0. Thus it follows that Ω0 does not contain the trivial
character, hence h belongs to S(n).
Therefore by Lemma 22, the virtual endomorphism (Γ0, h) is a good self-similar datum. Thus by Lemma 4, Γ is a level transitive self-self-similar group.
Let N be a CSC nilpotent Lie group N and let Γ be a cocompact lattice. The manifold M = N/Γ is called a nilmanifold.
The diffeomorphism f : M ! M is called an Anosov diffeomorphism if (i) there is a continuous splitting of the tangent bundle T M as T M = Eu⊕ Es which is invariant by df , and
(ii) there is a Riemannian metric relative to which df |Es and df
−1| Eu are
contracting.
For any x ∈ M f induces a group isomorphism f∗ : π1(X, x)! π1(M, f (x)).
Indeed we have π1(X, x) ' π1(M, f (x)) ' Γ, but these isomorphisms are only
unique up to an inner automorphism. Since any isomorphisms of Γ extends to an automorphism of nR = LieN by Lemma 21, the corresponding
auto-morphism f∗ ∈ Aut nR is defined up to an inner automorphism. Since f∗ is
defined modulo the unipotent radical of Aut nR, the set Spec f∗of eigenvalues
f∗ is well defined.
Manning’s Theorem. The set Spec f∗ contains no root of unity.
Originally the theorem has been proved in [18]. Later on, A. Manning proved a much stronger result. Indeed Spec f∗ contains no eigenvalues of
ab-solute value 1, and f is topologically conjugated to an Anosov automorphism, see [19].
7.3 A Corollary for nilmanifolds with an Anosov diffeomorphim
Corollary 1. Let M be a nilmanifold. If M admits an Anosov diffeomor-phism, then its fundamental group π1(M ) is level transitive self-similar.
Proof. By definition, we have M = N/Γ, where N is a CSC nilpotent Lie group and Γ ' π1(M ) is a cocopact lattice. Set nR = Lie N and nC =
C ⊗ nR. By Manning’s Theorem and Lemma 9, nChas a very special grading. Therefore Γ is level transitive self-similar by Theorem 2.
7.4 Characterisation of fractal FGTF nilpotent groups The fractal FGTF nilpotent groups are characterized by the next Theorem 3. The proof follows the same pattern than Theorem 2, but, indeed, every step is elementary.
Let nQ be a finite dimensional nilpotent Lie algebra and let Γ be a
Theorem 3. The following assertions are equivalent (i) The group Γ is fractal
(ii) nC admits a non-negative special grading.
(iii) nQ admits a non-negative special grading.
Proof. It follows from Lemma 7 that Assertions (ii) and (iii) are equivalent. Proof that (i) ⇒ (ii). By assumption, there is a fractal datum (Γ0, f ). Let g : Γ! Γ0 be the inverse of f . We will also denotes by f and g the automor-phisms of n extending f and g.
Let Λ ⊂ n be the additive subgroup generated by Γ. By Lemmas 19, Λ is an additive lattice. Since we have g(Λ) ⊂ Λ, it follows that all eigenvalues of g are algebraic integers.
Moreover (Γ0, g−1) is a self-similar datum, thus Spec g−1|z contains no
root of unity. Therefore, by Lemma 11, Assertion (ii) holds. Proof that (iii) ⇒ (i). Let’s assume Assertion (iii) and let
nQ = ⊕k≥0nQk
be a non-negative special grading of nQ.
By Lemma 19, there are lattices Λ ⊃ Λ0 of nQ such that and Γ ⊂ Λ and
Γ is an union of Λ0-cosets. Since it is possible to enlarge Λ, we can assume that
Λ = ⊕k≥0Λk,
where Λk= Λ ∩ nQk. There is an integer d ≥ 1 such that d.Λ ⊂ Λ0. Therefore
Λ ⊃ Γ and Γ is an union of dΛ-cosets.
Let g be the automorphism of nQ defined by g(x) = (d + 1)kx if x ∈ nQ k.
We claim that g(Γ) ⊂ Γ. Let x ∈ Γ and let x = P
k≥0 xkbe its decomposition
into homogenous components. We have g(x) = x +P
k≥1((d + 1)
k− 1)x k.
By hypothesis each homogenous component xkbelongs to Λ. Since (d+1)k−1
is divisible by d, we have g(x) ∈ x + dΛ ⊂ Γ and the claim is proved.
8
Examples of not self-similar FGTF
nilpo-tent groups
This section provides an example of a FGTF nilpotent group which is not even self-similar, see subsection 8.6. The end of the section is about the Milnor-Scheuneman conjecture.
8.1 FGTF nilpotent groups with rank one center
Let Γ be a FGTF nilpotent group and let Z(Γ) be its center.
Lemma 27. Let’s asssume that Γ is self-similar and Z(Γ) ' Z. Then Γ is transitive self-similar.
Proof. Assume that Γ admits a faithful self-similar action on some A∗, where A is a finite alphabet. Let a1, . . . , ak be a set of representatives of A/Γ, where
k is the number of Γ-orbits on A. For each 1 ≤ i ≤ k, let Γi be the stabilizer
of ai. For any h ∈ Γi, there is hi ∈ Γ such that
h(aiw) = aifi(h)(w),
for all w ∈ A∗. Since the action is faithfull hi is uniquely determined and the
map fi : Γi ! Γ, h 7! hi is a group morphism.
Let nQ = Q ⊗ Γ, and let z be its center, and let z 6= 0 be a generator
of ∩iZ(Γi). By Lemma 21, the group morphism fi extends to a Lie algebra
morphism ˜fi : nQ ! nQ, For any 1 ≤ i ≤ k. Since z = Q ⊗ Z(Γ) is one
dimensional, it follows that either ˜fi is an isomorphism or ˜fi(z) = 0. In any
case, we have ˜f (z) = xiz, for some xi ∈ Q. However Zz is not invariant by
all ˜fi, otherwise it would be in the kernel of the action. It follows that at
least one xi is not an integer.
For such an index i, the fi-core of Γi is trivial, and the virtual morphism
(Γi, fi) is a self-similar datum for Γ. Thus Γ is transitive self-similar.
8.2 Small representations
Let N be a CSC nilpotent Lie group with Lie algebra nR and let Γ be a
cocompact lattice.
Lemma 28. If Γ is transitive self-similar, then there exists a faithfull nR
Proof. By hypothesis, Γ is transitive self-similar. By Theorem 2, zC admits
a special grading
nC = ⊕
n∈ZnCn.
Let δ : nC ! nC be the derivation defined by δ(x) = nx if x ∈ n
n. Since
δ|zC is injective, it follows that there is some ∂ ∈ Der nR such that ∂|zR is
injective.
Set mR = R∂ n nR. Relative to the adjoint action, mR is a faithfull zR
-module. Therefore mRis a faithfull nR-module with the prescribed dimension.
8.5 Filiform nilpotent Lie algebras
Let n be a nilpotent Lie algebra over Q. Let Cnn be the decreasing central
series, which is inductively defined by C0n = n and Cn+1n = [n, Cnn]. The
nilpotent Lie algebra n is called filiform if dim C0n/C1n= 2 and dim Ckn/Ck+1n ≤ 1 for any k > 0. Set n = dim n. It follows from the definition that dim Ckn/Ck+1n= 1 for any 0 < k ≤ n − 2 and Ckn = 0 for any k ≥ n − 1.
Lemma 29. Let n be a filiform nilpotent Lie algebra over Q, with dim n ≥ 3. Then its center z has dimension one.
Proof. Let z ∈ n be non-zero. Let k be the integer such that z ∈ Ckn\Ck+1n.
Since Ckn= Ck+1n⊕ Qz
Ck+1n = [n, Ckn] = [n, Ck+1n] + [n, z] = Ck+2n.
It follows that Ck+1n = 0. Therefore z lies in Ckn, which is a one dimensional ideal.
8.6 Benoist Theorem
Benoist’s Theorem. There is a nilpotent Lie algebra nR
B of dimension 11
over R, with the following properties (i) The Lie algebra nR
B has no faithfull representations of dimension 12,
(ii) the Lie algebra nR
B is defined over Q, and
(iii) the Lie algebra nR
B is filiform.
The three assertions appear in different places of [3]. Indeed Assertion (i), which is explicitely stated in Theorem 2 of [3], hold for a one-parameter family of eleven dimensional Lie algebras, which are denoted a−2,1,t in section
when t is rational, a−2,1,t is defined over Q. Therefore the Benoist Theorem
holds for the Lie algebras nB = a−2,1,t where t is any rational number.
8.7 A FGTF group which is not self-similar
Let NB the CSC nilpotent Lie group with Lie algebra nRB. Since nRB is defined
over Q, NB contains some cocompact lattice.
Corollary 2. Let Γ be any cocompact lattice in NB. Then Γ is not
self-similar.
Proof. Let’s assume otherwise. By Benoist Theorem and Lemma 29, the center of nR
B is one dimensional. Thus the center of Γ has rank one, and by
Lemma 27, Γ is transitive self-similar. By Lemma 28, nR
B admits a faithfull
representation of dimension 12, which contradicts Benoist Theorem. Therefore Γ is not self-similar.
8.8 On the Scheuneman-Milnor conjecture
A smooth manifold M is called affine if it admits a torsion-free and flat connection. Scheuneman [28] and Milnor [23] asked the following question
is any nilmanifold M affine?
The story of the Scheuneman-Milnor conjecture is quite interesting. For many years, there are been a succession of proofs followed by refutations, but there was no doubts that the conjecture should be ultimalely proved... until a counterexample has been found by Benoist [3]. Indeed it is an easy corollary of his previously mentionned Theorem.
The following question is still open
if π1(M ) is transitive self-similar, is the nilmanifold M affine?
I believe that the answer should be negative. However the proof in [3] is very specific and difficult. Therefore, I doubt that it can be easily adapted to provide a counterexample to this refined version of the Scheuneman-Milnor conjecture.
9
Absolute Complexities
For the whole chapter, N will be a CSC nilpotent Lie groups, with Lie algebra nR. It is assumed that that nC admits a special grading and that N contains
