Nilpotent Lie Groups and Hyperbolic Automorphisms ”

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Proof. Sice G is semisimple and defined over Q, it has no non-trivial character defined over Q. Then it follows from Theorem 9.4 of [2] that the set of Z-points of G, denoted by G(Z) is a lattice (i.e., a discrete subgroup of G such that G/G(Z) carries a G-invarient finite measure) in G. Note that if we rewrite the representation ρ in terms of the generating set of Γ, it remains a polynomial representation defined over Q because of the rationality assumption on Γ. From now on we are going to consider ρ as this representation only. Now it follows from Proposition 10.13 of [12] that there is a finite index subgroup Gm(Z) of G(Z) such that ρ(Gm(Z)) preserves the lattice Γ. Our theorem will be proved if we can show that Gm(Z) contains some hyperbolic automorphism of N. Let D be a maximal R-split torus in G and let C be the centralizer of D in G. Then C is a Cartan subgroup of G and it follows from Theorem 2.8 of [11] that up to conjugation by an element of G, L = C ∩ Gm(Z) is a uniform lattice in C. Since G is semisimple, without loss of generality, we may assume that C = KD, where K is compact, K and D commute. Then if π denotes the projection map from C to D, π(L) is a uniform lattice in D. By Borel density theorem (see [1] for details), L being a lattice, is Zariski dense in D. Let Dh be the set of elements of D, which acts as hyperbolic automorphisms on N.

Now suppose α ∈ D and as an element of SL(q, R), let α1,...,αq be the e eigenvalues of α. If we consider the action of α as automorphism of N, then it is easy to see that the eigenvalues of α are given by the set

S = {αi1 αi2 ...αin }1≤n≤k. e e e Since α preserves the subspace A and N =∼ N/A, it follows that there is a subset S1 of S such that for the action of α on N, the eigenvalues are given by the elements of the set S1. Let us denote the elements of S1 by s1,s2,...sN . Then

h D = {α ∈ D|si 6=1, 1 ≤ i ≤ N}.

2 Since we have assumed that all the weights of G (which means all the weights of a maximal R-split torus of G) are non-trivial, it follows that Dh is non-empty. Clearly, Dh is a Zariski open subset of D. Then there is some α ∈ L ∩ Dh and the theorem is proved. Corollary 0.2. Along with all the assumptions of Theorem 0.1, let Γ′ be another lattice in N k given by Γ′ = ρ(g)Γ for some g ∈ G. Then there is a hyperbolic automorphism α′ of N such that α′ preserves Γ′.

Proof. By the above theorem, let α be the automorphism of N which preserves Γ. Let α′ = gαg−1. Then it is clear that α′ is a hyperbolic automorphism of N which preserves Γ′. Remark 0.3. If N is a free k-step nilpotent Lie algebra on q generators, then SL(q, R) ⊂ Aut(N) and, therefore, for any rationally deﬁned full latice Γ in N k there is a hyperbolic automorphism of N preserving Γ. Therefore, the corre- sponding Lie group admits expansive automorphisms. If N is a 2-step nilpotent Lie algebra then it admits a positive graduation and then it is easy to produce hyperbolic automorphisms (see [15] for details) of N, and therefore the simply connected Lie group whose Lie algebra is N admits hyperbolic automorphisms. But all simply connected nilpotent Lie groups do not admit hyperbolic automorphism. In [3], an example of a 3-step nilpotent Lie algebra g is discussed, all of whose derivations are nilpotent. Thus, if G is the simply connected nilpotent Lie group with Lie algebra g, then the connected component of Aut(G) is an unipotent group. Since Aut(G) is an algebraic group, it has only ﬁnitely many connected components and hence no automorphism of G is hyperbolic.

Now we discuss an example where the hypothesis of Theorem 0.1 is veriﬁed and which is diﬀerent from the trivial examples discussed in the above remark. Let N be a 2-step nilpotent Lie algebra such that [N,N] has dimension p and U has dimension q. Following [4] and [5], we call N a 2-step nilpotent Lie algebra of type (p, q). It was shown in [4] that any 2-step nilpotent Lie algebra of type (p, q) is isomorphic to a Lie algebra N = Rq ⊕ W , which is called a standard metric nilpotent Lie algebra of type (p, q) (see [4] for details) where W is a p-dimensional subspace of so(q, R). If W = so(q, R), then N is the free 2-step nilpotent Lie algebra on q generators. SL(q, R) acts on Rq ⊕ so(q, R) by automorphism where the action is given as follows: for g ∈ SL(q, R) and (v, w) ∈ Rq ⊕ so(q, R), g(v, w) = (gv,gwgt). Also t SL(q, R)W = {g ∈ SL(q, R): gwg ∈ W, whenever w ∈ W } q acts on R ⊕ W by automorphism. We denote SL(q, R)W by GW . The action of SL(q, R) on so(q, R) given by

g(w)= gwgt (1) for g ∈ SL(q, R) and w ∈ so(q, R), gives rise to the representation ρ of SL(q, R) in SL(so(q, R)), which is an algebraic representation deﬁned over Q. Let g ∈

3 SL(q, R) be given by the matrix (gij ). Let us try to write the matrix of ρ(g) in the standard orthonormal basis of so(q, R). ρ(g) is a d × d matrix where 1 d = 2 q(q − 1). Each row (column) of ρ(g) corresponds to a member of the double indexed set I = {ij : 1 ≤ i < j ≤ q} which has d number of elements q because of the following: if {e1, ..., eq} denotes the standard basis of R , then

eij := [ei,ej ] (1) is one of the elements of the standard orthonormal basis of so(q, R) with respect to the inner product described in [4], and each element of the standard orthonormal basis of so(q, R) is determined by 1. Now let ρ(g)ij be an entry of the matrix ρ(g). Then i corresponds to two rows of the matrix g and we denote them by i1 and i2. Also, j corresponds to two columns of the matrix g which we denote by j1 and j2. Then a straightforward calculation shows that

ρ(g)ij = gi1j1 gi2j2 − gi2j1 gi1j2 . That is ρ(g)ij is the determinant of the 2 × 2 minor of g formed by the iith, i2th rows and j1th, j2th columns of g. We call a p-dimensional subspace W to be standard subspace if W is the span of p elements from the standard orthonormal basis of so(q, R). Now let p = 3 and q be such that q − 3 ≥ 3. Let W be the standard 3- dimensional subspace of so(q, R) spanned by the ﬁrst 3 elements of the standard orthonormal basis of so(q, R) i. e., W is spanned by e12, e13 and e23. Also let Γ= Ze12 ⊕ Ze13 ⊕ Ze23. Then it is easy to see that G = SL(3, R) × SL(q − 3, R) is a subgroup of GW satisfying the hypothesis of Theorem 0.1. Therefore, there is a hyperbolic automorphism α of N = Rq ⊕ W such that α preserves the lattice Γ. Hence the nilpotent Lie group N/Γ= Rq ⊕ W/Γ admits a hyperbolic automorphism. Acknowledgement . The authors would like to thank Prof. S. G. Dani for suggesting [11] and Prof. E. Vinberg for pointing out the mistakes in the previous version. The ﬁrst named author acknowledges the support of National Board for Higher Mathematics, India, through NBHM post-doctoral fellowship during which the major portion of the work is done.

Summary of revisions over the previous version

Note that some mistakes were found in the previous version of this paper titled “ 2-step nilpotent Lie groups and hyperbolic automorphisms ”. The mistakes were found in Lemma 4.2. It was also found out that we need some extra assumption in Theorem 1.1. In the present version, we have put this extra assumption in Theorem 0.1, but we have not restricted ourselves to 2-step nilpotent Lie groups, rather we have considered k-step (k ≥ 2) nilpotent Lie groups. As Lemma 4.2 of the previous version fails to hold, in the present version, we do not use the density arguments as we did in the previous version, to achieve same kind of results for a more general class of nilpotent Lie groups apart from those mentioned in Theorem 0.1 and Corollary 0.2. Therefore, we think the results in Section 3 of the previous version regarding convergence in the space of closed subgroups and the use of them in Section 4 become irrelevant for the present version. So, we have not included those results in the present version, we plan to put those results in a suitable place of a seperate article.

4 References

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5 • Manoj Choudhuri, Institute of Infrastructure Technology Research and Management, Near Khokhara Circle, Maninagar (East), Ahmedabad 380026, Gujarat, India. email: [email protected] • C. R. E. Raja, Statistics and Mathematics Unit, Indian Statistical Institute Bangalore Center, 8th Mile, Mysore Road, R. V. C. E. Post, Bangalore 560059, Karnataka, India. email: [email protected]