Rigidity of certain solvable actions on the torus

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1 Rigidity of certain solvable actions on the torus Masayuki ASAOKA July 24, 2014 Abstract An analog of the Baumslag-Solitar group BS(1, k) acts on the torus naturally. The action is not locally rigid in higher dimension, but any perturbation of the action should be homogeneous. 1 Introduction For integers n 1 and k 2, let Γ n,k be the finitely presented group given by Γ n,k = a, b 1,..., b n ab i a 1 = b k i, b i b j = b j b i for any i, j = 1,..., n. The group Γ 1,k is just the Baumslag-Solitar group BS(1, k) = a, b aba 1 = b k. It acts on the projective line RP 1 = R { } by a x = kx and b x = x+1, where we set c = and + t = for any c 0 and t R. This action preserves the standard projective structure on RP 1. In [2], Burslem and Wilkinson proved a classification theorem of smooth 1 BS(1, k)-action on RP 1. As a corollary, they obtained the following rigidity result. Theorem 1.1 (Burslem and Wilkinson [2]). Any real analytic BS(1, k)-action on RP 1 is locally rigid. In particular, the above projective action is locally rigid. Recall the definition of local rigidity of a smooth action of a discrete group. Let Γ be a discrete group and M a smooth closed manifold. The group Diff(M) of smooth diffeomorphisms is endowed with the C -topology. A Γ-action is a homomorphism from Γ to Diff(M). For a Γ-action ρ and γ Γ, we write ρ γ for the diffeomorphism ρ(γ). By A(Γ, M), we denote the set of smooth Γ-actions on M. This set is endowed with the topology generated by the open basis {O γ,u = {ρ A(Γ, M) ρ γ U}}, where γ and U run over Γ and all open subsets of Diff(M). We say two Γ-actions ρ 1 and ρ 2 are smoothly conjugate if there exists a diffeomorphism h of M such Partially supported by JSPS Grant-in-Aid for Scientific Research (C) 1 The term smooth means of C in this paper. 1

2 that ρ γ 2 = h ργ 1 h 1 for any γ Γ. An Γ-action ρ 0 is locally rigid if it admits a neighborhood in A(Γ, M) in which any action is smoothly conjugate to ρ 0. The above projective BS(1, k)-action on RP 1 can be generalized to Γ n,k - actions on the sphere S n. Let B = (v 1,..., v n ) be a basis of R n. We define an BS(n, k)-action ρ B on S n = R n { } by ρ a B (x) = k x and ρbi B (x) = x + v i for x R n, where c = and + v = for any c 0 and v R n. The sphere S n admits a natural conformal structure and the action ρ B preserves it. In [1], the author of this paper proved that the action ρ B is not locally rigid but it exhibits a weak form of rigidity. Proposition 1.2 ([1]). ρ B and ρ B are smoothly conjugate if and only if there exists a conformal linear transformation T of R n such that T B = B. In particular, ρ B is not locally rigid if n 2. Theorem 1.3 ([1]). There exists a neighborhood of ρ B in A(Γ n,k, S n ) in which any action ρ is smoothly conjugate to ρ B with some basis B. In particular, any Γ n,k -action close to ρ B preserves a smooth conformal structure on S n. In this paper, we prove analogous results for another generalization of the projective BS(1, k)-action on RP 1. Let B = (v 1,..., v n ) be a basis of R n with v j = (v ij ) n i=1. We define a Γ n,k-action ρ B on the n-dimensional torus T n = (R { }) n by ρ a B(x 1,..., x n ) = (k x 1,..., k x n ) ρ b j B (x 1,..., x n ) = (x 1 + v 1j,..., x n + v nj ). Remark that the point x = (,..., ) T n is a global fixed point of the action ρ B. Let G be the subgroup of GL n R consisting of linear transformations f which have the form f(x 1,..., x n ) = (a 1 x σ(1),..., a n x σ(n) ) with real numbers a 1,..., a n 0 and a permutation σ on {1,..., n}. The aim of this paper is to show that the action ρ B is not locally rigid if n 2, but it exhibits rigidity like the above Γ n,k -action on S n. Proposition 1.4. Two actions ρ B and ρ B are smoothly conjugate if and only if B = gb for some g G. In particular, ρ B is not locally rigid if n 2. Theorem 1.5. There exists a neighborhood of ρ B in A(Γ n,k, T n ) in which any action is smoothly conjugate to ρ B for some basis B of R n. The theorem is proved by an application of the method used in [1]. Firstly, we show persistence of the global fixed point x. Next, we reduce the theorem to the corresponding theorem for local actions at the global fixed point. The same argument as in [?], we can see that the theorem for local actions follows from exactness of a finite dimensional linear complex. The exactness can be checked by an elementary computation. 2

3 2 Proof of Theorem Reduction from global to local Let Γ be a discrete group and M a smooth closed manifold. We say that a point x M is a global fixed point of a Γ-action ρ on M if ρ γ (x) = x for any γ Γ. We can apply the following general result on persistence of a global fixed point of Γ n,k -action to the action ρ B. Lemma 2.1 ([1, Lemma 2.10]). Let M be a manifold and ρ be a Γ n,k -action on M. Suppose that ρ 0 has a global fixed point p 0 such that (Dρ a 0) p0 = k 1 I and (Dρ b i 0 ) p 0 = I for any i = 1,..., n. Then, there exists a neighborhood U Hom(Γ n,k, Diff(M)) of ρ 0 and a continuous map ˆp : U M such that ˆp(ρ 0 ) = p 0 and ˆp(ρ) is a global fixed point of ρ for any ρ U. The action ρ B and its global fixed point x satisfy the assumption of the lemma. Hence, any action ρ close to ρ B admits a global fixed point x ρ close to x. A Γ-action with a global fixed point induces a local Γ-action. We define the space of local actions on R n as follows. Let D be the group of germs of local diffeomorphisms of R n fixing the origin. For F D and r 1, we denote the F. It is an element of the vector space S r,n of symmetric r-multilinear maps from (R n ) r to R n. We define a norm (r) on S r,n by r-th derivative of F at the origin by D (r) 0 F (r) = sup{ F (ξ 1,..., ξ r ) ξ 1,..., ξ r R n, ξ i 1 for any i}, where is the Euclidean norm on R n. We also define a pseudo-distance d r on D by r d r (F, G) = D (i) 0 F D(i) 0 G (i). i=1 The pseudo-distance on D induces a topology on D, which we call the Cloc r - topology. Remark that it is not Hausdorff. Let Hom(Γ, D) the set of homomorphisms from Γ to D, which can be regarded as the set of local Γ-actions on (R n, 0). The Cloc r -topology on D induces a topology on Hom(Γ, D) like A(Γ, M). We also call this topology on Hom(Γ, D) the Cloc r -topology. We say two local Γ-actions P 1 and P 2 are smoothly conjugate if there exists H D such that P γ 2 = H P γ 1 H 1 for any γ Γ. Let φ be the local coordinate of T n at x given by φ(x 1,..., x n ) = ( 1 x 1,..., where 1/ = 0. For a basis B of R n, we define a local Γ n,k -action P B by P γ B = φ ργ B φ 1. For each Γ n,k -action ρ close to ρ B, we can take a local coordinate φ ρ close to φ with φ(x ρ ) = 0 so that a local Γ n,k -action given by Pρ γ = φ ρ ρ γ B φ 1 ρ is Cloc 3 -close to ρ B. 3 1 x n ),

4 The following proposition reduces Theorem 1.5 to the corresponding result for local actions. Proposition 2.2. Let ρ be a Γ n,k -action on T n close to ρ B. Suppose that the induced local action P ρ is smoothly conjugate to P B for some basis B of R n. Then, the action ρ is smoothly conjugate to ρ B. The rest of this subsection is devoted to the proof of the proposition. Let B = (v 1,..., v n ) be a basis of R n such that P ρ is smoothly conjugate to P B. For each σ = (σ 1,..., σ n ) {±1} n, there exist integers m σ 1,..., m σ n such that σ i n j=1 mσ j v ij > 0 for any i = 1,..., n. Set b σ = b mσ b mσ n n and vi σ = n j=1 mσ j v ij. Then, we have ρ b σ B (x 1,..., x n ) = (x + v σ 1,..., x n + v σ n). (1) Let m be the maximum of { m i σ σ {±1 n }, i = 1,..., n} and put S = {a ±1 } {b l b l n l i m}. By the assumption of the proposition, there exists a neighborhood U of x and a diffeomorphism h : U h(u) such that x U, h(x ) = x ρ and h ρ γ B = ργ h on U (ρ γ B ) 1 (U) for any γ S. Take an open interval I RP 1 \{0} such that x I n U γ S (ργ B ) 1 (U). Put I 1 = {x I x = or x > 0}, I 1 = {x I x = or x < 0}. and U σ = I σ1 I σn for σ = (σ 1,..., σ n ) {±1} n. Equation (1) implies that ρ b σ B (x)(u σ ) U σ, T n = n 0 (ρb σ B ) n (U σ ), and lim n (ρ b σ B ) n (x) = x for any x T n. For σ {±1} n, let m(x, σ) be the minimal integer m such that (ρ b σ B ) m (x) is contained in U σ. We define a map h σ : T n T n by h σ (x) = (ρ bσ ) m(x,σ) h (ρ b σ B ) m(x,σ) (x). We will show that h σ does not depend on the choice of σ and it is a smooth conjugacy between ρ B and ρ. Lemma 2.3. h σ (x) = (ρ b σ ) m h (ρ b σ B ) m (x) for any m m(x, σ). Proof. Proof is done by induction of m. Suppose that the equation holds for some m m(x, σ). Since (ρ b σ B ) m (U σ ) U σ I n, we have (ρ b σ ) (m+1) h (ρ b σ B ) m+1 (x) = (ρ b σ ) (m+1) (h ρ b σ B ) (ρ b σ B ) m (x) Hence, the required equation holds for m + 1. Lemma 2.4. The map h σ is injective. = (ρ b σ ) (m+1) (ρ b σ h) (ρ bσ B )m (x) = (ρ b σ ) m h (ρ bσ B )m (x). Proof. Take x 1, x 2 T 2 and m = max{m(x 1, σ), m(x 2, σ)}. Then, we have h σ (x i ) = (ρ b σ ) m h (ρ b σ B ) m (x i ). for i = 1, 2. The map in the right-hand side is injective. 4

5 Lemma 2.5. h σ ρ γ B = ργ h σ for any γ Γ. Proof. Fix x T n and take m m(x, σ) such that m m(ρ γ B (x), σ) for any γ S. It is sufficient to show that h σ ρ γ B = ργ h σ for γ {a, b 1,..., b n }. For any i = 1,..., n, the identity b i b j = b j b i implies that h σ ρ b i B (x) = (ρbσ ) m h (ρ bσ ) m ρ b i B (x) The identity ab i = b k i a also implies = (ρ b σ ) m (h ρ b i B ) (ρb σ ) m (x) = (ρ bσ ) m (ρ bi h) (ρ bσ ) m (x) = ρ bi (ρ bσ ) m h (ρ bσ ) m (x) = ρ b i h σ (x). h σ ρ a B (x) = (ρbσ ) km h (ρ bσ ) km ρ a B (x) = (ρ b σ ) km (h ρ a B ) (ρb σ ) m (x) = (ρ b σ ) km (ρ a h) (ρ b σ ) m (x) = ρ a (ρ b σ ) m h (ρ b σ ) m (x) = ρ a h σ (x). For a diffeomorphism f on a manifold M and a hyperbolic fixed point p of f, we denote the unstable manifold of p by W u (p, f) (see e.g., [3] for the definitions and basic results on hyperbolic dynamics). By Fix(f), we denote the set of fixed point of f. For l 0, let Fix l (f) be the set of hyperbolic fixed point of f whose unstable manifold is l-dimensional. The diffeomorphisms ρ a B and ρa B are Morse-Smale diffeomorphisms with the fixed point set {0, } n. For each fixed point p = (p 1,..., p n ) {0, } n, W u (p, ρ a B ) = W 1 W n with W j = R if p j = 0 and W j = { } if p j =. If ρ is sufficiently close to ρ B, then ρ a is a Morse-Smale diffeomorphism and Fix l (ρ a ) has the same cardinality as Fix l (ρ a B ), and hence, as Fix l(ρ a B ) for any l = 0,..., n. By Lemma 2.4 and Lemma 2.5, h σ maps Fix(ρ a B ) to Fix(ρa ) bijectively. Lemma 2.6. For any l = 0,..., n and p Fix l (ρ a B ), h σ(p) is a point in Fix l (ρ a ). Moreover, the restriction of h σ to W u (p, ρ a B ) is a diffeomorphism onto W u (h σ (p), ρ a ). Remark that W u (q, ρ a ) is an (embedded) submanifold diffeomorphic to R l for q Fix l (ρ a ) since ρ a is Morse-Smale. Proof. Take l = 0,..., n and p Fix l (ρ a B ). Notice that W u (p, ρ a B ) U σ is an open subset of W u (p, ρ a B ). Thus, there exists a neighborhood V u of p in W u (p, ρ a B ) such that (ρbσ B )m(p,σ) (V u ) U σ. We have m(y, σ) m(p, σ) for 5

6 any y V u. This implies that h σ = (ρ b σ ) m(p,σ) h (ρ bσ B )m(p,σ) on V u. In particular, the restriction of h σ to V u is a diffeomorphism onto h σ (V u ). Since W u (p, ρ a B ) = m 0 (ρa B )m (V u ), h σ ρ a B = ρa h σ, and h σ is injective, the restriction of h σ to W u (p, ρ a B ) is a diffeomorphism onto h σ(w u (p, ρ a B )). For x W u (p, ρ a B ), we have (ρ a ) m (h σ (x)) = h σ (ρ a B ) m (x) m h σ (p). This implies that h σ (W u (p, ρ a B ) is a subset of W u (h σ (p), ρ a ). In particular, the dimension of W u (h σ (p), ρ a ) is at least l. Since h σ maps the finite set Fix(ρ a B ) to Fix(ρ a ) bijectively and the sets Fix j (ρ a B ) and Fix j(ρ a ) have the same cardinality for each j = 0,..., n, we obtain that h σ maps Fix l (ρ a B ) to Fix l(ρ a ) bijectively. The set h σ (W u (p, ρ a B )) is a ρa -invariant open subset of W u (h σ (p), ρ a ) which contains h σ (p). It should coincide with W u (h σ (p), ρ a ), and hence, the restriction of h σ to W u (p, ρ a B ) is a diffeomorphism onto W u (h σ (p), ρ a ). Lemma 2.7. h σ (p) does not depend on the choice of σ for any p Fix(ρ a B ). Proof. Take l = 0,..., n and p = (p 1,..., p n ) Fix l (ρ B ). Put b p = p i = b i. Then, p is the unique element in Fix l (ρ a B ) which is fixed by ρbp B. By the identity ρ b p h σ = h σ ρ b p B, h σ(p) is the unique element in Fix l (ρ a ) = h σ (Fix l (ρ a B )) which is fixed by ρ bp. Lemma 2.8. The map h σ does not depend on the choice of σ. Proof. Take σ, σ {±1} n and put g = h 1 σ h σ. It is sufficient to show that the restriction g p of g to W u (p, ρ a B ) is the identity map for each p = (p 1,..., p j ) Fix(ρ a B ) = {0, }n. By the above lemmas, g p (p) = p and the restriction of g p is a diffeomorphism of W u (p, ρ a B ) which commutes with ρa B. Recall that ρ a B (x) = kx and W u (p, ρ a B ) is naturally identified with a vector space p i =0R. Under the identification, we have (Dg p ) 0 x = g p (k m x) lim m + k m = ρ a B g p (ρ a B ) 1 (x) = g p (x). In particular, the map g p is an linear isomorphism. The linear map g p commutes with ρ b j B for any j = 1,..., n. This implies that g p(π p (v j )) = π p (v j ), where π p : R n pi =0 R is the natural projection. Since (π p (v j )) n j=1 spans p i =0R, the map g p is the identity map on W u (p, ρ a B ) for each p Fix(ρa B ). Since I n = σ {±1} n I σ and h σ = h on I σ, the above lemma implies that h σ = h on I n. For any x T n and σ {±1} n, the point (ρ b σ B ) m(x,σ) (x) is contained in I n. Take a neighborhood U x of x such that (ρ bσ B )m(x,σ) (U x ) I n. By Lemma 2.5, we have h σ (y) = (ρ bσ ) m(x,σ) h σ (ρ a B )m(x,σ) (y) for any y T n. This implies that h σ (y) = (ρ b σ ) m(x,σ) h (ρ a B )m(x,σ) (y) for any y U x. Hence, h σ is a local diffeomorphism. Since h σ is injective, it is a diffeomorphism of h σ. By Lemma 2.5, it is a smooth conjugacy between two actions ρ B and ρ. 6

7 2.2 Rigidity of local actions Fix a basis B = (v 1,..., v n ) of R n with v j = (v ij ) n i=1. Let P B be the local Γ n,k -action defined in the previous section. In this subsection, we show the local version of Theorem 1.5. Theorem 2.9. If a local action P Hom(Γ n,k, D) is sufficiently close to P B in C 3 loc -topology, then it is smoothly conjugate to P B for some basis B of R n. Combined with Proposition 2.2, the theorem implies Theorem 1.5. The above theorem follows from the same argument as in [1]. Firstly, we prove the stability of the linear part of the local action. Secondly, we show exactness of a linear complex and see that existence of B follow from it. For w = (w 1,..., w n ) n i=1 Rn, we define a map Q w : R n R n R n by Q w (x, y) = n (w j x j y j )e j j=1 for x = (x i ) n i=1 and y = (y i) n i=1, where (e 1,..., e n ) be the standard basis of R n. Then, the local action P B satisfies that P a B(x 1,..., x n ) = k 1 x P b j B (x 1,..., x n ) = x Q vj (x, x) + O( x 3 ). Let I be the identity matrix of size n. stability of the linear part of P bi. We recall a lemma in [1] concerning Lemma 2.10 ([1, Lemma 2.2]). Let P be a local action in Hom(Γ n,k, D). Suppose that D (1) 0 P a = k 1 I and D (1) 0 P bi = I for any i = 1,..., n. Then, there exists a Cloc 1 -neighborhood U of P in Hom(Γ n,k, D) such that D (1) 0 P b i = I for any P U and i = 1,..., n. Hence, D (1) = I for any j = 1,..., n if P is sufficiently Cloc 1 -close to P B, where I is the identity map on R n. The following is essentially proved in Lemma 2.3 of [1]. Lemma Let P be a local action in Hom(Γ n,k, D) such that D (1) 0 P a = k 1 I and D (1) = I for any j = 1,..., n. Suppose that there exists δ > 0 such that max A D(2) 0 P bj 2D (2) 0 P bj (A, I) (2) δ A (1), j=1,...,n for any linear map A : R n R n. Then, P a = k 1 I for any P which is sufficiently C 2 loc -close to P B. Proof. Let U be a C 2 loc -open neighborhood of P consisting of P U such that 3 D (2) D (2) (2) + D (1) 0 P a k 1 I D (2) (2) < δ/2 7

8 for any j = 1,..., n. Fix P U and put A = D (1) 0 P a k 1 I, B j = D (2) D (2) 0 P bj, C j = A D (2) 2D (2) (A, I). We will show that A = 0. The identity P a P b j = P bk j P a implies that (k 1 I + A) (D (2) + B j ) = k (D (2) + B j ) (k 1 I + A, k 1 I + A). Thus, we have that C j (2) = A B j 2B j (A, I) (D (2) + B j ) (A, A) (2) ( A (1) 3 B j (2) + A (1) D (2) (2)) (δ/2) A (1) for any j = 1,..., n. By assumption, max j=1,...,n C j (2) δ A (1). Hence, we obtain that A = 0. We apply the lemma for P B. Lemma The local action P B satisfy the assumption of Lemma In particular, P a = k 1 I for any P Hom(Γ n,k, D) which is sufficiently C 2 loc -close to P B. Proof. Take a square matrix A = (a ij ) of size n and put Then, C j = A D (2) 2D (2) (A, I) = 2 { A Q vj 2Q vj (A, I) }. C j (e i, e i ) = 2{A Q vj (e i, e i ) 2Q vj (Ae i, e i )} = 2{A( v ij e i ) 2( v ij a ii e i )} = 2v ij a iie i a ki e k. k i This implies that C j (e i, e i ) = 2 v ij (a ki ) n k=1, and hence, max C j (2) 2 max v ij (a ki ) n k=1. j=1,...,n j=1,...,n Since (v 1,..., v n ) is a basis of R n, there exists δ > 0 such that max j=1,...,n v ij δ for any i = 1,..., n. We also have A (1) n max i=1,...,n (a ki ) n k=1. This implies that max j=1,...,n C j (2) (2δ/n) A (1). 8

9 Recall that S r,n is the vector space of symmetric r-multilinear maps from (R n ) r to R n. Elements of S 1,n are just linear endomorphisms of R n. For Q, Q S 2,n, we define [Q, Q ] S 3,n by [Q, Q ](ξ 0, ξ 1, ξ 2 ) = 2 Q(ξ k, Q (ξ k+1, ξ k+2 )) Q (ξ k, Q(ξ k+1, ξ k+2 )), k=0 where we set ξ 3 = ξ 0 and ξ 4 = ξ 1. We also define linear maps L 0 B : S1,n (S 2,n ) n and L 1 B : (S2,n ) n (S 3,n ) n(n 1)/2 by L 0 B(A, B ) = (A Q vi Q vi (A, I) Q vi (I, A ) + Q Bei ) n i=1, L 1 B(q 1,..., q n ) = ([q i, Q vj ] [q j, Q vi ]) 1 i j n. It is not hard to check that Im L 0 B Ker L1 B. We can obtain the next proposition by the exactly same argument as in p of [1]. Proposition If Ker L 1 B = Im L0 B, then Theorem 2.9 holds. We will prove the proposition in the next subsection. 2.3 Proof of Proposition 2.13 Recall that I = (e 1,..., e n ) is the standard basis of R n. As shown in Lemma 2.11 of [1], it is enough to prove Proposition 2.13 for the case B = I. Set L 0 = L 0 I and L1 = L 1 I. For v, w R n, let v, w be the standard inner product of v and w, i.e., v, w = n i=1 v iw i. Let W be the subspace of (S 2,n ) n consisting of elements (q i ) n i=1 such that q i(e i, e i ) = 0 and q i (e j, e j ), e j = 0 for any i, j = 1,..., n. The following formula on Q i is useful for computation; { v, e i e i (i = j) Q v (e i, e j ) = Q ei (v, e i ) = 0 (i j) for v R n and i, j = 1,..., n. In particular, Q ei (e j, e k ) = e i if i = j = k and Q ei (e j, e k ) = 0 otherwise. Lemma W + Im L 0 = (S 2,n ) n. Proof. Take (q i ) n i=1 (S2,n ) n. We put a ii = q i (e i, e i ), e i, a ji = q i (e i, e i ), e j, and b ii = 0, b ji = q i (e j, e j ), e j for mutually distinct i, j = 1,..., n. Let A and B be square matrices of size n whose (i, j)-entries are a ij and b ij,respectively. Then, L 0 (A, B) = (q A,B i ) n i=1 satisfies that q A,B i (e i, e i ) = A Q ei (e i, e i ) 2Q ei (Ae i, e i ) + Q Bei (e i, e i ) = Ae i 2a ii e i + b ii e i = q i (e i, e i ), q A,B i (e j, e j ) = A Q ei (e j, e j ) 2Q ei (Ae j, e j ) + Q Bei (e j, e j ) = b ji e j = q i (e j, e k ), e j e j. 9

10 Hence, q i q A,B i is an element of W. Lemma Ker L 1 W = {0}. Proof. Take (q i ) n i=1 Ker L1 W. Since (q i ) n i=1 W, we have q i(e i, e i ) = 0 and q i (e j, e j ), e j = 0 for any i, j = 1,..., n. If i j, [q i, Q ej ](e j, e j, e j ) = 3 { q i (e j, e j ) Q ej (e j, q i (e j, e j )) } = 3 {q i (e j, e j ) q i (e j, e j ), e j e j }, = 3 q i (e j, e j ), [q j, Q ei ](e j, e j, e j ) = 3 {q j (e j, e j ) Q ei (e j, q j (e j, e j ))} = 0. Since [q i, Q ej ] [q j, Q ei ] = 0, we obtain that q i (e j, e j ) = 0. If i j, [q i, Q ej ](e i, e j, e j ) = q i (e i, e j ) 2Q ej (e j, q i (e i, e j )) = q i (e i, e j ) 2 q i (e i, e j ), e j e j [q j, Q ei ](e i, e j, e j ) = Q ei (e i, q j (e j, e j )) = Q ei (e i, 0) = 0. Since [q i, Q ej ] [q j, Q ei ] = 0, we obtain that q i (e i, e j ) = 0. For mutually distinct i, j, k = 1,..., n, [q i, Q ej ](e j, e j, e k ) = q i (e k, e j ) 2Q ej (e j, q i (e j, e k )) [q j, Q ei ](e j, e j, e k ) = 0. = q i (e j, e k ) 2 q i (e j, e k ), e j e j, Since [q i, Q ej ] [q j, Q ei ] = 0, we obtain that q i (e j, e j ) = 0. Now, we have q i (e j, e k ) = 0 for any (q i ) n i=1 Ker L1 W and any i, j, k = 1,..., n. Now, we prove Proposition Since Im L 0 is a subspace of Ker L 1, we have (S 2,n ) n = W Im L 0 by the above lemmas. By Im L 0 Ker L 1 and Ker L 1 W = {0} again, we obtain that Ker L 1 = Im L 0. 3 Proof of Proposition 1.4 It is easy to see that any linear isomorphism g G of R n can be extended uniquely to a diffeomorphism h g of T n = (R { }) n and the diffeomorphism h g is a conjugacy between ρ B and ρ gb. Suppose that ρ B and ρ B are smoothly conjugate by a diffeomorphism h. We will show that h = h g for some g G. The conjugacy h preserves the unique repelling fixed point (0,..., 0) of ρ a B and ρa B and their unstable manifold R n T n = (R { }) n. The restriction h R of h to R n commutes with the linear 10

11 map x kx. As in the proof of Lemma 2.8, the map h R is linear. Take (a ij ) n i,j=1 such that h R (e j ) = n i=1 a ije i. We set V j = {(x 1,..., x n ) T n x j =, x i if i j} for i = 1,..., n. Each V i is a submanifold of V j which is diffeomorphic to R n 1. Since h is continuous, we have h(v j ) V i. a ij 0 Since h is a diffeomorphism of T n, there exists a unique σ(j) {1,..., n} such that a ij 0 for each j = 1,... n. Since the linear transformation h R is invertible, σ is a permutation of {1,..., n}. Therefore, h R is an element of G. References [1] M.Asaoka, Rigidity of certain solvable actions on the sphere. Geom. Topol. 16 (2012), (electronic). [2] L.Burslem and A.Wilkinson, Global rigidity of solvable group actions on S 1. Geom. Topol. 8 (2004), (electronic). [3] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge,