Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Let $M\stackrel{\rho_0}{\curvearrowleft}S$ be a $C^\infty$ locally free action of a connected simply connected solvable Lie group $S$ on a closed manifold $M$. Roughly speaking, $\rho_0$ is parameter rigid if any $C^\infty$ locally free action of $S$ on $M$ having the same orbits as $\rho_0$ is $C^\infty$ conjugate to $\rho_0$. In this paper we prove two types of result on parameter rigidity. First let $G$ be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathrm{SO}_0(n,1)$ $(n\geq2)$ or $\mathrm{SU}(n,1)$ $(n\geq2)$, and let $\Gamma$ be an irreducible cocompact lattice in $G$. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma\backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu and Kleiner-Leeb on the quasiisometries of Riemannian symmetric spaces of noncompact type. Secondly we show, if $M\stackrel{\rho_0}{\curvearrowleft}S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho_0$ with certain coefficients must vanish. This is a partial converse to the results in the author's [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157-191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.


Parameter rigidity of the action of AN on Γ\G
Let M ρ0 S be a C ∞ locally free (ie the isotropy subgroup of every point is discrete) action of a connected simply connected solvable Lie group S on a closed C ∞ manifold M . Let F be the set of all orbits of ρ 0 , which is called the orbit foliation of ρ 0 and actually is a C ∞ foliation of M . We say ρ 0 is parameter rigid if every C ∞ locally free action M ρ S with the same orbit foliation as that of ρ 0 is parameter equivalent to ρ 0 .
(We do not assume that ρ is close to ρ 0 in some topology.) Here parameter equivalence between ρ and ρ 0 means the following. There exist a diffeomorphism F of M and an automorphism Φ of S such that: • F (ρ 0 (x, s)) = ρ (F (x), Φ(s)) for all x ∈ M and s ∈ S • the map F preserves each leaf of F , that is, F (L) ⊂ L for all L ∈ F • the map F is C 0 homotopic to the identity map of M through C ∞ maps which preserve each leaf of F .
For example a linear flow on a torus is parameter rigid if and only if the velocity vector satisfies the Diophantus condition.
Theorem 1 (Katok-Spatzier). Let G be a connected semisimple Lie group with finite center of real rank at least 2 without compact factors nor simple factors locally isomorphic to SO 0 (n, 1) (n ≥ 2) or SU(n, 1) (n ≥ 2), and let Γ be an irreducible cocompact lattice in G. Let G = KAN be an Iwasawa decomposition. Then the action Γ\G A by right multiplication is parameter rigid. This is proved using representation theory of semisimple Lie groups and has lead to a number of subsequent research. In this paper we prove the following, based on the above theorem and applying large scale geometry.

Theorem 2. Under the same assumptions as Theorem 1, the action Γ\G
AN by right multiplication is parameter rigid.
We give a proof of this theorem in Section 3 and Section 4 after recalling the results in Maruhashi [17] in Section 2. The proof is a combination of the following three steps: 1. vanishing of cohomology ⇒ parameter rigidity. This is the sufficient condition for parameter rigidity proved in [17]. In the current article this is Theorem 4 2. cohomology vanishing results. These are by Katok-Spatzier [11], [12] and Kanai [10]. See Theorem 10 and Corollary 12 in this paper 3. bridging the gap between Step 1 and Step 2. This is because the cohomology vanishing results are available only for finitely many coefficients, while the sufficient condition for parameter rigidity requires vanishing of cohomology for seemingly much more coefficients. Here we use Proposition 6, which shows the relevance to large scale geometry. Then the main point is that our acting group AN is isometric to G/K by Iwasawa decomposition G = AN K. So we can use the rigidity theorems of Pansu [18] and Kleiner-Leeb [13] on quasiisometries of symmetric spaces, and a certain rigidity property of quasiisometries of hyperbolic spaces proved in Farb-Mosher [5] and Reiter Ahlin [19].
Theorem 2 shows a contrast between the higher rank case and PSL(2, R), the universal cover of PSL(2, R), for which Asaoka [1] gives (generally) nontrivial orbitpreserving deformations of the actions of AN by right multiplication.
Theorem 3 (Asaoka [1]). Let Γ be a cocompact lattice in PSL(2, R) and let A = a a −1 a > 0 , N = Let Φ Γ be the flow on Γ\ PSL(2, R) defined by the action of A by right multiplication, P be the set of oriented periodic orbits of Φ Γ and τ (γ) be the period of γ for γ ∈ P. Consider which is an open neighborhood of 0 in H 1 Γ\ PSL(2, R); R . Then there exists an analytic locally free action ρ a of AN on Γ\ PSL(2, R) for each a ∈ ∆ Γ with the following properties: • The action ρ 0 is defined by the right multiplication.
• All the ρ a 's have the same orbit foliation F .
• Actions ρ a and ρ a ′ are not parameter equivalent if a = a ′ .
• Any C ∞ locally free action of AN whose orbit foliation is F is parameter equivalent to ρ a for some a ∈ ∆ Γ .
• The action ρ a does not preserve any C 0 volume form on Γ\ PSL(2, R) except when a = 0.
We also know how the action ρ a is controlled by the cohomology class a, but we refer the reader to [1] for that and more information. Note that the above deformation is different from the nonorbit-preserving deformation coming from the deformation of the lattice, whose deformation space has the dimension equal to that of Teichmüller space, because such deformations are necessarily C 0 volume preserving.

Preliminaries
This section is a summary of the results we need later, proved in Maruhashi [17]. See [17] for the detail. In this paper Lie algebras are denoted by the corresponding lowercase Fraktur of the corresponding Lie groups. The symbol Γ( · ) denotes the set of all C ∞ sections of a vector bundle.

Leafwise cohomology
Let M ρ0 S be a C ∞ locally free action of a connected simply connected solvable Lie group S on a closed manifold M with the orbit foliation F . Let ω 0 ∈ Γ (Hom(T F , s)) denote the canonical 1-form of ρ 0 , ie (ω 0 ) x : T x F → s for x ∈ M is defined as the inverse of the derivative at the identity of the map S → M , s → ρ 0 (x, s). Let be the leafwise exterior derivative of F , defined by the same formula as the usual exterior derivative. Then ω 0 satisfies the Maurer-Cartan equation and for X, Y ∈ Γ(T F ). Let s π V be a representation of s on a finite dimensional real vector space V . Then πω 0 ∈ Γ (Hom (T F , End(V ))) satisfies We regard πω 0 as the connection form of a flat F -partial connection ∇ of the trivial vector bundle M × V → M relative to any global frame of the bundle which has where our definition of exterior product is The square of this operator is zero by the flatness. The cohomology H * F ; s π V of this complex is the leafwise cohomology of F with coefficient π. Recall that the cohomology H * s; s π V of the Lie algebra s with coefficient π is obtained from the complex Hom * s, V . We have an injective cochain map where ω * 0 is the pullback by ω 0 . Then by Lemma 2.1.3 of [17], the induced map is injective and we see H * s; s π V as a subspace of H * F ; s π V .

A property from large scale geometry
Let ρ ∈ A(F , S) and let a ρ : M × S → S be the unique C ∞ map satisfying for all x ∈ M and s ∈ S. The map a ρ is defined since ρ 0 and ρ have the same orbit foliation. It is known that a ρ is a cocycle over ρ 0 . Let X, B be metric spaces. A surjective map p : X → B is a distance respecting projection if and d H denotes the Hausdorff distance. Let p : X → B and p ′ : is fiber respecting or f is fiber respecting over ϕ if f and ϕ are maps and there exists a Proposition 5. Let G be a connected Lie group and H a connected normal closed subgroup of G. Take an inner product of g. Endow g/h with the inner product for which the restriction h ⊥ ∼ → g/h of the projection g → g/h is an isometry. Give G and G/H left invariant Riemannian metrics corresponding to these inner products. Then the projection p : G → G/H is a distance respecting projection.
Proof. This follows from Lemma 4.1.1 of [17] by noting that H Ad g/h is trivial.
Assume H 1 (F ) = H 1 (s) for an action M ρ0 S and let ρ ∈ A(F , S) and ϕ ρ : s → s/h, a ρ : M × S → S as above. Let K ρ and H be the Lie subgroups corresponding to ker ϕ ρ and h. Then S/K ρ and S/H are vector groups. Letφ ρ : S/K ρ → S/H be the linear isomorphism with differential ϕ ρ : s/ ker ϕ ρ ≃ s/h.

Reduction of the proof of Theorem 2 to Proposition 13
Let G be a connected semisimple Lie group. Fix a Cartan decomposition g = k ⊕ p and a maximal abelian subspace a of p. Let Σ be the restricted root system of (g, a) and fix a positive system Σ + of Σ. Let n = λ∈Σ+ g λ , where is a restricted root space. Let K, A and N be the Lie subgroups corresponding to k, a and n. Then G = KAN is an Iwasawa decomposition. The group AN is a connected simply connected solvable Lie group and its Lie algebra is an = n ⋊ a.
It is easy to show that n is the nilradical of an and n = [an, an]. So we must take h = n to apply Theorem 4. Then To apply Theorem 4, we must show H 1 (F ) = H 1 (an) and then calculate cohomology with coefficient for any ρ ∈ A(F , AN ), where ϕ ρ : an → a. Note that ker ϕ ρ = n and ϕ ρ | a ∈ GL(a). The g λ -component an R. Therefore, we get the following.

Lemma 7.
Let G be a connected semisimple Lie group. Fix a Cartan decomposition g = k ⊕ p, a maximal abelian subspace a of p with the associated restricted root system Σ and a positive system Σ + of Σ. Let F be the orbit foliation of a C ∞ locally free action M ρ0 AN on a closed manifold M . If and for any λ ∈ Σ + and ρ ∈ A(F , AN ), then ρ 0 is parameter rigid.
Before proving Theorem 2 we remark that the same result but with a stronger assumption of real rank at least 3 follows easily from the following result of Kononenko [15,Theorem 8.2].
Theorem 8 (Kononenko [15]). Let G be a connected semisimple Lie group with finite center of real rank at least 3 whose simple factors are of real rank at least 2, and let Γ be an irreducible cocompact lattice in G. Let G = KAN be an Iwasawa decomposition. Take µ : an → R be any nonzero linear function which vanishes on n, and letμ : AN → GL(1, R) be the homomorphism with differential µ. Then anỹ µ-twisted C ∞ cocycle over the action Γ\G AN by right multiplication is C ∞ cohomologous to a constant cocycle. Equivalently we have H 1 F ; an µ R = H 1 an; an µ R .

Corollary 9.
Let G be a connected semisimple Lie group with finite center of real rank at least 3 whose simple factors are of real rank at least 2, and let Γ be an irreducible cocompact lattice in G. Let G = KAN be an Iwasawa decomposition. Then the action Γ\G AN by right multiplication is parameter rigid.
Proof. Under the assumptions of this corollary, (2) in Lemma 7 follows from the case λ = 0 in Corollary 12 below and (3) in Lemma 7 follows from Theorem 8, which imply parameter rigidity of the action.
But this does not cover the case of real rank 2. In this case we only know vanishing of the cohomology with coefficients corresponding to restricted roots. Theorem 10. Let G be a connected semisimple Lie group with finite center of real rank at least 2 without compact factors or simple factors locally isomorphic to SO 0 (n, 1) (n ≥ 2) or SU(n, 1) (n ≥ 2), and let Γ be an irreducible cocompact lattice in G. Fix a Cartan decomposition g = k ⊕ p and a maximal abelian subspace a of p with the associated restricted root system Σ. Let F A be the orbit foliation of the action Γ\G A by right multiplication. Then we have: for any λ ∈ Σ.
Remark 11. In Theorem 2.2 (2) of [10] it is written that u (a notation from [10]) is C ∞ if the conditions (i) and (ii) from that paper are satisfied, but those conditions (i) and (ii) are always satisfied, so that we get the above result.

Corollary 12.
Let G be a connected semisimple Lie group with finite center of real rank at least 2 without compact factors or simple factors locally isomorphic to SO 0 (n, 1) (n ≥ 2) or SU(n, 1) (n ≥ 2), and let Γ be an irreducible cocompact lattice in G. Fix a Cartan decomposition g = k ⊕ p, a maximal abelian subspace a of p with the associated restricted root system Σ and a positive system Σ + of Σ. Let F be the orbit foliation of the action Γ\G AN by right multiplication. Then we have for any λ ∈ Σ ∪ {0}, where λ : a → R is regarded as λ : an → R by extending it as 0 on n.
By Corollary 12, the proof of Theorem 2 reduces to the following proposition.
Proposition 13. Let G be a connected semisimple Lie group. Fix a Cartan decomposition g = k ⊕ p, a maximal abelian subspace a of p with the associated restricted root system Σ and a positive system for all λ ∈ Σ + , where λ : a → R is regarded as λ : an → R by extending it linearly as 0 on n, then ρ 0 is parameter rigid.
Note that we need no assumption on the simple factors of G in this proposition.
If 1 in Theorem 10 is true without the assumption ( * ), then 2 in Theorem 10 and Corollary 12 are true without the assumption ( * ). Hence Theorem 2 will be true without the assumption ( * ) by Proposition 13.

Proof of Proposition 13
To prove Proposition 13, it suffices to show that λ • ϕ ρ | a ∈ Σ + for any λ ∈ Σ + and any ρ ∈ A(F , AN ) by Lemma 7. At this moment we know ϕ ρ | a is only an element of GL(a), so it is not clear whether ϕ ρ | a preserves Σ + . To prove it we need rigidity of quasiisometries of symmetric spaces. For the proof of Proposition 13 we may assume that G has no compact factors, since this does not change AN . Recall that Inn(g) = Ad(G) = G/Z(G), where Z(G) denotes the center of G, and G/Z(G) has the trivial center. Replacing G with G/Z(G) also does not change AN , so we may assume G = Inn(g) as well.
The mapping an → anK gives a canonical diffeomorphism AN ≃ G/K by the Iwasawa decomposition. Henceforth we identify AN with G/K in this way. This is AN equivariant.
Recall that the identification p ≃ T K G/K is by X → d dt e tX K t=0 . In the following K denotes the subgroup K or the point K in G/K depending on the context. G-invariant Riemannian metrics on G/K are in one-to-one correspondence with inner products on p invariant under K Ad p. We equip G/K with a G-invariant Riemannian metric g corresponding to the restriction of B θ to p, where θ is the Cartan involution associated with the Cartan decomposition g = k ⊕ p, B the Killing form of g and The restriction of B θ to p is the same as the restriction of B to p. We give AN the Riemannian metric which makes the identification AN ≃ G/K an isometry. This Riemannian metric is AN invariant. Geodesics in G/K passing K at time 0 are of the form e tX K (t ∈ R) for X ∈ p. Note that e tX K (t ∈ R) for X ∈ g \ p is not a geodesic in general. In AN curves of the form ne tH (t ∈ R) for fixed n ∈ N and H ∈ a are geodesics. The decomposition g = k ⊕ p is orthogonal with respect to the positive definite symmetric bilinear form B θ . Let g ′ λ be the orthogonal projection to p with respect to g = k ⊕ p of g λ for λ ∈ Σ. The space g ′ λ has the same dimension as g λ since k = ker(θ − id) and θg λ = g −λ . This orthogonal projection maps an isomorphically to p by the Iwasawa decomposition g = k ⊕ a ⊕ n. Therefore, Note that a ⊥ n ′ since a ⊥ g λ for λ ∈ Σ and a ⊥ k with respect to B θ . Observe that the differentiation an p = a ⊕ n ′ at 1 of the identification AN ≃ G/K maps an to p by the orthogonal projection with respect to g = k ⊕ p. Therefore, a maps identically to a and n maps isomorphically to n ′ . So a ⊥ n in an.
For any ρ ∈ A(F , AN ) and x ∈ M , consider the diagram where p is the natural projection. We give A a left invariant Riemannian metric for which the restriction a → a of the natural projection an → a to n ⊥ = a becomes an isometry, ie we consider the restriction of B to a. Then p is a distance respecting projection by Proposition 5 and a ρ (x, · ) is a fiber respecting biLipschitz diffeomorphism overφ ρ by Proposition 6. Since G = Ad(G), we have G = G 1 × · · · × G ℓ , where G i is a connected noncompact simple Lie group with trivial center. Since any two maximal compact subgroups of G are conjugate by an inner automorphism of G, we have Since maximal abelian subspaces in p are conjugate by Ad(k) for some k ∈ K and Ad(k) preserves each p i , we have a = a 1 ⊕ · · · ⊕ a ℓ for some maximal abelian subspace a i of p i . Let be the restricted root space decomposition of g i . Then The metric g on AN decomposes as The same kind of decomposition holds for the metric on A.

In this way we identify
is fiber respecting and to complete the proof of Proposition 13 it suffices to show λ • ϕ i ∈ Σ i+ for any λ ∈ Σ i+ . We consider the following two cases separately: • The group G i is of real rank at least 2 or locally isomorphic to Sp(n, 1) (n ≥ 2) or F −20 4 .
• The group G i is of real rank 1. in either cases. From now on we will no longer consider the original objects G, K, A, N , g, Σ, ϕ ρ etc and we will focus only on the decomposed objects Hence we will drop all the subscripts i to simplify the notations. So we have but we do not have M , ρ and a ρ . Recall that G = Ad(G), g is the restriction of B at T K G/K = p, AN is equipped with a Riemannian metric by the identification AN ≃ G/K, the Riemannian metric of A is the one which makes p * | a : a ≃ a an isometry, and is fiber respecting. Under these conditions we must prove λ • ϕ ∈ Σ + for any λ ∈ Σ + . , there exists a homothety F :

The case where
is an isometry. Since the isometry group of G/K acts transitively, there exists the minimum K 0 ∈ (−∞, 0) of the sectional curvature of (G/K, g). Then cK 0 is the minimum of the sectional curvature of (G/K, cg). Since they are isometric we must have K 0 = cK 0 hence c = 1.
where L denotes the left multiplication. Since L a0n0 is an isometry, is also fiber respecting. Note that f is an isometry and f (1) = 1.

Lemma 16. The mapφ is an isometry.
Proof. There exists a constant C > 0 such that d H (f (aN ),φ(a)N ) < C for all a ∈ A. Then we have for all a ∈ A. Hence for all t > 0 and H ∈ a we have
Now we regard f as f : G/K → G/K and p : G/K → A. Consider
Proof. Take any H ∈ a. Let f * H = X + Y for some X ∈ a and Y ∈ n ′ . Since we have H ≥ X . Because e tH K (t ∈ R) is a geodesic and f is an isometry, f e tH K (t ∈ R) is also a geodesic and f e tH K = e tX+tY K. Let t > 0.
Since f is fiber respecting overφ, there exists a constant C > 0 such that Since e tX+tY K = f e tH K ∈ f p −1 e tH and by the definition of the Hausdorff distance, there exists x ∈ p −1 φ e tH such that d e tX+tY K, x < C.
The map p is distance decreasing since d(a, a ′ ) = d p −1 (a), p −1 (a ′ ) for all a, a ′ ∈ A. So d e tX ,φ e tH = d p e tX+tY K , p(x) ≤ d e tX+tY K, x < C.
By the triangle inequality we have for all t > 0. This forces H = X and then Y = 0 by the equation For the second assertion we have by (6) d e tf * H , e tϕH < C for any t ∈ R. This implies f * H = ϕH.

Proposition 18.
Let g be a real semisimple Lie algebra and let G = Inn(g). (Recall that the Lie algebra of G is naturally isomorphic to g and G is the identity component of Aut(g).) Fix a maximal compact subgroup K of G: 1. Let ψ ∈ Aut(g) and consider Ψ ∈ Aut(G) defined by Ψ(g) = ψgψ −1 . The automorphism Ψ permutes the maximal compact subgroups of G. Identifying the set of all maximal compact subgroups of G with G/K by gKg −1 ↔ gK, the map I ψ : G/K → G/K induced by Ψ is an isometry with respect to the G-invariant Riemannian metric defined by the restriction of the Killing form to the orthogonal complement of the Lie algebra of K.
2. Suppose g has no compact simple factor. Then the mapping ψ → I ψ is an isomorphism from Aut(g) to Isom(G/K).
Proof. This is Exercise 7 in Chapter VI of Helgason [7]. A proof can be found in Solutions to Exercises.
By Proposition 18 there exists ψ ∈ Aut(g) such that f = I ψ . Since f (K) = K, we have Ψ(K) = K. This implies f (gK) = Ψ(g)K for all g ∈ G. We have ψ(k) = k. Since and B is ψ-invariant, we also have ψ(p) = p. Hence f * = ψ| p : p → p by (7) and ψ(a) = a by Lemma 17. Therefore, ψ| a = ϕ : a → a again by Lemma 17. Since ψ is an isomorphism of g which preserves a, we have ψ −1 g λ = g λ•ψ|a for any λ ∈ Σ. Thus For a Weyl chamber C in a, let be the positive system corresponding to C, let n C = λ∈ΣC g λ , and let N C be the Lie subgroup corresponding to n C . Let C 0 ⊂ a be the Weyl chamber corresponding to Σ + , ie Then C 1 = ψC 0 is a Weyl chamber in a. We have λ ∈ Σ + if and only if λ • (ψ| a ) −1 ∈ Σ C1 . Thus ψn = n C1 . By (7) we have f (N K) = N C1 K. Therefore, the Hausdorff distance between N C1 K and N K is finite.

Lemma 19.
If C and C ′ are distinct Weyl chambers in a, then the Hausdorff distance between N C K and N C ′ K is infinite.
Proof. Take λ ∈ Σ C \ Σ C ′ . Hence g λ ⊂ n C and g −λ ⊂ n C ′ . We will prove that e g −λ K contains arbitrarily far points from N C K. Let H λ ∈ a be the element defined by λ(H) = B (H λ , H) for all H ∈ a. By Knapp [14, Proposition 6.52] there exists nonzero X λ ∈ g λ such that: • the subspace RθX λ ⊕RH λ ⊕RX λ is a Lie subalgebra of g isomorphic to sl(2, R). The isomorphism is given by For any x ∈ R we have This can be regarded as an equation of elements in the universal cover SL(2, R) of SL(2, R). We rewrite it using the exponential map: Mapping the above equation by the homomorphism SL(2, R) → G, we get Note that Thus (8) gives the Iwasawa decomposition of exp xX ′ −λ as an element of G = N C AK. Therefore, This shows N C ′ K contains points arbitrarily far from N C K.

The case where G is of real rank
is fiber respecting, f is a quasiisometry and h is a map, then h is close to the identity map.
The mapφ is close to the identity map by this proposition. But sinceφ is a homomorphism,φ must be the identity map. Hence λ • ϕ = λ ∈ Σ + for all λ ∈ Σ + and this concludes the proof of Proposition 13. Proposition 20 is Proposition 5.8 of Farb-Mosher [5] when G is locally isomorphic to SO 0 (n, 1). For the other cases it is basically Theorem 33 of Reiter Ahlin [19] but the proof there seems incomplete. To get the conclusion of Proposition 20 we need to argue at some point in the same manner as Farb-Mosher do. Here we give a proof of Proposition 20 following the arguments by Farb-Mosher and Reiter Ahlin.
We have Σ + = {λ} for G locally isomorphic to SO 0 (n, 1) and Σ + = {λ, 2λ} for the other cases. Accordingly n = g λ in the former case and n = g λ ⊕ g 2λ in the latter case. Take H ∈ a such that λ(H) = 1. Hence a = RH. We identify A with R by e tH → t.
We write the proof for the case of Σ + = {λ, 2λ} but no change is needed when we have Σ + = {λ} except notational one.
Let g t be the Riemannian metric on N induced from g by the embedding N ֒→ AN , x → xe tH . Let d and d t be the metrics induced from g and g t respectively. Since x ye tH = (xy) e tH , ie the embedding N ֒→ AN is N -equivariant, g t is a left invariant Riemannian metric on N . Let · j be a norm on g jλ (j = 1, 2) and set Lemma 21. There exists K 1 ≥ 1 such that for all t ∈ R and x, y ∈ N .
Proof. Since e tH x = φ t (x)e tH , φ t : (N, g 0 ) → (N, g t ) is an isometry. Hence It is known that there exists a constant K 1 ≥ 1 such that for all x ∈ N . See for example Breuillard [3, Proposition 4.5]. Therefore, Corollary 22. There exists K 2 ≥ 1 such that for any fixed t 0 ∈ R we have under the above conditions. In particular We get the conclusion from these two inequalities.
Lemma 23. The embedding (N, d t ) ֒→ (AN, d) is uniformly proper for each t ∈ R and the uniformity data are independent of t. In fact there exists a function ρ : for all x, y ∈ N and t ∈ R.
Proof. The second inequality is obvious. For the first inequality, define ρ 1 : Then ρ 1 is strictly increasing and lim R→∞ ρ Lemma 24. Let X, Y , S, T be geodesic spaces, let f : X → Y be a quasiisometry, and let σ : S → X, τ : T → Y be uniformly proper maps such that d H (f σ(S), τ (T )) < ∞. Take any map g : S → T satisfying sup x∈S d (f σ(x), τ g(x)) < ∞. Then g is a quasiisometry and the quasiisometry constants depend only on the quasiisometry constants for f , the uniformity data for σ and τ , and sup x∈S d (f σ(x), τ g(x)).
We identify h : A → A with h : R → R by h e tH = e h(t)H . Define f t : (N, d t ) → N, d h(t) by f xe tH = f t (x)e u(x,t)H . Then f t satisfies the property of Lemma 24.
In fact since f is fiber respecting over h, there exists a constant C 1 > 0 such that d H f p −1 e tH , p −1 e h(t)H < C 1 for all t ∈ R. Hence there exists y ∈ N such that d f xe tH , ye h(t)H < C 1 . Therefore, for all x ∈ N and t ∈ R. By Lemma 23 and Lemma 24,  Proof. Assume the contrary: ∂f (∞) = x ∈ N . Take y ∈ N with y = (∂f ) −1 (∞). Let γ be the directed geodesic connecting y and ∞. Then the Hausdorff distance between f (γ) and the directed geodesic γ ′ connecting ∂f (y) and x is finite. Hence the height of f (γ) is bounded above. Since h is a quasiisometry, we can choose t 0 ∈ R so that h(t 0 ) is as large as we wish. Therefore, the height of f p −1 e t0H is also large. But we always have f ye t0H ∈ f p −1 e t0H ∩f (γ) = ∅, which is impossible.
For any x ∈ N , xe tH (t ∈ R) is a geodesic of AN connecting x ∈ ∂AN and ∞. Then f xe tH (t ∈ R) is a quasigeodesic of AN . By Lemma 25 there exists a constant There exists s(x, t) ∈ R such that d f xe tH , ∂f (x)e s(x,t)H < C 2 . We have By (10) and (11) we get Therefore, Hence Namely f t and ∂f are close and the constant of closeness is independent of t. Thus ∂f : (N, d t ) → N, d h(t) is a quasiisometry with constants independent of t, so there exists a constant K 3 ≥ 1 such that for all x, y ∈ N and t ∈ R.
Lemma 26. For any fixed t 0 ∈ R we have for all t ≤ t 0 and x, y ∈ N with x −1 y > e t0 K 1 2K 2 3 + K 1 .
Proof. If t ≤ t 0 and x −1 y > e t0 K 1 2K 2 3 + K 1 , we have Hence It is easy to show that h is a quasiisometry of R. See Farb-Mosher [5, Lemma 5.1].
Lemma 27. There exists L > 0 such that for any
Lemma 28. For any fixed t 0 ∈ R, we have Proof. If t ≤ t 0 − L and and h(t) ≤ h(t 0 ). So we get the desired inequality by Corollary 22.
Lemma 29. There exists C 3 > 0 such that for any t 0 ∈ R and t ≤ t 0 , we have Proof. Fix t 0 and take x, y ∈ N with x −1 y large enough so that we can apply Corollary 22, Lemma 26 and Lemma 28. Then for any t ≤ t 0 − L, we have .
by a constant independent of t 0 . Hence the claim is proved.
Letf : AN → AN be a coarse inverse of f , ief is a quasiisometry such thatf • f and f •f are close to the identity map. Leth : R → R be a coarse inverse of h. It is easy to show thatf is fiber respecting overh. Apply Lemma 29 tof andh rather than f and h. Then there exists C ′ 3 > 0 such that for all s ≤ s 0 . Now we can argue completely in the same way as in Farb-Mosher [5, page 167 just after Claim 5.9] to prove that h is close to the identity map.

Necessary conditions for parameter rigidity
From this section we consider necessary conditions for parameter rigidity. (For the definition of parameter rigidity, see the beginning of Section 1.) These necessary conditions are given by certain vanishing of zeroth and first cohomology of the orbit foliation. The main results are Theorem 30 and Theorem 35. Let M ρ0 S denote a C ∞ locally free action of a connected simply connected solvable Lie group S on a closed C ∞ manifold M , with the orbit foliation F . Recall that a connected simply connected solvable Lie group S is called of exponential type if the exponential map exp : s → S is a diffeomorphism, or equivalently, every eigenvalue of ad X either is 0 or has nonzero real part for each X ∈ s. For a proof of this equivalence, see Dixmier [4,Théorème 3] or Saito [20]. A derivation of a Lie algebra is called an outer derivation if it is not an inner derivation.
The first necessary condition is the following.
Theorem 30 (Vanishing of H 0 ). Assume that S is of exponential type and there is an outer derivation of s. If M ρ0 S is parameter rigid, then M is connected and We will prove Theorem 30 in Section 6.
Corollary 31. Let N = 1 be a connected simply connected nilpotent Lie group and let M ρ0 N be a parameter rigid action. Then M is connected and H 0 (F ) = H 0 (n).
Proof. Every nonzero nilpotent Lie algebra over any field has an outer derivation. See Jacobson [9].
Note that H 0 (F ) consists of real valued leafwise constant C ∞ functions on M and H 0 (s) (as a subspace of H 0 (F )) consists of real valued constant functions on M . Hence we have H 0 (F ) = H 0 (s) if and only if leafwise constant C ∞ functions are constant. This is satisfied if there is a dense leaf of F . In the proof of Theorem 30 we don't prove the existence of a dense leaf of F . We prove H 0 (F ) = H 0 (s) somewhat algebraically without studying dynamical properties of the foliation F . Remark 32. The author does not know whether Theorem 30 remains true if we drop one of the two assumptions on S. One possibility of constructing counterexamples which are parameter rigid but H 0 (F ) is huge is the following. Take a connected simply connected solvable Lie group S and a cocompact lattice Γ in S such that: • S has no outer automorphisms • Γ is a rigid lattice in S, which means, if Γ ′ is a lattice in S and α : Γ → Γ ′ is an isomorphism, then α extends to an automorphism of S. (This terminology is taken from Starkov [21].) The author does not know the existence of such S and Γ. But if we had such a pair, Proposition 6.1.2 in Maruhashi [17] says, the action Γ\S S defined by right multiplication is parameter rigid because in this case parameter rigidity is equivalent to the rigidity of the lattice Γ. Then the action S 1 ×Γ\S S defined by (x, y)s = (x, ys) is perhaps parameter rigid by the first condition, whereas H 0 (F ) is now identified with the space of all real valued C ∞ functions on S 1 .
Recall the following theorem.
Theorem 33 (Maruhashi [16]). Let N be a connected simply connected nilpotent Lie group, and let M ρ0 N be a C ∞ locally free action. Then the following are equivalent: • The action ρ 0 is parameter rigid and H 0 (F ) = H 0 (n).
Hence we have the following.
Corollary 34. Let N be a connected simply connected nilpotent Lie group, and let M ρ0 N be a C ∞ locally free action. Then the following are equivalent: • The action ρ 0 is parameter rigid.
Proof. This is true even if N = 1.
If we have vanishing of H 0 for the trivial coefficient, then we can deduce vanishing of H 0 for various nontrivial coefficients by an easy argument. This will be done in Lemma 42 in Section 6.
The second necessary condition is on vanishing of H 1 . The following will be proved in Section 7.
Theorem 35 (Vanishing of H 1 ). Let V ⊂ s be an ad-invariant subspace (ie an ideal of s) for which n ad V is trivial. Assume that any eigenvalue of ad X on s/V either is 0 or has nonzero real part for any X ∈ s. If M ρ0 S is parameter rigid, then we have Note that the assumption is weaker than the assumption that S is of exponential type, as it allows ad X : V → V to have purely imaginary nonzero eigenvalues.
Here an element is represented by a leafwise constant form, that is, represented by a form φ • ω 0 for some C ∞ leafwise constant map φ : M → Hom(s, V ). If we assume also that s has an outer derivation, then by Theorem 30, the conclusion simplifies to Let us consider the coefficients appearing Theorem 35. We have V ⊂ n, thus V is contained in the center of n, and is an abelian ideal of s. (For the first part, if not, take X ∈ V \ n, then n + RX would be a nilpotent ideal of s which is larger than the nilradical n.) As an example of a coefficient V satisfying the property, we can take V = n s , where n ⊃ n 2 ⊃ · · · ⊃ n s ⊃ 0 is the lower central series of n.
As a more concrete example, we consider the 2-dimensional solvable Lie algebra ga = RX ⊕ RY defined by [X, Y ] = Y . Then the 1-dimensional representation ga ad RY satisfies the condition of Theorem 35, but the trivial representation s ga/RY does not satisfy the condition.

Vanishing of H 0 -proof of Theorem 30
The proof of Theorem 30 is immediate after proving Lemma 40, whose proof is the main part of this section. Several lemmas before Lemma 40 prepare an "integration" map µ, which will be used in the proof of Lemma 40. Sublemma 6 inside Lemma 40 is similar to Lemma 45 in the next section and the same kind of argument already appeared in Maruhashi [16] when the vanishing of H 1 was proved under the assumption of parameter rigidity together with the vanishing of H 0 for actions of nilpotent Lie groups.
Let M ρ0 S be a C ∞ locally free action of a connected simply connected solvable Lie group S on a closed C ∞ manifold M , with the orbit foliation F and the canonical 1-form ω 0 . Let s π V be a finite dimensional real representation, and let S Π V denote the representation whose differentiation is π. Then the trivial bundle M × V → M is an S-equivariant vector bundle with the action defined by Let for s ∈ S, ξ ∈ Γ blc (V ) and x ∈ M . We equip V with a norm coming from an inner product. Then Γ blc (V ) is a Banach space with the supremum norm. Let Γ lc (V ) be the closed subspace of Γ blc (V ) which consists of bounded leafwise constant sections.

Lemma 36.
There is an S-equivariant continuous linear map which is the identity on Γ lc (V ).
Proof. Since S is amenable, by one of the characterizations of amenability, we have a bi-invariant mean µ 0 : C b (S) → R on the space C b (S) of all bounded continuous real valued functions on S. See Page 26-29 of Greenleaf [6]. Recall that µ 0 (1) = 1 and its operator norm is 1. Take a basis v 1 , .
Then this is independent of a choice of a basis of V . We have by left invariance, and by right invariance. We also have µ(ξ) = ξ for ξ ∈ Γ lc (V ) since µ 0 (1) = 1. By taking v 1 , . . . , v n to be an orthonormal basis and using µ 0 = 1, we see Lemma 37. For v ∈ V , x 0 ∈ M and sufficiently small s ∈ S, the locally defined section of M × V → M on the leaf containing x 0 , is a parallel section for ∇, that is, ∇ξ 0 = 0.
Proof. For any y = ρ 0 (x 0 , s 0 ) with small s 0 ∈ S and any X ∈ s, we have Therefore, the directions of orbits of the action M × V S are horizontal for the leafwise connection ∇. By the expression of covariant derivative by parallel transport, we have for any ξ ∈ Γ(V ), X ∈ s and x ∈ M . Note that X ∈ s is regarded as X ∈ Γ(T F ) using the locally free action ρ 0 .
where Θ denotes the left Maurer-Cartan form of S. (In [2], Φ is referred to as an endomorphism, but it is the same Φ appearing in the definition of parameter equivalence which we saw in Section 1, so Φ can be taken as an automorphism. It is easy to see P * Θ is equivalent to the expression P −1 d F P in [2]. There is a small difference between our definition of parameter equivalence and the one in [2], since in [2] the map F is assumed to be homotopic to the identity through diffeomorphisms. But this does not cause any problem here.) Let a denote both projections s → s/n and S → S/N , where n is the nilradical of s and N is the Lie subgroup corresponding to n. By projecting (13), we get since s/n is abelian. For any x ∈ M , X ∈ s and T > 0, we integrate (14) over the curve ρ 0 x, e tX for 0 ≤ t ≤ T . Then noting Ψ being leafwise constant, T aΨ x * X = T aΦ * X + aP ρ 0 x, e T X − aP (x).
Since aP is bounded due to the compactness of M , we must have aΨ x * X = aΦ * X and aP is leafwise constant. Hence there exists a leafwise constant C ∞ map R : M → S such that Q = R −1 P : M → N . Since R is leafwise constant, we have and (13) becomes Let n ⊃ n 2 ⊃ · · · ⊃ n s ⊃ 0 be the lower central series of n. Recall that exp : n → N is a diffeomorphism and log : N → n is defined.

Sublemma.
Assume that there exist a C ∞ map Q : M → N and a leafwise constant C ∞ map R : M → S such that: Then we can find a C ∞ map Q ′ : M → N and a leafwise constant C ∞ map R ′ : M → S such that: Proof. Take subspaces V 0 , . . . , V s such that s = V 0 ⊕ n and n i = V i ⊕ n i+1 for i = 1, . . . , s. We can write Q = exp ( First note that In fact, for all X = d dt x(t) t=0 ∈ T x F , Let s π 0 k V k be the representation obtained from s ad n k /n k+1 by the identification We take s π k V k as s π V considered in the beginning of this section; we let ∇ be the leafwise connection defined by π k , and we let µ : Γ blc (V k ) → Γ lc (V k ) be the map in Lemma 36. Write Take the V k components of (16) to get and π 0 k vanishes on n, we have π k ω 0 = π 0 k β 0 . Therefore, Hence α k (X) = β k (X) + ∇ X Q k for any X ∈ s. Note that α k (X) and β k (X) are leafwise constant because R is leafwise constant. Applying µ and using Lemma 39, we get Therefore, representation with the derivative s π k V k . Then for any t ∈ R and x ∈ M , we have for all t ∈ R. Note that Q ′ k ρ 0 x, e tX is bounded with respect to t. Take a basis of V k which turns −π k (X) = −π 0 k (Φ * X) into a real Jordan normal form. Since any eigenvalue of ad X : s → s for any X ∈ s either is 0 or has nonzero real part by our assumption that s is of exponential type, the same is true for π 0 k (X) : V k → V k for all X ∈ s. Therefore, each Jordan block of −π k (X) = −π 0 k (Φ * X) has the eigenvalue which either is 0 or has nonzero real part. For a Jordan block whose eigenvalue has the nonzero real part, the corresponding components of e −tπ k (X) Q ′ k (x) have the following forms:    e ta * . . .
if the eigenvalue a is real, and    where R t = cos tb sin tb − sin tb cos tb if the eigenvalue a + bi is not real. Since this must be bounded for all t ∈ R, c 1 = · · · = c m = 0, which implies the corresponding components of Q ′ k ρ 0 x, e tX must be constant.
On the other hand, for a Jordan block with the eigenvalue 0, the corresponding where the entries in the * part of the matrix are now polynomials in t. Since bounded polynomial functions must be constant, we see the corresponding components in Q ′ k ρ 0 x, e tX are also constant. So Q k is leafwise constant. Put Q ′ = e −Q k Q. Then log Q ′ has values in n k+1 and Applying Sublemma 6 to (15) repeatedly, we finally get Q = 1 and therefore for some R. Therefore, Ψ x is equal to Φ modulo inner automorphisms.
Theorem 30 is restated and proved here.
Theorem 41. Assume that S is of exponential type and there is an outer derivation of s. If M ρ0 S is parameter rigid, then M is connected and H 0 (F ) = H 0 (s).
Proof. Since there is an outer derivation of s, the outer automorphism group Out(S) of S is nontrivial, hence M is connected. Take an outer derivation ϕ of s and set Φ t = e tϕ ∈ Aut(S). For any f ∈ H 0 (F ), consider a map M → Aut(S) defined by x → Φ f (x) . Since this is leafwise constant, x → Φ f (x) ∈ Out(S) is constant by Lemma 40. Let Inn(S) denote the inner automorphism group of S. This is a connected normal Lie subgroup of Aut(S). We must be a bit careful because Inn(S) might not be closed in Aut(S) in general. See Hochschild [8]. But the cosets of Inn(S) defines a foliation on Aut(S) and Φ t is a curve transverse to the foliation. Since the automorphisms Φ f (x) for all x ∈ M are contained in a single leaf of F and M is connected, Φ f (x) must be constant with respect to x. This implies f is constant over M .
Finally we see vanishing of H 0 with nontrivial coefficients.

Vanishing of H 1 -proof of Theorem 35
Here we prove the following (a restatement of Theorem 35).
Theorem 43. Let V ⊂ s be an ad-invariant subspace (ie an ideal of s) for which n ad V is trivial. Assume that any eigenvalue of ad X on s/V either is 0 or has nonzero real part for any X ∈ s. If M ρ0 S is parameter rigid, then we have Proof. Take any [ω] ∈ H 1 F ; s ad V . Let ω 0 be the canonical 1-form of ρ 0 . Fix an ǫ > 0 and put η := ω 0 + ǫω ∈ Γ (Hom (T F , s)). Let us see η satisfies the Maurer-Cartan equation. As we saw in Section 5, V is abelian and then But this is zero because ω satisfies d F ω + (ad ω 0 ) ∧ ω = 0 and (ad ω 0 ) ∧ ω = [ω 0 , ω] + [ω, ω 0 ]. Since M is compact, we can assume η x : T x F → s is bĳective for all x ∈ M by taking ǫ > 0 small enough. Then there exists a unique action ρ of S on M whose orbit foliation is F and whose canonical 1-form is η. See Asaoka [2, Proposition 1.4.3]. By parameter rigidity, ρ is parameter equivalent to ρ 0 . Thus by Proposition 1.4.4 of [2], there exist a C ∞ map P : M → S and an automorphism Φ of S satisfying ω 0 + ǫω = Ad P −1 Φ * ω 0 + P * Θ, where Θ is the left Maurer-Cartan form of S. By seeing this equation modulo n, we get ω 0 ≡ Φ * ω 0 + d F P mod n, where bar denotes the projection S → S/N . The same argument as in the proof of vanishing of H 0 yields ω 0 ≡ Φ * ω 0 mod n, d F P ≡ 0 mod n.
So we can take a leafwise constant C ∞ map R : M → S such that Q := R −1 P ∈ N . Then Equation (18) becomes where Ψ * = Ad R −1 Φ * is leafwise constant.

Lemma 44. There exists a filtration
where W i 's are ideals of s such that [n, W i ] ⊂ W i+1 .
Proof. The proof is similar to the proof of Sublemma 6. Take complementary subspaces V i 's so that s = V 0 ⊕ n and W i = V i ⊕ W i+1 . Write We have

Equation (19) gives
If k = s, we have If ∇ denotes the covariant derivative defined from s ad V , then by [n, V ] = 0 we have Therefore, ω is cohomologous to ǫ −1 (β s − α s ) which is leafwise constant since so are ω 0 and Ψ * ω 0 . If k < s, then Let s π k V k denote the representation obtained from s ad W k /W k+1 by the identification W k /W k+1 ≃ V k , and let ∇ be the leafwise connection defined by π k . Recall that ∇Q k = d F Q k + π k ω 0 Q k . Since we have α k ≡ β k + d F Q k + π k ω 0 Q k mod W k+1 , which implies By the same argument starting from Equation (17) in the proof of vanishing of H 0 , using the assumption on the eigenvalues of ad X, we can conclude that Q k is leafwise constant. Define Q ′ : M → N by Q = e Q k Q ′ . Then Equation (19) becomes where Ψ ′ * = Ad e −Q k Ψ * . Now we have log Q ′ ∈ W k+1 and Applying Lemma 45 repeatedly, we see that ω is cohomologous to a leafwise constant cocycle. Note that we have used the assumption on the eigenvalues only on V 1 , . . . , V s−1 , but not on V s = V .