Rigidity of trivial actions of abelian-by-cyclic groups
aa r X i v : . [ m a t h . D S ] A ug RIGIDITY OF TRIVIAL ACTIONS OFABELIAN-BY-CYCLIC GROUPS
ANNE E. MCCARTHYDepartment of MathematicsFort Lewis CollegeDurango, CO, 81301
Abstract.
Let Γ A denote the abelian-by-cyclic group associatedto an integer-valued, non-singular matrix A . We show that if A has no eigenvalues of modulus one, then there are no faithful C perturbations of the trivial action ι : Γ A → Diff ( M ), where M isa compact manifold. introduction The question of existence and stability of global fixed points for groupactions has been studied in many different contexts. In the setting ofactions of Lie groups it was shown by Lima [8] that n commuting vectorfields on a genus g surface, Σ g , of non-zero Euler characteristic havea common singularity. This implies that any action of the abelian Liegroup R n on Σ g has a global fixed point. It was later shown by Plante[10] that any action of a nilpotent Lie group on a surface with non-zeroEuler characteristic has a global fixed point.The study of stability of global fixed points for group actions is alsorelated to the study of foliations. Given a foliation of a manifold witha compact leaf L and a transverse disk D, the holonomy map along L defines an action of π ( L ) on the disk D . Perturbations of groupactions are related to the study of foliations, for given a nearby leaf L ′ diffeomorphic to L , the holonomy along L ′ defines a new action thatis a perturbation of the original. The Thurston stability theorem givesconditions for local stability of C foliations of a compact manifold.Methods of Thurston were then modified by Langevin and Rosenberg[7], Stowe [11] [12], and Schweitzer [13] to establish results regardingstability of global fixed points, and leaves of fibrations.Inspired by the ideas of Lima [8], Bonatti [1] used methods similar toThurston’s to show that any Z n action on surfaces with non-zero Eulercharacteristic generated by diffeomorphisms C close to the identityhas a global fixed point. Using similar techniques, Druck, Fang and Firmo [4] proved a discrete version of Plante’s theorem. The dynamicsof group actions generated by real analytic diffeomorphisms close tothe identity were studied by Ghys [6], who showed that such actionswere either recurrent, or displayed a property similar to solvability ofthe group.We examine how a particular family of solvable groups acts via dif-feomorphisms close to the identity on a compact manifold. We showthat such actions display near-rigidity, in the sense that there are nofaithful actions with generators close to the identity. It is an easy con-sequence that actions close to the identity of these groups on a compactsurface of non-zero Euler characteristic have a global fixed point.Let A be an invertible n × n matrix with integer entries. To thematrix A one can associate the solvable groupΓ A = h a, b , ...b n | b i b j = b j b i , ab i a − = Y j b A ij j i . For instance, when A = (cid:18) (cid:19) the associated group isΓ A = h a, b , b | b b = b b , ab a − = b b , ab a − = b b i . In the case where A is a 1 × A = [ n ], the associated group isthe Baumslag-Solitar group , BS (1 , n ) = h a, b | aba − = b n i . These groups have a geometric interpretation as the fundamental groupof the space T n × [0 , / ∼ where ends are glued via the toral endomor-phism induced by A . A group Γ is said to be abelian-by-cyclic if thereexists an exact sequence1 → A → Γ → Z → , where the group A is abelian, and Z is an infinite cyclic group. Notethat the commutator subgroup [Γ , Γ] is contained in A , so all such Γare solvable groups. The class of all finitely presented, torsion free,abelian-by-cyclic groups is exactly given by groups of the form Γ A . See[5] for a nice proof of this.We are interested in actions of the groups Γ A on compact manifoldsin the case where the matrix A does not have an eigenvalue of modulusone. Given a finitely generated group Γ and a manifold M , a C r actionof Γ on M is a homomorphism ρ : Γ → Diff r ( M ). We commonly referto this homomorphism as a representation (into Diff r ( M )) and use theassociated language. The representation ρ is said to be faithful if ρ is IGIDITY OF TRIVIAL ACTIONS OF ABELIAN-BY-CYCLIC GROUPS 3 injective. We denote by R r (Γ , M ) the collection of all representationsof Γ into Diff r ( M ).We now fix some notations that we use for the remainder of thisexposition. Let M be a compact manifold embedded in R ℓ . Let k · k denote the standard Euclidean metric on R ℓ . Given a C diffeomor-phism h , we will define k h k = sup x ∈ M {k h ( x ) k + k D x h k} . With this we define the C distance d ( h, g ) = k h − g k . The collection of C representations R (Γ , M ) carries a topology.For the group Γ, fix a generating set h γ , ..., γ k i , and let d C ( ρ , ρ ) = sup γ ,...,γ k d C ( ρ ( γ i ) , ρ ( γ i )) . This distance depends on the choice of generating set for the groupΓ. However, given any two generating sets the associated metrics areequivalent. We now state our main theorem.
Theorem 1.1.
Let M be a compact manifold and A be a non-singular n × n matrix with integer entries. Let h a, b , . . . b n i be generators Γ A . If A has no eigenvalue of modulus one, then there exists ǫ > such thatany C action ρ : Γ A → Diff ( M ) with d C ( ρ, id ) < ǫ is not faithful.In particular, ρ ( b i ) = id for all i = 1 , , · · · n . Remark:
This can be viewed as a rigidity result in the followingsense. Consider the trivial representation ι : Γ A → Diff ( M ) , given by ι ( a ) = ι ( b i ) = id . Note that any assignment ρ : Γ A → Diff ( M ) with ρ ( a ) = f and ρ ( b i ) = g i = id determines an action. Therefore, thereare infinitely many perturbations of the trivial representation of theform ρ ( g i ) = id . The main theorem states that representations with ρ ( b i ) = id are in fact the only C perturbations of the trivial action.The compactness hypothesis is necessary for this theorem, see remarksat the end of Section 3 for further discussion.This result is not true for C perturbations. The groups Γ A arediscrete subgroups of the affine group. The standard representationsare given by α ( a )( x ) = λx , and α ( b i )( x ) = x + v i , where λ is aneigenvalue of A with corresponding eigenvector v . An example of Lima[8] gives two vector fields for which the associated flows give an actionof the affine group on the sphere S . This example produces actionsarbitrarily close to the identity in R (Γ A , S ), by taking sufficiently ANNE E. MCCARTHY small time- t maps of the flows by which the affine group acts. In fact,this action has no global fixed point.Under the hypotheses of Theorem 1.1, we characterize global fixedpoints: Corollary 1.2. If d C ( ρ − id ) < ǫ, then any fixed point of ρ ( a ) is aglobal fixed point for ρ. Acknowledgments.
This project grew out of dissertation work com-pleted at Northwestern University. Special thanks are extended toAmie Wilkinson for her guidance and many helpful conversations dur-ing the course of this project. I would also like to thank ChristianBonatti for discussions conveying valuable intuitions regarding the be-havior of diffeomorphisms close to the identity.2.
Preliminary Tools
Central to Thurston’s argument is that given a non-trivial holonomymap H about a compact leaf L , if the linear action dH : π ( L ) → GL ( k, R k ) is trivial then there is non-trivial representation u : π ( L ) → R k . Stowe [11] later showed that in the case where the linear action isnon-trivial, there is an analogous cocyle u : G → R k . Further results re-garding how this representation behaves under perturbation have beenproved by Langevin and Rosenberg [7], Stowe [11], [12], and Schweitzer[13]. This body of work establishes the following lemma. Lemma 2.1 (Thurston) . Let G be a group with finitely many gener-ator g , g , . . . g r , and let M be a manifold embedded in R ℓ . Supposethere exist an action ρ : G → Diff ( M ) and a point p ∈ M for which ρ ( g )( p ) = p for all g ∈ G , and D p ρ ( g ) = id for all g ∈ G . Then either (1) There is a neighborhood of p consisting of points that are fixedby all actions near ρ in R ( G, M ) or (2) There exist actions ρ k → ρ in R ( G, M ) and points x k → p in M with x k not fixed by ρ k for which the sequence u k : G → R ℓ given by u k ( g ) = ρ k ( g ) x k − x k max {k ρ k ( g j ) x k − x k k : 1 ≤ j ≤ r } converges to a non-trivial homomorphism u : G → T p M . The vector f ( x ) − x in R ℓ indicates how the point x is moved by f .The above lemma characterizes such displacement vectors . The homo-morphism u associates to each g ∈ G a normalized displacement vector.For points x near the fixed point p , we note that the displacement of x by ρ ( g g ) is close to the vector sum of the displacements associated IGIDITY OF TRIVIAL ACTIONS OF ABELIAN-BY-CYCLIC GROUPS 5 to ρ ( g ) and ρ ( g ). Bonatti [2] modified the ideas of Thurston to pro-vide estimates on how much the assignment g ρ ( g )( x ) − x can differfrom a homomorphism when the image of ρ is close to the identity in R ( G, M ). These modifications do not require a global fixed point withtrivial linear action.
Lemma 2.2 (Bonatti) . For all η > , N ∈ N , there exists ǫ > suchthat the following property holds: Let f , f , ...f N be diffeomorphisms f i : M → M such that d C ( f i , id ) < ǫ . Then for all x ∈ M, (cid:13)(cid:13)(cid:13) ( f ◦ f ◦ · · · ◦ f N − id ) ( x ) − N X i =1 ( f i − id ) ( x ) (cid:13)(cid:13)(cid:13) < η sup i k ( f i − id ) ( x ) k . As we proceed, we will consider displacements in R ℓ of n differentfunctions simultaneously using an n × ℓ displacement matrix. The fol-lowing elementary bound concerning entries of a matrix will be useful. Lemma 2.3.
Let A be an n × ℓ matrix, with row vectors a i and columns v j . Then for all ≤ i ≤ n , k a i k ≤ √ ℓ sup {k v j k , ≤ j ≤ ℓ } . Proof.
Note that n X i =1 k a i k = ℓ X j =1 k v j k , since both of these quantities give the sum of the square of all theentries of A . Therefore, for all 1 ≤ i ≤ n , k a i k ≤ n X i =1 k a i k = ℓ X j =1 k v j k ≤ ℓ sup {k v j k , ≤ j ≤ ℓ } . (cid:3) It will also be useful to rewrite the composition a k b i a − k in terms ofthe generators b , . . . , b n . Here ( A k ) ij is the entry of A k in the ( i, j )position. Lemma 2.4.
For generators a and b i of Γ A , a k b i a − k = Y j a ( A k ) ij j . ANNE E. MCCARTHY
Proof.
We use induction on k , where the group relation ab i a − = Q j b A ij establishes the base case. Suppose that a k b i a k = Q b ( A k ) iℓ ℓ .This implies that a k +1 b i a − ( k +1) = f Y ℓ b ( A k ) iℓ ℓ ! a − = Y ℓ ( ab ℓ a − ) ( A k ) iℓ . Applying the group relation to ab ℓ a − we get that a k +1 b i a − ( k +1) = Y ℓ Y j b A ℓj j ! ( A k ) iℓ = Y j b P ℓ ( A k ) iℓ A ℓj j . Which shows that a k +1 b i a − ( k +1) = Q j b ( A k +1 ) ij j . (cid:3) Overview of Proof of Theorem 1.1
Suppose ρ : Γ A → Diff ( M ) is an action. Let ρ ( a ) = f and ρ ( b i ) = g i denote the images of the generators of Γ A within Diff r ( M ). For theaction to be close to the identity means that the functions f and g i ,1 ≤ i ≤ n all have distance to the identity in R ( G, M ) less than ǫ .We apply our understanding of behavior of diffeomorphisms close tothe identity to examine how displacements by the functions g i vary aswe move from a point x to f ( x ).In particular, we show that g i -displacements of the point x are relatedto those at the point f ( x ) by the matrix A that defines the group. Thisis seen fairly easily in the case that the matrix A = [ n ] is a 1 × y = f ( x ) , then k g ( x ) − x k = k gf − ( y ) − f − ( y ) k≈ k D g ( x ) f k k f gf − ( y ) − ( y ) k ≈ k g n ( y ) − y k≈ n k g ( y ) − y k The above approximations use that the derivative of f has norm closeto one, and that the displacement resulting from a composition canbe viewed as a sum of displacements. More generally, we establish forthe functions g , . . . , g n that if we collect all displacements of a point x ∈ R ℓ into an n × ℓ matrix D ( x ), then D ( x ) ≈ AD ( y ) . We then examine the consequences of having the displacement bythe functions g , . . . , g n at different points related via the applicationof a hyperbolic matrix. On the one hand, by assuming that the actionis close to the identity, we are assuming that all the displacements bythe functions g j are uniformly small. On the other hand, displacementsof the point x are related to those of the point f ( x ) by application of IGIDITY OF TRIVIAL ACTIONS OF ABELIAN-BY-CYCLIC GROUPS 7 the hyperbolic matrix A . In order to conclude the proof, use compact-ness to choose x to be the point with a sort of maximal displacement.Follow displacements along a partial orbit x, f ( x ) , . . . , f k ( x ) until dis-placements have been expanded by A . These expanded displacementscontradict the choice of x unless the original displacements were zero.Note that the estimates provided in Lemma 2.2 depend on the numberof functions that are composed, so we must fix the number of iteratesneeded to detect expansion.It is of interest that this argument is very different from those thatuse Thurston’s stability theorem. The contradiction in this argumentoccurs at a place of maximal displacement for ρ ( b i ), and not a globalfixed point. In fact, one can locally define a faithful action ρ with aglobal fixed point p for which D ( ρ ( g ))( p ) = id : The standard affineaction ρ ( a )( x ) = λx , ρ ( b i )( x ) = x + v i fixes infinity. On a positive half-neighborhood of infinity we modify this action using ideas of Navas [9].Conjugate F = φ − ◦ f ◦ φ and G i = φ − ◦ g i ◦ φ , where φ = e x . The resultis an action defined on [0 ,
1) that fixes 0, with F ′ (0) = G ′ i (0) = 1. Thiscan be extended by defining the functions G , . . . , G n to be the identityon ( − ,
0) and any appropriate choice of F on ( − ,
0) to produce anaction on the interval ( − , . g i -Displacement Along Orbits of f Before proceeding, we fix a number of constants. Given an n × n matrix A with no eigenvalues of modulus one there exists an A -invariantsplitting R n = E u ⊕ E s and constants k ∈ N , θ u >
1, and 0 < θ s < k A k v k > θ u k v k for all v ∈ E u and k A k v k < θ s k v k for all v ∈ E s . We fix k to be the least positive integer that ensures this expansion andcontraction, and fix the corresponding θ u and θ s . We also fix N ∈ N to based on the number of composed functions to which we will applyLemma 2.2. In particular, set N > max i X j (cid:12)(cid:12) A kij (cid:12)(cid:12) . Next, fix α < / C k = k A k k + α − α ANNE E. MCCARTHY where k A k k denotes the operator norm. Select 0 < η < α √ nℓ smallenough to ensure that2 η √ nℓ (2 C k + 1) < min { θ u − , − θ s } . We now choose η such that d ( f, id ) < η implies that d ( f k , id ) <η , and d ( f, id ) < η . For a C function, we know by definition that f ( x + y ) − f ( x ) = Df ( y ) + o ( y ) . Furthermore we an find an ǫ suchthat if k y k < ǫ , then o ( y ) < η k y k . Let ǫ be selected so that Lemma2.2 holds for N and η as selected above. Finally, set ǫ = min { η , ǫ , ǫ } . This will be the ǫ for which the theorem holds.We now introduce some notation. All preliminary estimates applyto displacement vectors of the form h ( x ) − x, where h can be any of thegenerating diffeomorphisms f or g i , and h ( x ) − x is a vector in R ℓ . Forthis reason, we write H ( x ) = h ( x ) − x. We think of H ( x ) as a vectorthat will sometimes be written in components as an ℓ -dimensional rowvector. To each point x in M , there is an associated n × ℓ matrix, D ( x ) , of displacements by the functions g i , D ( x ) = G ( x )... G n ( x ) . Specifically, the i th row of D is the ℓ -dimensional row vector g i ( x ) − x ,corresponding to the displacement of x by the diffeomorphism g i .Next, we examine how the displacement matrix D ( x ) varies as wemove along orbits of f . We will see that D ( x ) ≈ AD ( f ( x )). Lemma 4.1.
Let ( D ( x )) j denote the j th column of the displacementmatrix D ( x ) at the point x . If y = f k ( x ) , then k ( D ( x )) j − A k ( D ( y )) j k < η √ nℓ (cid:18) j k ( D ( x )) j k + sup j k ( D ( y )) j k (cid:19) . for all ≤ j ≤ ℓ .Proof. Suppose we are given f ∈ Diff ( M ) with d C ( f k , id ) < η andlet f − k ( y ) = x . By the definition of the derivate, we know that f k ( g i ( x )) = f k ( x ) + D x f k ( G i ( x )) + o ( k G i ( x ) k ) . From the selection ǫ , we know that o ( k G i ( x ) k ) < η k G i ( x ) k , thus allowing us to bound k f k g i ( x ) − f k ( x ) − D x f k ( G i ( x )) k < η k G i ( x ) k . IGIDITY OF TRIVIAL ACTIONS OF ABELIAN-BY-CYCLIC GROUPS 9
Letting x = f − k ( y ), and using that k D x f k − Id k < η this implies that k ( f k g i f − k ( y ) − y ) − ( G i ( x )) k < η k G i ( x ) k , which, by Lemma 2.4 is equivalent to the bound (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y j g ( A k ) ij j − id ! ( y ) − G i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < η k G i ( x ) k . By Lemma 2.2, we know that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y j g ( A k ) ij j − id ! ( y ) − X j ( A k ) ij ( g j − id )( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < η sup j {k ( g j − id )( y ) k} . So that for each i = 1 , , ...n , and for all x ∈ M, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A k G ( y )... G n ( y ) i − G i ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < η k G i ( x ) k + η sup j k G j ( y ) k . Specifically, this inequality tells us that the i th row of D ( x ) differsvery little in norm from the i th row of A k D ( y ). This implies that thespecific entries ( D ( x )) ij and ( A k D ( y )) ij must also satisfy (cid:12)(cid:12) ( D ( x )) ij − ( A k D ( y )) ij (cid:12)(cid:12) < η (cid:18) i k G i ( x ) k + sup j k G j ( y ) k (cid:19) . Therefore the columns must satisfy k ( D ( x )) j − A k ( D ( y )) j k < η √ n (cid:18) i k G i ( x ) k + sup j k G j ( y ) k (cid:19) . By application of Lemma 2.3, we can bound the norms k G i ( x ) k and k G i ( y ) k using the columns of the corresponding displacement matrices: k ( D ( x )) j − A k ( D ( y )) j k < η √ nℓ (cid:18) j k ( D ( x )) j k + sup j k ( D ( y )) j k (cid:19) . (cid:3) In order to prove the theorem we must modify the estimates ofLemma 4.1, so that our upper bound depends on displacements atonly one point.
Lemma 4.2.
There exists a constant C k > such that for all x ∈ M sup i k ( D ( x )) i k < C k sup j k ( D ( f k ( x ))) j k Proof.
It suffices to show for η < α √ nℓ . Let x ∈ M be given. Select1 ≤ j ≤ ℓ such that k ( D ( x )) j k = sup j k ( D ( x )) j k . Applying Lemma 4.1 to x , j and y = f k ( x ) we see that k ( D ( x )) j − A k ( D ( y )) j k < η √ nℓ (cid:18) k ( D ( x )) j k + sup j k ( D ( y )) j k (cid:19) . Which implies that(1 − α ) k ( D ( x )) j k < k A k ( D ( y )) j k + α sup j k ( D ( y )) j k . So we can conclude that(1 − α ) k ( D ( x )) j k < ( k A k k + α ) sup j k ( D ( y )) j k . Setting C k = k A k k + α − α we get the desired result. (cid:3) Proof of Theorem 1.1
We now conclude the proof of Theorem 1.1.
Proof.
Choose z and j to be the values that attain the supremumsup x ∈ M, ≤ j ≤ ℓ {k π u (( D ( x )) j ) k , k π s (( D ( x )) j ) k} . We consider two different cases. If the supremum is attained by k π u (( D ( z )) j ) k = sup x ∈ M, ≤ j ≤ ℓ {k π u (( D ( x )) j ) k , k π s (( D ( x )) j ) k} , we note that k ( D ( z )) j k ≤ k π u (( D ( z )) j ) k + k π s (( D ( z )) j ) k ≤ k π u (( D ( z )) j ) k . Consider the point x = f − k ( z ), and apply Lemmas 4.1 and 4.2. Wesee that k π u (( D ( x )) j ) − π u ( A k ( D ( z )) j ) k < η √ nℓ (2 C k + 1) k π u (( D ( z )) j ) k . We conclude that k π u (( D ( x )) j ) k > ( θ u − η √ nℓ (2 C k +1)) k π u (( D ( z )) j ) k > k π u (( D ( z )) j ) k , where the last inequality follows from our choice of η .Likewise if the supremum is attained by k π s (( D ( z )) j ) k , we set y = f k ( x ), and are able to conclude k π s (( D ( y )) j ) k > ( θ s +2 η √ nℓ (2 C k +1)) k π s (( D ( z )) j ) k > k π s (( D ( z )) j ) k . So we conclude D ( x ) = 0 for all x , thus proving the theorem. (cid:3) IGIDITY OF TRIVIAL ACTIONS OF ABELIAN-BY-CYCLIC GROUPS 11
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