Anosov diffeomorphisms on infra-nilmanifolds associated to graphs
aa r X i v : . [ m a t h . D S ] A ug ANOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDSASSOCIATED TO GRAPHS
JONAS DER´E AND MEERA MAINKAR
Abstract.
Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems ex-hibiting uniform hyperbolic behavior. Therefore their properties are intensively studied, including whichspaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov dif-feomorphism is homeomorphic to an infra-nilmanifold, i.e. a compact quotient of a 1-connected nilpotentLie group by a discrete group of isometries. This conjecture motivates the problem of describing whichinfra-nilmanifolds admit an Anosov diffeomorphism.So far, most research was focused on the restricted class of nilmanifolds, which are quotients of 1-connected nilpotent Lie groups by uniform lattices. For example, Dani and Mainkar studied this questionfor the nilmanifolds associated to graphs, which form the natural generalization of nilmanifolds modeled onfree nilpotent Lie groups. This paper further generalizes their work to the full class of infra-nilmanifoldsassociated to graphs, leading to a necessary and sufficient condition depending only on the induced actionof the holonomy group on the defining graph. As an application, we construct families of infra-nilmanifoldswith cyclic holonomy groups admitting an Anosov diffeomorphism, starting from faithful actions of theholonomy group on simple graphs. Introduction
A diffeomorphism f : M → M on a closed Riemannian manifold M is called Anosov if the tangent bundle
T M has a continuous splitting
T M = E u ⊕ E s preserved by Df : T M → T M such that Df exponentiallyexpands E u and exponentially contracts E s with respect to the Riemannian metric. This property does notdepend on the choice of metric on M and hence we will just talk about Anosov diffeomorphisms on a closedmanifold. The first example of an Anosov diffeomorphism was Arnolds’ cat map , which is the map inducedby the matrix on the 2-torus T = R (cid:30)Z . Similarly it is easy to find for every n ≥ n, Z ) which induce Anosov diffeomorphisms on the torus T n = R n (cid:30)Z n , whereas the circle S does notadmit such a map. Note that the n -dimensional torus is exactly the nilmanifold modeled on the abelian Liegroup R n .In his seminal paper [Sma67], S. Smale gave the first example of a non-toral Anosov diffeomorphism andraised the question of classifying the closed manifolds M admitting such an Anosov diffeomorphism. Aftermore than 50 years, the only known examples are on spaces homeomorphic to infra-nilmanifolds, which wewill introduce in full detail in Section 2.1. In short, an infra-nilmanifold is a compact quotient of a simply Mathematics Subject Classification.
Primary: 37D20; Secondary: 22E25, 20F34.The first author was supported by a postdoctoral fellowship of the Research Foundation – Flanders (FWO). connected nilpotent Lie group N by a discrete subgroup of isometries. Nowadays, it is conjectured thatevery manifold admiting an Anosov diffeomorphism is in fact homeomorphic to an infra-nilmanifold with anAnosov diffeomorphism, but unfortunately there is no recent progress towards a proof. Hence most researchfocuses on the problem of describing the infra-nilmanifolds admitting an Anosov diffeomorphism.Even for the restricted class of nilmanifolds, this is a hard question, leading to a variety of techniques forconstructing and classifying nilmanifolds admitting such diffeomorphisms. For example, the papers [Lau03,Pay09, Der15] give different methodes for constructing lattices in nilpotent Lie groups, leading to Anosovdiffeomorphisms satisfying additional properties. Some of these constructions are general enough to give acomplete list of possibilities, e.g. [LW09] gives a classification of Anosov diffeomorphisms on nilmanifolds ofdimension ≤
8, which was slightly corrected by the second author in [Der15].One particular paper in this direction is [DM05] which gives a full classification in the special case ofnilmanifolds associated to graphs, generalizing the result of [Dan78] which treated the free nilpotent Liegroups. In the general case of infra-nilmanifolds, a lot less is known, and the only full characterizationof Anosov diffeomorphisms is in the case of infra-nilmanifolds modeled on free nilpotent Lie groups in[DV11, DD14], which extended the classical result of Porteous for flat manifolds [Por72]. In this paper, wegeneralize the latter result to the class of infra-nilmanifolds modeled on Lie groups associated to graphs.To state the main result, we first introduce some notations. Every infra-nilmanifold induces a uniquerational Lie algebra n Q and a representation of a finite subgroup ρ : H → Aut( n Q ) of automorphisms,which is called the rational holonomy representation. In the case of infra-nilmanifolds associated to a graph G , this rational holonomy representation induces an action on the coherent components of the graph G and representations ρ i : H i → GL( V λ i ) for some coherent components λ i . The subgroups H i ≤ H are thestabilizers of the coherent components λ i for the action of H . We show that these representations completelydetermine the existence of an Anosov diffeomorphism. Theorem 1.1.
Let Γ \ N G be an infra-nilmanifold associated to a graph G with rational holonomy represen-tation ρ : H → Aut( n Q G ) . If ρ i : H i → GL( V λ i ) are the induced representations on the coherent components,then the following are equivalent.The infra-nilmanifold Γ \ N G admits an Anosov diffeomorphism. m For every ≤ i ≤ k , every Q -irreducible component of ρ i that occurs with multiplicity m splits in more than c ( H · λ i ) m components over R . The numbers c ( H · λ i ) are either 1 or 2 and depend only on the orbit H · λ i of the coherent component λ i under the action of H . A more detailed explanation of the notation in this theorem will follow in Section 4.In the special case where H acts faithfully on the coherent components, each group H i is trivial and thecriterion only depends on the number of elements in the component. We use this observation to give families NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 3 of examples starting from faithful actions on finite graphs, leading to several families of infra-nilmanifoldsmodeled on graphs admitting an Anosov diffeomorphism.We start by giving some general results about Anosov diffeomorphisms on infra-nilmanifolds in Section2, where the crucial ingredient is the characterization of [DV11] in terms of rational Lie algebras and therational holonomy representation. Section 3 gives then a full description of the automorphism group of Liealgebras associated to graphs, including an induced action of finite subgroups on the coherent components.Next, we apply these results to get the main result in Section 4 and present the construction for examples inSection 5. Finally, we state some open questions in Section 6, related to other rational forms of Lie algebrasassociated to graphs. 2.
Anosov diffeomorphisms on infra-nilmanifolds
In this section, we introduce the necessary definitions about infra-nilmanifolds and recall the charac-terization in [DV11] of infra-nilmanifolds admitting an Anosov diffeomorphism depending on the rationalholonomy representation. Just as in the case of free nilpotent Lie groups, this will be the starting point foranalyzing the existence of Anosov diffeomorphisms on Lie groups associated to graphs.2.1.
Infra-nilmanifolds.
Let N be a connected and simply connected (hereinafter 1-connected) nilpotentLie group and let Aut( N ) denoted the group of Lie group automorphisms of N . The affine group Aff( N ) isdefined as the semi-direct product N ⋊ Aut( N ) and acts on the Lie group N on the left as follows:( n, φ ) · x = nφ ( x ) for all ( n, φ ) ∈ Aff( N ) , x ∈ N. Let C be a compact subgroup of Aut( N ) and let Γ be a discrete, torsion-free subgroup of N ⋊ C suchthat the quotient Γ \ N is compact. We note that such a Γ is known as an almost-Bieberbach group and since C is compact, the group Γ can be seen as a subgroup of isometries on N for a suitable Riemannian metric.The quotient space Γ \ N is a closed manifold and is called an infra-nilmanifold modeled on the Lie group N .If C is trivial, then Γ ≤ N is a uniform lattice and in this case the manifold Γ \ N is called a nilmanifold.It is known that the normal subgroup M = Γ ∩ N of Γ is a uniform lattice in N and that the quotientΓ (cid:30) M is finite by [Aus60], where we identify N with the subgroup of left translations N × { } of N ⋊ C . Thisimplies that the infra-nilmanifold Γ \ N is finitely covered by the nilmanifold M \ N with H = Γ (cid:30) M as thegroup of covering transformations. Let p : Γ → C denote the natural projection on the second component,then H = p (Γ) ≈ Γ (cid:30) M is a finite group which is known as the holonomy group of Γ. An infra-nilmanifoldis a nilmanifold if and only if the holonomy group is trivial. The group Γ fits in the following short exactsequence: 1 / / M / / Γ p / / H / / . (2.1)Without loss of generality, one can assume that C = p (Γ) = H . The infra-nilmanifolds modeled on theadditive group R m are exactly the closed flat Riemannian manifolds. JONAS DER´E AND MEERA MAINKAR
Let α = ( n, φ ) ∈ Aff( N ) be an affine transformation such that α Γ α − = Γ. Then α induces a diffeomor-phism ¯ α on the infra-nilmanifold Γ \ N , which is defined by¯ α : Γ \ N → Γ \ N Γ x ¯ α (Γ x ) := Γ ( α · x ) = Γ nφ ( x ) . A diffeomorphism of an infra-nilmanifold as above is called an affine infra-nilmanifold automorphism . Themap ¯ α is an Anosov diffeomorphism if and only if the linear part φ of α is hyperbolic, i.e. all the eigenvaluesof φ are of absolute value different from 1. The eigenvalues of φ are defined as the eigenvalues of thecorresponding automorphism on the Lie algebra corresponding to N , so the eigenvalues of the differential of φ at the identity.The only known examples of Anosov diffeomorphisms are those which are topologically conjugate to affineinfra-nilmanifold automorphisms. In fact, it is conjectured that every Anosov diffeomorphism is topologicallyconjugate to an affine infra-nilmanifold automorphism, see [Dek12, Sma67]. Therefore an important questionis to study which infra-nilmanifolds admit an affine hyperbolic infra-nilmanifold automorphism. Note thatan infra-nilmanifold admits an Anosov diffeomorphism if and only if it admits a hyperbolic affine infra-nilmanifold automorphism by [Dek12].2.2. Rational holonomy representation.
For infra-nilmanifolds modeled on the abelian Lie group R m ,the short exact sequence (2.1) splits and thus gives rise to a natural representation of the holonomy group ρ : H → Aut( Z m ) = GL( m, Z ) . For general infra-nilmanifolds, this is not the case and therefore we introduce the rational holonomy repre-sentation associated to an infra-nilmanifold Γ, see [DV11].Let n denote the nilpotent Lie algebra associated to the nilpotent Lie group N . It is known that if N is 1-connected, then the exponential map exp : n → N is a diffeomorphism, see for example [Rag72]. Letlog : N → n denote the inverse of the map exp. As before we take M = Γ ∩ N as uniform lattice in N and let n Q denote the Q -span of log( M ), which is a rational Lie algebra by [Seg83]. We define M Q as exp( n Q ) andthis group is known as the rational Mal’cev completion or radicable hull of M . It is the unique torsion-freeradicable nilpotent group containing M , such that every element of M Q has a positive power lying in M ,see [Seg83]. The rational subalgebra n Q is called a rational form of the real Lie algebra n .To obtain the rational holonomy representation, one embeds the lattice M into its rational Mal’cevcompletion M Q . Since every automorphism of M uniquely extends to an automorphism of M Q , there existsa group Γ Q which fits in the following commutative diagram: NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 5 / / M / / (cid:15) (cid:15) (cid:15) (cid:15) Γ / / (cid:15) (cid:15) (cid:15) (cid:15) H / / / / M Q / / Γ Q / / H / / , where the bottom exact sequence splits, see [Dek96, Section 3.1] for the details. By fixing a splittingmorphism s : H → Γ Q , we define the rational holonomy representation ρ : H → Aut( M Q ) by ρ ( h )( n ) = s ( h ) ns ( h ) − . Equivalently, since Aut( M Q ) ≈ Aut( n Q ) under the exponential map, we can consider the representation as ρ : H → Aut( n Q ) into the automorphisms of the corresponding Lie algebra. We will not distinguish betweenthese representations and call them both the rational holonomy representation of the almost-Bieberbachgroup Γ. The map ρ is always injective, so the rational holonomy representation is faithful. Often we willidentify the holonomy group with its image under the rational holonomy representation.The rational holonomy representation does depend on the choice of s , but this dependence is not relevantin the study of Anosov diffeomorphisms. In fact, by the work of [DV09a, Theorem A] the rational holonomyrepresentation contains all information about the existence of Anosov diffeomorphisms. Theorem 2.1.
Let Γ \ N be an infra-nilmanifold with rational holonomy representation ρ : H → Aut( M Q ) .Then Γ \ N admits an Anosov diffeomorphism if and only if there exists an integer-like hyperbolic automor-phism of M Q which commutes with every element of ρ ( H ) . Here, integer-like means that the characteristic polynomial of the corresponding automorphism on the Liealgebra n Q has coefficients in Z and constant term ±
1. An automorphism ϕ ∈ Aut( M Q ) satisfying the con-ditions of the theorem is called Anosov and every rational Lie algebra admitting an Anosov automorphism iscalled an
Anosov Lie algebra . Therefore studying Anosov diffeomorphisms on infra-nilmanifolds is equivalentto studying Anosov automorphisms of rational Lie algebras commuting with certain finite subgroups of theLie algebra. In this paper, we restrict ourselves to the case of Lie algebras associated to graphs, for whichwe give a full description of the automorphism group in the next section.3.
The automorphism group of Lie groups associated to graphs
In this section we describe the automorphism group of Lie groups, or equivalently Lie algebras, associatedto graphs, improving the result of [DM05] which describes only the connected component of the identityelement. The methods lead to an induced action of the holonomy group on the coherent components of thegraph, which is crucial for our final results. We start by recalling the construction of a 2-step nilpotent Liealgebra n K G associated to a finite simple graph G as in [DM05]) over any field K of characteristic 0.3.1. Lie algebra associated to graphs.
Let G = ( S, E ) denote a finite simple graph where S is the set ofvertices and E is the set of edges. We associate with G a 2-step nilpotent Lie algebra n K G over any field K of JONAS DER´E AND MEERA MAINKAR characteristic 0 in the following way. The underlying vector space of n K G is V ⊕ W where V is the K -vectorspace with basis S and W is the subspace of V V spanned by { α ∧ β | αβ ∈ E } . The Lie bracket structureon n K G is given by the following relations:(1) [ α, β ] = α ∧ β for all α, β ∈ S with αβ ∈ E ,(2) [ α, β ] = 0 for all α, β ∈ S with αβ / ∈ E , and(3) [ u, v ] = 0 for all u, v ∈ n K G with either u ∈ W or v ∈ W .Recall that the automorphism group of a Lie algebra over the field K is a K -linear algebraic group, i.e.it is given by the K -rational points of a subgroup G of the general linear group over a complex vector spacewhich is defined as the zero set of polynomial equations over K . We denote the subgroup of K -rationalpoints as G ( K ). For more details and terminology about these groups, we refer to [Bor91, Hum81]. In theremainder of this paper we consider only the Zariski topology on linear algebraic groups, meaning that forexample GL( V ) is connected.If we denote by T = { ϕ ∈ Aut( n K G ) | ϕ ( V ) = V } and U = { ϕ ∈ Aut( n K G ) | ∀ x ∈ n K G : ϕ ( x ) − x ∈ W } , then both T and U are K -linear algebraic subgroups of Aut( n K G ). The automorphism group is equal tothe semidirect product Aut( n K G ) = U ⋊ T by [DM05, Proposition 2.1.]. Note that U consists of unipotentelements and hence forms a subgroup of the unipotent radical of Aut( n K G ).Under the natural projection map p : T → GL( V ) = GL (cid:18) n K G (cid:30) [ n K G , n K G ] (cid:19) , we get that the image G = p ( T )is a K -linear algebraic group containing the diagonal matrices. In fact as shown in [DM05], one can seethat this property characterizes the class of 2-step nilpotent Lie algebras associated to graphs. Moreover,the map p is injective and hence every element g ∈ G uniquely corresponds to an automorphism ϕ ∈ T with p ( ϕ ) = g . The groups T and G are hence isomorphic as K -linear algebraic groups. Example 3.1. If G is the complete graph on n vertices, then n K G is isomorphic to the free 2-step nilpotentLie algebra on n generators. If G is the discrete graph on n vertices, then n K G ≈ K n is an abelian Lie algebra.Hence Lie algebras associated to graphs intermediate between free nilpotent Lie algebras and abelian Liealgebras. Note that in both these cases the map p : T → GL( V ) is surjective.In the next paragraphs, we study general linear algebraic groups containing the subgroup of diagonalmatrices and afterwards apply these results to the specific case of Lie algebras associated to graphs.3.2. Linear algebraic groups containing D . In this section, G ≤ GL( V ) is a linear algebraic groupdefined over a field K of characteristic 0, where V is a finite dimensional complex vector space. At the endof the section, we will interpret the results for the K -rational points of the group G , which corresponds tovector spaces over the field K . NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 7
We fix a basis S for the vector space V and denote by D = D S the subgroup of GL( V ) consisting of all the diagonal endomorphisms with respect to S , i.e. the linear endomorphisms given by the diagonal matrices withrespect to the basis S . Then D S is a maximal torus of GL( V ), i.e. a maximal connected abelian subgroupconsisting of semisimple elements. The identity map on V is denoted as I V . The goal is to describe thegroups G with D S ≤ G or equivalently the linear algebraic groups over K which contain a K -split torus ofmaximal rank.We start by recalling the work of [DM05] about these connected linear algebraic groups, in particularthe definition of the partial order relation ≺ associated to them. Using the relation ≺ , we compute thenormalizer in GL( V ) of these groups in terms of permutation matrices, leading to a general description. The connected case . For every α, β ∈ S , we denote by E αβ ∈ End( V ) the linear map which is defined by E αβ ( γ ) = α if γ = β, γ = β, for every basis vector γ ∈ S . Let g denote the Lie algebra of G , which we consider as a Lie subalgebra ofEnd( V ).First we define a relation ≺ on S , using the maps E αβ , as follows :For α, β ∈ S, we say that α ≺ β whenever E αβ ∈ g . It is easy to see that ≺ is a reflexive (since D S ≤ G ) and transitive relation on S . We start by giving anequivalent definition for the relation ≺ depending on the linear algebraic group G itself. Proposition 3.2.
Let G ≤ GL( V ) be a linear algebraic group containing the subgroup D S of all the diagonalendomorphisms with respect to the basis S of V . Let α, β ∈ S with α = β . The following statements areequivalent: (1) α ≺ β (2) I V + tE αβ ∈ G ( K ) for every t ∈ K .Proof. If α ≺ β and α = β , then by definition of ≺ , we have that E αβ ∈ g and hence exp( tE αβ ) ∈ G for t ∈ K . We know that ( E α,β ) = 0 as α = β which implies that I V + tE αβ = exp( tE αβ ) ∈ G ( K ) for every t ∈ K .Conversely, assume that I V + tE αβ ∈ G for every t ∈ K and α = β . Since Q ⊂ K and Q is dense in R ,we have exp( tE αβ ) ∈ G for all t ∈ R and hence E αβ ∈ g . (cid:3) This shows that the relation ≺ does not depend on the field K we are working with. The group ofpermutations on S preserving ≺ is important for the remainder of the paper. Definition 3.3.
Let A denote a finite set with a relation R and let Perm( A ) denote the set of all permutationsof A . A permutation σ ∈ Perm( A ) preserves the relation R if for x, y ∈ A , σ ( x ) Rσ ( y ) whenever xRy . ByPerm( A, R ) we denote the set of all permutations of A preserving the relation R . JONAS DER´E AND MEERA MAINKAR
The set Perm(
A, R ) forms a group under composition because the set A is assumed to be finite.We define now an equivalence relation ∼ on S as follows. For α, β ∈ S , we say that α ∼ β if either α = β or both E αβ and E βα are in g . Equivalently, α ∼ β if and only if α ≺ β and β ≺ α . Let { S λ } λ ∈ Λ denote theset of all equivalence classes under this equivalence relation and will be called coherent components . Withabuse of notation, we will denote a coherent component S λ by λ . By construction the relation ≺ induces apartial order ≺ on Λ defined as follows: λ ≺ µ for λ, µ ∈ Λ if α ≺ β for some (and hence every) α ∈ λ and β ∈ µ . Moreover, the elements of Λ can be enumerated as λ , . . . , λ k such that if λ i ≺ λ j then i < j . Anypermutation σ ∈ Perm( S, ≺ ) induces a permutation ¯ σ ∈ Perm(Λ , ≺ ) on the equivalence classes.For λ ∈ Λ, we denote a subspace of V spanned by the elements of the equivalence class λ by V λ . Notethat V = M λ ∈ Λ V λ . We embed the subgroup GL( V λ ) into GL( V ) by taking the automorphism as the identityon each V µ , λ = µ ∈ Λ. Theorem 3.4 ([DM05]) . Let G ≤ GL( V ) be a connected linear algebraic group containing the subgroup D of diagonal matrices, then (3.1) G = Y λ ∈ Λ GL( V λ ) ! M where M is the unipotent radical of G . The group M is generated by the elements I V + tE α,β with α ≺ β and β ⊀ α . We note that the subgroup L = k Y i =1 GL( V λ i ) is a maximal reductive subgroup of G , which is also called a Levi subgroup of G . Theoriginal result in [DM05] is for connected Lie groups, but it implies the above result for linear algebraicgroups.The following lemma is useful for studying the full automorphism group of Lie algebras associated tographs. Let V be any finite-dimensional vector space given as a direct sum V = M i ∈ I V i , then we writeˆ V j = M i ∈ Ii = j V i . Again, we embed GL( V j ) as a subgroup of GL( V ) in the natural way by taking identity onthe other components, so the elements d ∈ GL( V j ) are exactly the ones for which d ( V j ) = V j and d is theidentity on ˆ V j . Lemma 3.5.
Let V = M i ∈ I V i be any finite-dimensional vector space, given as a direct sum of subspaces. Forevery g ∈ GL( V ) , the following are equivalent: g GL( V j ) g − = GL( V k ) ⇐⇒ g ( V j ) = V k and g ( ˆ V j ) = ˆ V k . The proof is immediate, but we present it for completeness.
Proof.
First assume that g GL( V j ) g − = GL( V k ). Consider the map d : V → V given by d ( x ) = x for x ∈ ˆ V j and d ( y ) = 2 y for y ∈ V j . Since d ∈ GL( V j ), we have d ′ = gdg − ∈ GL( V k ) where d ′ has eigenvalues 1 and 2 NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 9 just as the map d . In particular, by considering the eigenspaces for eigenvalue 1 and 2, we have ˆ V k ⊂ g ( ˆ V j )and g ( V j ) ⊂ V k . The reverse inclusions follow from considering g − , thus the first implication follows.Now assume that g ( V j ) = V k and g ( ˆ V j ) = ˆ V k . It suffices to show that the inclusion g GL( V j ) g − ⊂ GL( V k )holds, since we can apply the same statement to the element g − . If d ∈ GL( V j ), then d ( V j ) = V j and so gdg − ( g ( V j )) = V k . Similarly since d is the identity map on ˆ V j , then gdg − is the identity map on g ( ˆ V j ) = ˆ V k and thus the inclusion follows. (cid:3) We will apply this lemma on the decomposition V = M λ ∈ Λ V λ for finding the induced action on Λ. The general case . For any permutation σ of S , we denote by P σ ∈ GL( V ) the element defined by P σ ( α ) = σ ( α )for all α ∈ S . The matrix of P σ with respect to S is the permutation matrix corresponding to the permutation σ . Note that at some places in literature, this is called the permutation matrix corresponding to thepermutation σ − . Denote by P : Perm( S ) → GL( V )the group homomorphism given by P ( σ ) = P σ . An easy computation shows that P σ ◦ E αβ ◦ P σ − = E σ ( α ) σ ( β ) for all σ ∈ Perm( S ) and for all α, β ∈ S . In particular, for σ ∈ Perm( S, ≺ ), we have that P σ ( V λ ) = V σ ( λ ) P σ ( ˆ V λ ) = ˆ V σ ( λ ) (3.2)for every coherent component λ , which we will need later in the paper. Here we recall that σ ∈ Perm(Λ , ≺ )is the permutation induced by σ and ˆ V λ = M µ ∈ Λ µ = λ V µ . The following fact about linear algebraic groups plays an important role.
Proposition 3.6.
Let G ≤ GL ( V ) be a linear algebraic group which contains D = D S as a subgroup. If G denotes the connected component of identity in G , then (3.3) G = G (cid:0) G ∩ P (Perm( S )) (cid:1) = (cid:0) G ∩ P (Perm( S )) (cid:1) G . Proof.
The second equality of equation (3.3) follows from the fact that G is a normal subgroup of G . Weprove the first equality by showing two inclusions. It is immediate that G (cid:0) G ∩ P (Perm( S )) (cid:1) ⊂ G , so itsuffices to show the reverse inclusion.First, for the sake of completeness, we include a proof of a standard known fact that the normalizer of D in GL( V ) is given by N GL ( V ) ( D ) = DP (Perm( S )) . (3.4) One inclusion is immediate, since for σ ∈ Perm( S ) and d ∈ D , we have P σ dP − σ ∈ D . For the other inclusion,suppose that g ∈ N GL ( V ) ( D ) and d ∈ D is given by d ( α ) = d α α for all α ∈ S with the property d α = d β if α = β , then we have gdg − = d ′ ∈ D . Since both d and d ′ have the same set of eigenvalues, the set ofdiagonal entries of d ′ is { d α : α ∈ S } . Denote by σ ∈ Perm( S ) satisfying d ′ ( σ ( α )) = d α σ ( α ) for each α ∈ S .Now d ′ g ( α ) = gd ( α ) = d α g ( α ) and hence g ( α ) is an eigenvector of d ′ corresponding to an eigenvalue d α foreach α ∈ S . Since the d α ’s are all distinct, we have g ( α ) = a α σ ( α ) for some a α = 0 for each α ∈ S . Thisproves that g = hP σ where h ( σ ( α )) = a α σ ( α ) for each α ∈ S , so h ∈ D and hence proving equation (3.4).Now take any element g ∈ G . The subgroup gDg − is a maximal torus in G and thus there exists h ∈ G with hgDg − h − = D or equivalently with hg ∈ N GL ( V ) ( D ). Because of equation (3.4), the element hg isof the form hg = dP σ with d ∈ D and σ ∈ Perm( S ). We conclude that g = h − dP σ ∈ G (cid:0) G ∩ P (Perm( S )) (cid:1) is of the desired form. (cid:3) By applying Proposition 3.6 to the normalizer of a connected linear algebraic group G = G , we get thefollowing theorem. Theorem 3.7.
Let G be a connected linear algebraic group which contains the diagonal matrices D = D S as a subgroup. The normalizer of G in GL( V ) is equal to N GL( V ) ( G ) = P (Perm( S, ≺ )) G. Proof.
Since the normalizer of an algebraic group in GL( V ) is itself an algebraic group by [Hum81, Page59], Proposition 3.6 implies that N GL( V ) ( G ) = P ( F ) G where P ( F ) is the subgroup of P (Perm( S )) whichnormalizes G . It remains to check that P ( F ) = P (Perm( S, ≺ )). We first assume that P σ ∈ P ( F ) or thus P σ GP σ − = G . Note that for every I V + E αβ , it holds that P σ ( I V + E αβ ) P σ − = I V + E σ ( α ) σ ( β ) and thus that I V + E αβ ∈ G if and only if I V + E σ ( α ) σ ( β ) ∈ G . From Proposition 3.2 it then follows that α ≺ β if and only if σ ( α ) ≺ σ ( β ) or thus that σ ∈ Perm( S, ≺ ).Conversely, asume that σ ∈ Perm( S, ≺ ), then we will show that P σ GP σ − = G from the explicit form for G in Theorem 3.4. First consider the generators of M which are given by I V + tE αβ with α ≺ β, β ⊀ α and t ∈ C . In this case P σ ( I V + tE αβ ) P σ − = I V + tE σ ( α ) σ ( β ) ∈ M, which shows that P σ M P σ − = M . On the other hand, for the subgroup GL( V λ ) with λ ∈ Λ, we get that P σ GL( V λ ) P σ − = GL( V ¯ σ ( λ ) ) by Lemma 3.5 and equation (3.2), where σ ∈ Perm(Λ , ≺ ) is the permutationinduced by σ . Hence the conclusion follows. (cid:3) In this way we get a description of all linear algebraic groups containing D . NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 11
Corollary 3.8.
Let G be a linear algebraic group containing D = D S and with G as the Zariski connectedcomponent of the identity. Then G is given by G = P ( F ) G where F ≤ Perm( S, ≺ ) is the finite subgroup given by F = { σ ∈ Perm( S, ≺ ) | P σ ∈ G } .Proof. This is an immediate consequence of Theorem 3.7. Indeed for if g ∈ G , then g = p σ h for some σ ∈ Perm( S, ≺ ) and h ∈ G . (cid:3) Remark . Theorem 3.1 and Corollary 3.8 allow us to order the basis S , by ordering the coherent classesas λ , . . . , λ k , such that the matrix of an element g ∈ G with respect to S is of the form P A λ A A · · · A k A λ A · · · A k A λ · · · A k ... ... ... . . . ... · · · A λ k , where P is a permutation matrix, A λ i ∈ GL( V λ i ) and A ij = , a zero matrix, if λ i ⊀ λ j .In order to fully understand the group G , we need a description of the subgroup P ( F ) ∩ G . Let Perm( S, ≺ )denote the subgroup of Perm(Λ , ≺ ) consisting of the permutations σ induced by σ ∈ Perm( S, ≺ ). Theintersection of P (Perm( S, ≺ )) and G is described by the following lemma. Lemma 3.10.
For every σ ∈ Perm( S, ≺ ) it holds that σ = 1 if and only if P σ ∈ G .Proof. First assume that σ ∈ Perm( S, ≺ ) such that σ = 1. Then for every λ ∈ Λ, we have P σ ( V λ ) = V σ ( λ ) = V λ . In particular, we have that P σ ∈ k Y i =1 GL( V λ i ) ⊂ G .For the other implication, assume that P σ ∈ G . Note that the transpose P Tσ = P σ − and therefore P Tσ ∈ G . By using Theorem 3.4, we get that P σ ∈ k Y i =1 GL( V λ i ), which implies that σ = 1. (cid:3) In other words this lemma states that G ∩ P (Perm( S, ≺ )) = P (cid:0) { σ ∈ Perm( S, ≺ ) | σ = 1 (cid:1) . The following is hence an immediate consequence.
Corollary 3.11.
Every linear algebraic group G containing the group of diagonal matrices D = D S satisfies G (cid:30) G ≈ ¯ F with ¯ F a finite subgroup of Perm( S, ≺ ) ≤ Perm(Λ , ≺ ) . Proof.
The group G is a subgroup of P (Perm( S, ≺ )) G = N GL( V ) ( G ), so every g ∈ G is of the form P σ h with σ ∈ Perm( S, ≺ ) and h ∈ G . Define the map π : G → Perm( S, ≺ ) which maps the element g = P σ h to σ . This map is well-defined, since if P σ h = P τ h ′ with σ, τ ∈ Perm( S, ≺ ) and h, h ′ ∈ G , then P σ − τ = P σ − P τ = hh ′− ∈ G and thus σ = τ by the previous lemma. One can check that π is a grouphomomorphism as G is normal in G . Therefore we have an induced map π : G (cid:30) G → Perm( S, ≺ ) which isan injective group homorphism and the corollary now follows immediately. (cid:3) Corollary 3.11 shows that there is a natural action of G on the equivalence classes Λ. We note that ingeneral Perm( S, ≺ ) ≤ Perm(Λ , ≺ ) is a strict subgroup as we will show in Example 3.15. We denote by π : G → Perm(Λ , ≺ ) the natural map defined by the previous corollary. It plays an important role in thenext sections for studying the actions of finite groups. Since σ = π ( P σ ) for every σ ∈ Perm( S, ≺ ), we havethat P σ ( V λ ) = V σ ( λ ) = V π ( P σ )( λ ) . Lemma 3.5 implies that P σ GL( V λ ) P − σ = GL( V σ ( λ ) ) = GL( V π ( P σ )( λ ) )for every P σ ∈ G . This alternative way of looking at the map π is how we will use it in the proof of the mainresult.Note that the previous results were for linear algebraic groups G ≤ GL( V ) over the complex numbersdefined over K , but the main application is the automorphism group Aut( n K G ) of a Lie algebra over asubfield K ⊆ C . Hence we are often interested in the subgroup of K -rational points G ( K ) for K ⊆ C . Since P ( F ) ⊆ G ( Q ), it follows immediately that G ( K ) = G ( K ) P ( F ) = k Y i =1 GL( V λ i , K ) ! M ( K ) P ( F ) . Full automorphism group of Lie algebras associated to graphs.
We now apply the previousresults to the automorphism group of a Lie algebra associated to a graph as introduced in [DM05]. We referto Subsection 3.1 for notations. Let G = ( S, E ) be a graph and n K G the corresponding Lie algebra over a field K of characteristic 0. Let Aut( G ) denote the group of all graph automorphisms of G .We recall two subgroups of Aut( n K G ) introduced as before: T = { ϕ ∈ Aut( n K G ) | ϕ ( V ) = V } and U = { ϕ ∈ Aut( n K G ) | ∀ x ∈ n K G : ϕ ( x ) − x ∈ W } , where W = Span { [ α, β ] | αβ ∈ E } . We will apply Corollary 3.8 to G = p ( T ) where p is the natural projectionmap p : T → GL( V ) = GL (cid:18) n K G (cid:30) [ n K G , n K G ] (cid:19) , but first we describe the elements P σ ∈ G . Lemma 3.12.
For every σ ∈ Perm( S ) , the following are equivalent: σ ∈ Aut( G ) ⇐⇒ P σ ∈ G = p ( T ) . NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 13
Proof.
First assume that σ ∈ Aut( G ) then P σ : V → V extends to an automorphism of n K G . In particular, P σ ∈ p ( T ) = G . For the other implication, note that P σ = p ( ϕ ) for ϕ ∈ T ⊂ Aut( n K G ), then ϕ ([ α, β ]) = [ P σ ( α ) , P σ ( β )] = [ σ ( α ) , σ ( β )] . Since we have the equivalence αβ ∈ E ⇐⇒ [ α, β ] = 0 and ϕ maps non-zero elements on non-zero elements,this implies that σ ∈ Aut( G ). (cid:3) Combining Corrolary 3.8 with the above lemma, we get the following description of the automorphismgroup of n K G . Corollary 3.13.
Let K be any field of characteristic , then the automorphism group Aut( n K G ) is equal to Aut( n K G ) = U T with T ≈ P (Aut( G )) G ( K ) = P (Aut( G )) k Y i =1 GL( V λ i , K ) M ( K ) , where the isomorphism is given by the natural projection p . So the automorphism group of n K G over any field K of characteristic 0 is completely described in terms of thegraph G . Note that the Levi subgroup L of G is also a Levi subgroup of Aut( n K G ) because U is a unipotentnormal subgroup.Consider the morphism π : G → Perm(Λ , ≺ ) as introduced under Corollary 3.8. Lemma 3.5 shows that g ( V λ ) = V π ( g )( λ ) for every g ∈ P (Perm( S, ≺ )). More general, this relation holds for every element in theLevi subgroup L = k Y i =1 GL(
K, V λ i ) P (Aut( G )) by Lemma 3.5, leading to an action L y Λ which is centralin the proof of the main result. In order to work with this action, we recall how we can find the relation ≺ directly from the graph G . Coherent components for graphs.
Let G = ( S, E ) denote a finite simple graph. We will now recall someobservations from [DM05] interpreting the relation ≺ on the set of vertices S . We will also introduce someterminology which would be useful in our applications later in the paper. For α ∈ S , we define the openneighborhood and closed neighborhood of α , respectively, as follows:Ω ′G ( α ) = { β ∈ S | αβ ∈ E } and Ω G ( α ) = Ω ′G ( α ) ∪ { α } . For α, β ∈ S , we have α ≺ β if and only if Ω ′G ( α ) ⊆ Ω G ( β ) by [DM05, Proposition 4.1]. Consequently, α ∼ β if and only if Ω ′G ( α ) ⊆ Ω G ( β ) and Ω ′G ( β ) ⊆ Ω G ( α ). As before, we denote the set of coherent components(equivalence classes) by Λ. It can be checked that the induced subgraph on each coherent component λ ∈ Λis either complete or discrete. More precisely, for every λ ∈ Λ, either αβ ∈ E for all distinct α, β ∈ λ or αβ / ∈ E for any α, β ∈ λ . Moreover, for λ = µ ∈ Λ, if there exist α ∈ λ, β ∈ µ with αβ ∈ E , then we have γδ ∈ E for all γ ∈ λ, δ ∈ µ . These properties lead to a notion of quotient graph G whose vertex set is Λ and the edge set E is given by E = { λµ | there exist α ∈ λ, β ∈ µ with αβ ∈ E } . Note that G might have vertices with loops. Indeed, if a subgraph of G induced on λ is complete and λ contains at least two vertices, then λλ ∈ E .The induced partial order ≺ on Λ is given by λ ≺ µ ⇐⇒ Ω ′G ( α ) ⊆ Ω G ( β ) for some α ∈ λ, β ∈ µ. We arrange the coherent components λ ′ s as λ , . . . , λ k such that if λ i ≺ λ j , then i ≤ j .In general, we have the inclusion of groups Aut( G ) ⊂ Perm( S, ≺ ) and Perm( S, ≺ ) ⊂ Perm(Λ , ≺ ). We givesome examples showing that these can be equalities or strict inclusions depending on the graph G . Example 3.14.
Consider the graph G with vertices { α , α , β , β , β , γ , γ } as follows. α α γ γ β β β Looking at the neighborhoods of the vertices, we get 3 coherent components, say λ, ν and µ , where λ = { α , α } , ν = { γ , γ } and µ = { β , β , β } . The corresponding quotient graph G is the graph λ ν µ Since Ω ′ ( α ) ⊂ Ω( β ), we can see that λ ≺ µ . The other pairs of the coherent classes are not comparablewith respect to the relation ≺ . Hence we can arrange the coherent classes as λ = λ, λ = µ and λ = ν sothat i ≤ j whenever λ i ≺ λ j . We could also arrange them in the order ν, λ, µ or λ, ν, µ .In this case, λ ≺ µ and no other pairs of coherent classes are comparable, hence the group Perm(Λ , ≺ )is trivial. The group Perm( S, ≺ ) consists of permutations σ of S satisfying σ ( λ i ) = λ i for all i ∈ { , , } .Thus Perm( S, ≺ ) = Aut( G ) in this case. Example 3.15.
Consider the graph G with a vertex set the same as in the Example 3.14 and the edge setas drawn below. α α γ γ β β β NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 15
There are three coherent classes in this graph, say λ = { α , α } , λ = { β , β , β } and λ = { γ , γ } , andthe quotient graph is as drawn below. λ λ λ The coherent classes satisfy λ ≺ λ , λ ≺ λ . We note that λ and λ are not comparable. Hence if λ i ≺ λ j ,then i ≤ j .Now we will show that Perm( S, ≺ ) is a proper subgroup of Perm(Λ , ≺ ) in this case. Consider a 2-cycle τ = ( λ λ ) ∈ Perm(Λ). Then τ ( λ ) = λ ≺ λ = τ ( λ ) and τ ( λ ) = λ ≺ λ = τ ( λ ). This shows that τ ∈ Perm(Λ , ≺ ). Now for every element σ ∈ Perm( S ), we have σ ( λ ) = λ as λ = λ . Hence τ isnot induced by any element of Perm( S, ≺ ). In particular it follows that not every element of Perm(Λ , ≺ ) isinduced by a graph automorphism. In fact, as in Example 3.14, we can show that Perm( S, λ ) = Aut( G ) byobserving that every permutation in Perm( S, ≺ ) stabilizes each coherent component. Example 3.16.
Consider a graph G as drawn below. α βη γδ There are five coherent classes and no two vertices are comparable under the relation ≺ . ThereforePerm( S, ≺ ) = Perm( S ) in this case. In particular, Aut( G ) is a proper subgroup of Perm( S, ≺ ).4. Anosov automorphisms commuting with finite subgroups
In this part we focus on the Lie groups corresponding to the Lie algebras of the previous section. For K = R , let N G be the connected and simply connected nilpotent Lie group with n R G as its Lie algebra. Wecall N G a 2-step nilpotent Lie group associated to the graph G . The rational Lie algebra n Q G corresponds toa lattice Γ ≤ N G , which is uniquely defined up to commensurability and the nilmanifold N G (cid:30) Γ is said to beassociated to the graph G . In the paper [DM05] the authors give an explicit form for such a lattice Γ, butthis is not necessary for our purposes. Note that there are in general other nilmanifolds modeled on a Liegroup associated to a graph, see Section 6. In this section, we study which infra-nilmanifolds covered by anilmanifold associated to a graph admit an Anosov diffeormorphism. Definition 4.1.
Let G be a graph and N G the Lie group associated to G . We say that an infra-nilmanifold N G / Γ is associated to the graph G if the Lie algebra corresponding to the radicable hull M Q of M = N ∩ Γis isomorphic to n Q G .By [Mai15] the graph G is uniquely determined by the rational Lie algebra n Q G .Theorem 2.1 states that for studying Anosov diffeomorphisms on infra-nilmanifolds associated to graphs,we have to study Anosov automorphisms commuting with finite subgroups H of the automorphism group Aut( n Q G ). We start by constructing induced representations on the coherent components before getting tothe proof of the main result. Induced representations on coherent components . Let L = P (Aut( G )) k Y i =1 GL( V λ i , Q )be the maximal reductive subgroup of G ( Q ) as above. We know that there is an algebraic isomorphismbetween T and G , so we can also consider L ≤ T ≤ Aut( n Q G ) as a Levi subgroup of Aut( n Q G ).If H is a finite subgroup of Aut( n Q G ), then it is reductive and thus lies in a maximal reductive subgroup L ′ . Since all maximal reductive subgroups of a linear algebraic group are conjugate in characteristic 0, thereexists g ∈ G ( Q ) such that gL ′ g − = L or thus gHg − ⊆ L . Without loss of generality, we can assume that H is a finite subgroup of L . The same argument shows that if there exists an Anosov automorphism on n Q G commuting with H , we can assume it lies in L as well. Indeed, if ϕ : n Q G → n Q G is an Anosov automorphism,then also its semisimple part ϕ s is Anosov and commutes with H . Now by considering the reductive subgroupgenerated by H and ϕ s , we can assume that both lie in L .Even stronger, if we consider the natural projection map ψ : Aut( n Q G ) → L by taking the quotient by theunipotent radical, then we get that ψ ( H ) is a finite subgroup of L . When we consider automorphisms asmatrices in the standard basis on n Q G , the map ψ is given by taking the block diagonal part of a matrix.If ϕ ∈ Aut( n Q G ) is an Anosov automorphism, then ψ ( ϕ ) will again be an Anosov automorphism lying in L with the same eigenvalues. Moreover, if H and ϕ commute, also ψ ( H ) and ψ ( ϕ ) commute. So without lossof generality, we can take the projection ψ in order to assume that both H and ϕ are elements of the Levisubgroup L . Even if we assume later that H is a subgroup of L , we can state our main theorem for generalrepresentation ρ : H → Aut( n Q G ), where we first apply the map ψ and get an Anosov automorphism for theoriginal representation as explained above.So from now on we assume that H is a subgroup in L . By restricting the action at the end of the previoussection, we get a map π : H → Perm(Λ , ≺ ) such that for every h ∈ H , we have h ( V λ ) = V π ( h )( λ ) . This gives an action of H on the coherent components Λ, which we denote by h · λ = π ( h )( λ ). For thisaction of H on Λ, we denote the orbits by κ i = H · λ i with 1 ≤ i ≤ k . Write c ( λ i ) = 1 if the subgraph ofthe quotient graph G (see Section 3) induced on the orbit κ i is an edgeless graph, otherwise c ( λ i ) = 2. Inother words, we look at the union of all coherent components in the orbit κ i which would give us a subsetof the vertex set S of G . If there are no two vertices in that union which are adjacent in G , then we write c ( λ i ) = 1, otherwise, we write c ( λ i ) = 2. In conclusion, c ( λ i ) = 2 if and only if there is an edge (possibly aloop) in the subgraph of G induced by the orbit κ i . Note that c ( λ i ) is equal to the nilpotency class of thesubalgebra generated by the elements in the coherent components of the orbit κ i . NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 17
Every orbit κ i = H · λ i also defines a subgroup H i ≤ H given by the stabilizer of λ i , i.e. H i = { h ∈ H | h ( V λ i ) = V λ i } . So the action of H on Λ determines k representations ρ i : H i → GL( V λ i ). These representations depend onthe choice of the elements λ i ∈ κ i , but this does not influence our results below. Example 4.2.
Consider a graph G with its quotient graph G drawn as below. α α γ γ δ δ β β λ λ λ λ Here the coherent classes are λ = { α , α } , λ = { β , β } , λ = { γ , γ } and λ = { δ , δ } . Suppose H isa subgroup of L generated by P σ where σ = Y i =1 ( α i β i )( γ i δ i ), which is hence of order 2. Then the action of H on Λ is given by P σ · λ = λ , P σ · λ = λ , P σ · λ = λ , P σ · λ = λ . Hence κ = { λ , λ } = κ and κ = { λ , λ } = κ . Consequently c ( λ ) = 1 = c ( λ ) and c ( λ ) = 2 = c ( λ ).The stabilizer H i is trivial for each i in this example. Remark . Although L is isomorphic to a subgroup of Aut( n Q G ), this isomorphism does not preserve thespectrum of an element. Take any g ∈ G ≤ GL( V ) which uniquely corresponds to an automorphism ϕ ∈ T ≤ Aut( n G ). Write n λ for the dimension of the subspace V λ with λ ∈ Λ. If g ∈ L ⊆ G and µ λ, , . . . , µ λ,n λ are the eigenvalues of g on V λ , then the eigenvalues of ϕ are equal to µ λ,i for λ ∈ Λ or µ λ,i µ λ ′ ,j for λ, λ ′ ∈ Λ. The latter eigenvalues only occur if either λ = λ ′ and these coherent components areconnected by an edge in the quotient graph or if λ = λ ′ , i = j and the coherent component λ is a completegraph. Main result . Recall that a matrix A ∈ GL( n, Q ) with eigenvalues µ , . . . , µ n is called c -hyperbolic if for all1 ≤ l ≤ c and 1 ≤ i j ≤ n the product l Y j =1 | µ i j | 6 = 1 . So 1-hyperbolic corresponds to the classical notion of hyperbolic matrices. A linear automorphism with a c -hyperbolic matrix is called c -hyperbolic automorphism. The action of H on the coherent components, theinduced representations ρ i together with c ( λ i ) contain all the information about the existence of Anosovdiffeomorphisms. Theorem 4.4.
Let G be a graph and n Q G be the rational Lie algebra associated to this graph. Let ρ : H → L ≤ Aut( n Q G ) be a representation of a finite group and ρ i : H i → GL( V λ i ) be the representations as introducedabove. The following statements are equivalent. There exists an Anosov automorphism ϕ ∈ Aut( n Q G ) commuting with every element of ρ ( H ) . m For every ≤ i ≤ k , there exists a c ( λ i ) -hyperbolic integer-like automorphism ϕ i ∈ GL ( V λ i ) such that ϕ i commutes with every element of ρ i ( H i ) . Recall that integer-like means that the characteristic polynomial has coefficients in Z and constant term ± Proof.
First assume that ϕ ∈ Aut( n Q G ) is an Anosov automorphism commuting with every element of ρ ( H ).As explained before we can assume that ϕ ∈ L ≤ T or thus in particular that ϕ ( V ) = V . Take g = p ( ϕ ) ∈ G the restriction of ϕ to V as introduced above. By taking a power of ϕ we can also assume that g ∈ G or thus ϕ ( V λ ) = V λ for every λ ∈ Λ. So by taking ϕ i the restriction of ϕ to the subspaces V λ i , we find inparticular that ρ i ( h ) ◦ ϕ i = ϕ i ◦ ρ i ( h ) for every h ∈ H i , giving the first condition on the ϕ i .For the second condition, we know that ϕ is hyperbolic so we only have to check it for λ i with c ( λ i ) = 2.Assume that µ , µ are two eigenvalues of ϕ i , then we have to show that | µ µ | 6 = 1. We can assume that µ = µ since otherways there is nothing to prove. There are two possibilities, either λ i is a complete graphor there exists λ ′ ∈ κ i such that λ i is connected to λ ′ .In the first case, we have that µ µ is an eigenvalue of ϕ by Remark 4.3. In the second case, take h ∈ H such that h · λ i = λ ′ . Because h ( V λ i ) = V λ ′ and ϕ ◦ h = h ◦ ϕ , we have that ϕ (cid:12)(cid:12) V λ ′ = h ◦ ϕ i ◦ h − and thus ϕ (cid:12)(cid:12) V λ ′ has the same eigenvalues as ϕ i . In particular, µ µ is again an eigenvalue of ϕ by Remark 4.3. So inboth cases the statement follows from the hyperbolicity of ϕ , finishing the first implication of the theorem.For the other implication, assume that ϕ i exists as in the theorem. Note that the eigenvalues of ϕ i arealgebraic integers. We now construct a hyperbolic element ϕ ∈ Aut( n Q G ) commuting with every elementin ρ i ( H i ). By taking some power of the ϕ i , we can assume that for all eigenvalues µ i , µ j of ϕ i and ϕ j respectively with i = j , we have that | µ i µ j | 6 = 1. Every λ ∈ Λ lies in some orbit κ i of the action of H on Λ and hence is of the form π ( h )( λ i ) for some i ∈ { , . . . , k } , meaning that ρ ( h )( V λ i ) = V λ . Now define g λ : V λ → V λ as g λ = ρ ( h ) ◦ ϕ i ◦ ρ ( h ) − .We claim that the definition of g λ does not depend on the choice of h such that π ( h )( λ i ) = λ . Indeed,assume that h , h ∈ H with π ( h )( λ i ) = π ( h )( λ i ) = λ , then h − h ∈ H i by definition. By assumption ϕ i commutes with ρ ( h − h ), so ρ ( h − h ) ϕ i = ϕ i ρ ( h − h ) on V λ i which implies the claim.Take g : V → V as the direct sum of all g λ : V λ → V λ . It is clear that g ∈ G and hence we can consider ϕ ∈ Aut( n Q G ) with p ( ϕ ) = g . Because of the assumption on the eigenvalues of the ϕ i and the fact that ϕ i is c ( λ i )-hyperbolic, we will show that ϕ is hyperbolic. Indeed, the eigenvalues of g are also eigenvalues of some ϕ i , since it is equal to a conjugate of ϕ i on V λ for every λ ∈ Λ. The eigenvalues of ϕ are hence eigenvalueof some ϕ i or a product of two eigenvalues µ λ,i µ λ ′ ,j for eigenvalues µ λ,i and µ λ ′ ,j of g λ and g λ ′ with λ and λ ′ connected. If λ and λ ′ lie in the same orbit, then we know that c ( λ ) = 2 and thus that the eigenvalue isdifferent from 1 in absolute value. If λ and λ ′ lie in a different orbit, then they correspond to different ϕ i NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 19 and ϕ j , hence the eigenvalue is different from 1 in absolute value by the extra condition mentioned before.We conclude that ϕ is a hyperbolic automorphism. In this proof we also showed that all the eigenvalues of ϕ are algebraic integers, therefore the automorphism ϕ is integer-like.We are left to check that ϕ commutes with the finite group ρ ( H ). It suffices to check that g commuteswith every element p ( ρ ( h )) ∈ p ( ρ ( H )) on every subspace V λ . Take λ i such that λ ∈ κ i and consider ˜ h ∈ H such that ρ (˜ h )( V λ i ) = V λ . For every x ∈ V λ , we get that g ( x ) = g λ ( x ) = ρ (˜ h ) ◦ ϕ i ◦ ρ (˜ h − )( x ) . Now ρ ( h ◦ ˜ h ) maps V λ i to ρ ( h )( V λ ), so the map g π ( h )( λ ) is given by ρ ( h ◦ ˜ h ) ◦ ϕ i ◦ ρ (cid:16) ( h ◦ ˜ h ) − (cid:17) ( y ) = ρ ( h ) ◦ g λ ◦ ρ ( h − )( y )for every y ∈ ρ ( h )( V λ ). The statement follows and this finishes the proof. (cid:3) In order to achieve a more workable condition, we recall the following result of [DD14].
Theorem 4.5.
Let ρ : G → GL( n, Q ) be a Q -irreducible representation. Then there exists a c -hyperbolic,integer-like matrix C ∈ GL mn ( Q ) which commutes with mρ = ρ ⊕ ρ ⊕ · · · ⊕ ρ | {z } m times if and only if ρ splits in strictlymore than cm components when seen as a representation over R . We refer to [DV09a] for more details. Combined with Theorem 4.5, this gives us the following result.
Theorem 4.6.
Let G be a graph and n Q G be the rational Lie algebra associated to this graph. Let ρ : H → Aut( n Q G ) be a representation of a finite group with ρ i : H i → GL( V λ i ) the representations as introduced before.Then the following are equivalent.There exists an Anosov automorphism ϕ ∈ Aut( n Q G ) commuting with every element of ρ ( H ) . m For every ≤ i ≤ k , every Q -irreducible component of ρ i that occurs with multiplicity m splits in strictly more than c ( λ i ) m components over R . For our purposes, we can only have c ( λ i ) ∈ { , } . In the case c ( λ i ) = 1, the condition states thatevery Q -irreducible component of ρ i which is also irreducible as a representation over R must occur at leasttwice in ρ i . If c ( λ i ) = 2, then the condition states that every Q -irreducible component which is irreducibleover R must occur at least three times, and if a Q -irruducible component splits in exactly 2 componentsover R it must occur at least twice. The Q -irreducible components which split in more components over R automatically satisfy the condition.The trivial representation of dimension 1 is both irreducible over Q and over R . In particular, if every ρ i is the trivial representation, we get the following consequence. Corollary 4.7.
Let G be a graph and n Q G be the rational Lie algebra associated to this graph. Let ρ : H → Aut( n Q G ) be a representation of a finite group such that the representation ρ i : H i → GL( V λ i ) introduced above are trivial. Then there exists an Anosov automorphism ϕ ∈ Aut( n Q G ) commuting with every elementof ρ ( H ) if and only if dim( V λ i ) ≥ if c ( λ i ) = 1 and dim( V λ i ) ≥ otherwise. So if H is trivial, we recover the main result of [DM05].5. Applications
In this section, we apply the previous results to construct new examples of Anosov diffeomorphisms oninfra-nilmanifolds. First we recall some machinery for constructing almost-Bieberbach groups given a rationalLie algebra n Q and a faithful representation ρ : H → Aut( n Q ). Next we apply this to give concrete examplesof graphs leading to infra-nilmanifolds admitting an Anosov diffeomorphism. Constructing almost-Bieberbach groups with given holonomy representation . An almost-Bieber-bach group Γ ≤ Aff( N ) induces a torsion-free radicable nilpotent group M Q ≤ N and a faithful representationof a finite group ρ : H → Aut( M Q ). But vice versa, given a torsion-free radicable nilpotent group M Q anda faithful representation ρ : H → Aut( M Q ) of a finite group, it is not always easy to check whether H corresponds to the rational holonomy representation of an almost-Bierberbach group Γ. The hard conditionto achieve is the torsion-freeness of the group Γ.In some special cases, this problem is completely solved though. The case where H is the trivial groupfollows immediately from the fundamental work of Mal’cev [Mal51], since the group M Q is always theradicable hull of some lattice M ≤ N in a 1-connected nilpotent Lie group N . The next interesting case isfor cyclic groups H , which was treated in [DV11] although it was not explicitly stated as a separate theoremtherein. The proof of [DV11, Proposition 5.2.] contains the necessary arguments and for the convenience ofthe reader, we sketch the proof. Theorem 5.1.
Let ρ : H → Aut( M Q ) be a faithful representation of a finite cyclic group H on a torsion-freeradicable nilpotent group M Q , where ρ ( H ) is generated by the element φ ∈ H . The following are equivalent:The representation ρ is a rational holonomy representation of an almost-Bieberbach group Γ . m The automorphism φ has eigenvalue .Proof. One direction is immediate, namely if ρ : H → Aut( M Q ) is the rational holonomy representation ofan almost-Bieberbach group Γ then φ has eigenvalue 1. Indeed, assume that Γ ≤ M Q ⋊ H is an almost-Bieberbach group with rational holonomy representation H . By definition there exists n ∈ M Q such that γ = ( n, φ ) ∈ Γ. Write m = γ | H | ∈ M \ { e } , then m = γmγ − = nφ ( m ) n − . So the automorphism M Q → M Q : x nφ ( x ) n − has eigenvalue 1, hence also the automorphism φ has eigenvalue 1, since bothautomorphism are conjugate.For the other direction, it is easy to find a finitely generated torsion-free nilpotent subgroup M such that M Q is the radicable hull of M and φ ( M ) = M . The automorphism φ induces automorphisms φ i ∈ GL( n i , Z )on the quotients M ∩ γ i ( M Q ) (cid:30) M ∩ γ i +1 ( M Q ) ≈ Z n i . By [DV11, Lemma 4.3.] there exists M ′ ≤ M Q NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 21 which contains M as a subgroup of finite index such that φ ( M ′ ) = M ′ and the induced representations φ i ∈ GL( n i , Z ) for M ′ are totally reducible. Since φ has eigenvalue 1, there exists i such that the inducedrepresentation φ i also has eigenvalue 1. Then [DV11, Proposition 5.1.] implies there exists a torsion-freeextension 1 → Z n i → E → Z | H | → φ i and hence [DV11, Theorem 4.1.] gives us the desiredresult. (cid:3) The extra ingredient used in the proof of [DV11, Proposition 5.2.] was showing that every automorphism offinite order on a free nilpotent Lie algebra of nilpotency class c ≥ Corollary 5.2.
Let G be a graph and φ ∈ Aut( G ) be a graph automorphism. The natural representation h φ i → Aut( n Q G ) is the rational holonomy representation of an almost-Bieberbach group.Proof. Consider the vector v = X α ∈ S α with S the set of vertices of the graph G . Since φ is an automorphismof the graph, the vector v is an eigenvector for eigenvalue 1 of the corresponding Lie algebra automorphism.The statement now follows immediately from Theorem 5.1. (cid:3) For the families of examples we will construct we will hence mainly focus on cyclic subgroups of graphautomorphisms.
Anosov diffeomorphisms for holonomy group Z n . We now give families of graphs G with free actionsof cyclic subgroups H ⊂ Aut( G ) by graph automorphism groups giving rise to infra-nilmanifolds admittingAnosov diffeomorphisms. The general strategy for constructing such examples is the following. We startwith a graph G , with a vertex set Λ, on which a finite group H acts freely, for example the Cayley graph of H for some finite generating set. We realize the graph G as a quotient graph of a simple graph G satisfyingthe following three conditions:(1) the action of H is induced by an action of H via graph automorphisms on G ,(2) each coherent class λ ∈ Λ is of size at least 2,(3) the size of λ is at least 3 whenever λµ is an edge in G for some µ in the orbit of λ .Using Corollary 4.7, if there exists an infra-nilmanifold associated to the graph G corresponding to the actionof H , it admits an Anosov diffeomorphism. Below we present examples for the cyclic groups Z n and becauseof Corollary 5.2, these always correspond to an infra-nilmanifold. Example 5.3 (Holonomy Z ) . Consider a graph G with 2 m coherent classes, m ≥ ℓ where ℓ ≥ λ λ · · · λ m − λ m · · · λ λ For m = 1, we get a complete bipartite graphs K ℓ,ℓ . Consider an automorphism φ of the graph G that isof order 2 and satisfies φ ( λ i ) = λ i +1 if i is odd. Let H denote a subgroup of Aut( G ) generated by φ so that H ∼ = Z . Then the stabilizers H i are all trivial and dim( V λ i ) = ℓ ≥
3. Hence by Corollaries 4.7 and 5.2, theassociated infra-nilmanifold with holonomy group Z admits an Anosov diffeomorphism.We can generalize the above family by varying the sizes of coherent classes and keeping the size of λ i and λ i +1 the same, for i odd. More precisely, we assume that the size of λ i and λ i +1 is ℓ i +12 for all odd i , where ℓ i +12 ≥ i = 2 m − ℓ m ≥
3. Note that we need ℓ m ≥ c ( λ m − ) = 2 (see Corollary 4.7). Usingthe same graph automorphism φ as above, we will get the associated infra-nilmanifolds admitting Anosovdiffeomorphisms with holonomy group Z . We note that the number of vertices in G is m X i =1 ℓ i and the numberedges in G is m − X i =1 ℓ i ℓ i +1 + ℓ m . Hence this family gives us examples of infra-nilmanifolds of dimensions ofthe form m X i =1 ℓ i + m − X i =1 ℓ i ℓ i +1 + ℓ m where m ≥ ℓ m ≥ ℓ i ≥ i = m . In particular, for m ≥ ℓ m = 3 and ℓ i = 2 for all 1 ≤ i < m , the dimension of the corresponding infra-nilmanifold is(4 m + 2) + (8 m + 5) = 12 m + 7. If m = 1, the dimension is 6 + 9 = 15. Remark . One can modify the Family I in many different ways. We describe one such a modification. Let m ≥ , ℓ = ℓ m = 3 and ℓ i = 2 for 2 ≤ i ≤ m −
1. The sizes of the coherent classes are the same as in thegeneralized family as described above, i.e., λ i = λ i +1 = ℓ i +12 for all odd i . We consider a graph G whosequotient graph is λ λ · · · λ m − λ m · · · λ λ In this case also by considering the same type of order 2 graph automorphism as in Example 5.3, we geta family of infra-nilmanifolds admitting Anosov diffeomorphisms with holonomy group Z . The number ofvertices of G is then 4 m +4 and the number of edges is 8 m +5+10. Hence the dimension of the corresponding2-step nilpotent Lie algebra is 12 m + 19. We observe that these dimensions are already covered by Example5.3. However, this graph and a graph from the Family I are not isomorphic and hence the associated2-step nilpotent Lie algebras are non isomorphic [Mai15]. This gives us examples of non isomorphic infra-nilmanifolds admitting Anosov diffeomorphisms with holonomy group Z and with dimension of the form12 m + 19 for m ≥ Example 5.5 (Holonomy Z n with n ≥ . Let n ≥ n = 4. We consider a graph G with n coherent classes { λ , . . . , λ n } all of size 3 and whose quotient graph G is a cycle graph with λ i λ ( i +1) mod n an edge in G for all i , 1 ≤ i ≤ n . Let φ denote the graph automorphism of G of order n satisfying φ ( λ i ) = λ ( i +1) mod n for all i and H be a cyclic subgroup of Aut( G ) generated by φ so that H ∼ = Z n . Then an infra-nilmanifold associatedto G admits an Anosov diffeomorphism with holonomy group Z n . The dimension of the infra-nilmanifoldis 3 n + 9 n = 12 n. This can be generalized by choosing the size of each coherent class to be ℓ ≥ ℓn + ℓ n . NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 23
For holonomy group Z , we consider a graph with 4 coherent classes each of size ℓ ≥ λ λ λ λ Here H would be a subgroup of graph automorphism group generated by an order 4 rotation mapping λ i to λ ( i +1)mod4 . The corresponding infra-nilmanifold is of dimension 4 ℓ + 2 ℓ ( ℓ −
1) + 4 ℓ . The minimumpossible such a dimension is 60 when ℓ = 3. Remark . In Family II described as above, we note that we had to deal with the Z case separately. Thisis because a simple cycle graph on 4 vertices can not be realized as a quotient graph of a graph. Here wewant to note that sufficient conditions for a given graph to be a quotient graph are given in [MPS18, Lemma4.6] 6. Open questions
Although Theorem 4.6 seems to give a full answer about the existence of Anosov diffeomorphisms on Liegroups associated to graphs, it only considers one type of uniform lattices of these Lie groups. However,if N is a 1-connected nilpotent Lie group with a uniform lattice M ≤ N , then the existence of an Anosovdiffeomorphism on the nilmanifold N (cid:30) M depends on the lattice M . For example, if N G is the Lie groupassociated to the graph G given as follows α βδγ i.e. N G is the direct sum of two real Heisenberg groups, then there exist uniform lattices M , M ≤ N G suchthat N G (cid:30) M admits an Anosov diffeomorphism but N G (cid:30) M does not. Equivalently, the Lie algebra n R G has twodistinct rational forms where one has an Anosov automorphism and the other does not [Mal00]. Hence whenstudying Anosov diffeomorphisms, even for nilmanifolds associated to graphs, the choice of rational formis crucial. For free nilpotent Lie groups, there is only one posible uniform lattice up to commensurability,hence this problem did not occur in the work [DV09b].Note that in this paper we only considered infra-nilmanifolds associated to graphs, which is by definitionequivalent to considering only the standard rational form n Q G ⊂ n R G of Lie algebras associated to graphs. It isstill an open problem to describe the other rational forms of n R G and characterize the ones which admit anAnosov automorphism. Question 1.
Is there a description of all rational forms m Q ⊂ n R G of real Lie algebras associated to graphs?Can we characterize the ones which admit an Anosov automorphism? Even if n Q G is not Anosov, their could be other rational forms which are Anosov, as the example G of thedirect sum of two Heisenberg algebras shows. But if n Q G is Anosov, all low-dimensional examples seem toimply the following conjecture. Conjecture.
Let n Q G be a Lie algebra associated to a graph G admitting an Anosov automorphism. If m Q isany rational form of the real Lie algebra n R G , then m Q is Anosov as well. If a full answer to the Question 1 would be known, the natural follow-up question would be the general-ization of Theorem 4.6 to other rational forms of n R G . Question 2.
Is there a characterization of the infra-nilmanifolds modeled on Lie groups associated to graphswhich admit an Anosov diffeomorphism?
Note that there is an important distinction between infra-nilmanifolds modeled on Lie groups associated tographs and infra-nilmanifolds associated to graphs, since by Definition 4.1 we assume that the rational Liealgebra of the latter is equal to n Q G . References [Aus60] Louis Auslander. Bieberbach’s Theorem on Space Groups and Discrete Uniform Subgroups of Lie Groups.
Ann. ofMath. (2) , 71((3)):pp. 579–590, 1960.[Bor91] Armand Borel.
Linear algebraic groups , volume 126 of
Graduate Texts in Mathematics . Springer-Verlag, secondedition, 1991.[Dan78] S.G. Dani. Nilmanifolds with Anosov automorphisms.
J. London Math. Soc. (2) , (18):pp. 553–559, 1978.[DD14] Karel Dekimpe and Jonas Der´e. Existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotentLie groups.
Topological Methods in Nonlinear Analysis , 2014.[Dek96] Karel Dekimpe.
Almost-Bieberbach Groups: Affine and Polynomial Structures , volume 1639 of
Lect. Notes in Math.
Springer–Verlag, 1996.[Dek12] Karel Dekimpe. What an infra-nilmanifold endomorphism really should be . . . .
Topol. Methods Nonlinear Anal. ,40(1):111–136, 2012.[Der15] Jonas Der´e. A new method for constructing Anosov Lie algebras.
Transactions of the American Mathematical Society ,2015.[DM05] S. G. Dani and Meera G. Mainkar. Anosov automorphisms on compact nilmanifolds associated with graphs.
Trans.Amer. Math. Soc. , 357(6):2235–2251, 2005.[DV09a] Karel Dekimpe and Kelly Verheyen. Anosov diffeomorphisms on infra-nilmanifolds modeled on a free 2-step nilpotentLie group.
Groups, Geometry and Dynamics , 3(4):pp. 555–578, 2009.[DV09b] Karel Dekimpe and Kelly Verheyen. Anosov diffeomorphisms on nilmanifolds modeled on a free nilpotent Lie group.
Dynamical Systems – an international journal , 24(1):pp. 117–121, 2009.[DV11] Karel Dekimpe and Kelly Verheyen. Constructing infra-nilmanifolds admitting an Anosov diffeomorphism.
Adv. Math. ,228(6):3300–3319, 2011.[Hum81] James E. Humphreys.
Linear Algebraic Groups . Graduate Texts in Mathematics. Springer-Verlag, New York, 1981.[Lau03] Jorge Lauret. Examples of Anosov diffeomorphisms.
J. Algebra , 262(1):201–209, 2003.[LW09] Jorge Lauret and Cynthia E. Will. Nilmanifolds of dimension ≤ Trans. Amer.Math. Soc. , 361(5):2377–2395, 2009.
NOSOV DIFFEOMORPHISMS ON INFRA-NILMANIFOLDS ASSOCIATED TO GRAPHS 25 [Mai15] M. G. Mainkar. Graphs and two-step nilpotent Lie algebras.
Groups Geom. Dyn. , 9(1):55–65, 2015.[Mal51] Anatoliˇı I. Mal’cev. On a class of homogeneous spaces.
Amer. Math. Soc. Translations , (39):pp. 1–33, 1951.[Mal00] Wim Malfait. Anosov diffeomorphisms on nilmanifolds of dimension at most six.
Geometriae Dedicata , 79((3)):291–298, 2000.[MPS18] Meera Mainkar, Matthew Plante, and Ben Salisbury. Counting Anosov graphs.
Ars Combin. , 141:29–51, 2018.[Pay09] Tracy L. Payne. Anosov automorphisms of nilpotent Lie algebras.
J. Mod. Dyn. , 3(1):121–158, 2009.[Por72] Hugh L. Porteous. Anosov diffeomorphisms of flat manifolds.
Topology , 11,:pp. 307–315, 1972.[Rag72] M. S. Raghunathan.
Discrete Subgroups of Lie Groups , volume 68 of
Ergebnisse der Mathematik und ihrer Grenzge-biete . Springer-Verlag, 1972.[Seg83] Daniel Segal.
Polycyclic Groups . Cambridge University Press, 1983.[Sma67] S. Smale. Differentiable dynamical systems.
Bull. Amer. Math. Soc. , 73,:pp. 747–817, 1967.
KU Leuven Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium
E-mail address : [email protected] Department of Mathematics, Pearce Hall, Central Michigan University, Mt. Pleasant, MI 48858, USA
E-mail address ::