aa r X i v : . [ m a t h . DG ] D ec Integral affine 3-manifolds.
Kozlov I.K.MSU, Moscow, [email protected]
Abstract
Affine manifolds are called integral if there is an atlas such that all transitionmaps are affine transformations with integer matrices of linear parts. In this paperwe describe all complete integral affine structures on compact three-dimensionalmanifolds up to a finite-sheeted covering. Also a complete list of integral affinestructures on the three-dimensional torus and compact three-dimensional nilman-ifolds was obtained.
Key words: affine manifolds, three-dimensional manifolds, nilmanifolds, solvmanifolds. An affine manifold is a smooth manifold with an atlas { ( U α , ϕ α ) } such that all maps ϕ − β ϕ α are affine transformations ~x A~x + ~b . An affine manifold is called integral ifthe matrices A of all transition maps are integer, i.e. A ∈ GL n ( Z ) .Integral affine manifolds naturally arise in symplectic geometry and the theory ofintegrable Hamiltonian systems. In particular, a base of any Lagrangian bundle withcompact connected fibers has a natural integral affine structure (see, for example, [1],[2]). In action-angle coordinates from the Liouville theorem, this affine structure on thebase is defined by the action coordinates.Earlier, in paper [2] classifying invariants of Lagrangian bundles were described(this work was based on [1] and [3]). The first invariant is an integral affine structureon the base. The remaining invariants are elements of some cohomology groups, andin [2] it was shown how they can be calculated if one knows the affine structure onthe base. Thus, in order to study Lagrangian bundles on (compact) six-dimensionalsymplectic spaces π : ( M , ω ) → B or integrable systems with degrees of freedom,a list of possible integral affine structures on (compact) three-dimensional manifoldsmay be useful. A similar list is described in this paper (see Theorems 2 and 4).Previously, two-dimensional Lagrangian bundles over orientable two-dimensionalsurfaces were classified by K. N. Mishachev in [3]. Moreover, he proved that any two-dimensional integral affine manifold is diffeomorphic to either the torus T or the Kleinbottle K and classified all integral affine structures on the two-dimensional torus T .Similar results for non-orientable surfaces (i.e., in the case of the Klein bottle K ) wereindependently (and almost simultaneously) obtained by I. K. Kozlov [2] and D. Sepe[4]. In this paper, we will continue these studies in the three-dimensional case.1t is well known that compactness of an affine manifold does not imply its (geodesic)completeness. So far, the Markus conjecture, which states that completeness of an affinemanifold is equivalent to existence of a parallel volume form (which in turn is equivalentto that for some atlas all the matrices of transition maps A ∈ SL n ( R ) ), has not beenproven. For simplicity, we will consider only those affine manifolds for which the Markusconjecture holds. We also assume that all manifolds are orientable so that all matricesof transition maps A ∈ SL n ( Z ) . Assumption . All integral affine manifolds considered in this paper are assumedto be complete and orientable.
Any complete affine manifold is a quotient E/ Γ , where Γ ⊂ Aff( E ) is a discretegroup of affine transformations that acts freely and properly discontinuously on anaffine space E . To be short we will often identify a complete affine manifold M ≈ E/ Γ with the corresponding group Γ ⊂ Aff( E ) .All three-dimensional affine crystallographic groups were classified in [5]. In partic-ular, all compact complete three-dimensional affine manifolds were described there. Theorem . Let M be a compact three-dimensional manifold. Then thefollowing conditions are equivalent:1. M admits a complete affine structure,2. M is finitely covered by a T -bundle over S ;3. π ( M ) is solvable and M is aspherical. A classification of integral affine structures is “finer” than the classification of affinecrystallographic groups in [5]. From the algebraic point of view, [5] studies conjugacyclasses of discrete subgroups Γ ⊂ Aff n ( R ) . In this paper we study conjugacy classes ofsubgroups Γ in the smaller group GL n ( Z ) ⋉ R n of affine transformations ~x A~x + ~b with an integer matrix A .In this paper, we describe all complete integral affine structures on compact three-dimensional manifolds up to a finite-sheeted covering (see Theorem 4). We considerstructures up to a covering so that not to get a too big final list. Also in Section 2we describe all integral affine structures on the three-dimensional torus and compactthree-dimensional nilmanifolds (see Theorem 2). Note that Theorem 2 is proved byelementary methods without using results from [5].In this paper we use the following notation and definitions. An affine transformation ~x A~x + ~b is denoted by ( A,~b ) . The linear and vector parts of an affine transformation g = ( A,~b ) will be denoted by L ( g ) = A and v ( g ) = ~b respectively. The group ofaffine transformations ( A,~b ) of a real n -dimensional space with an integral matrix A is denoted as AGL n ( Z ) . An affine frame Oe . . . e n of an affine space A n consists ofa point O ∈ A n and vectors e . . . e n that form a basis of R n . An affine frame willbe called integer if all vectors e i ∈ Z n . A diffeomorphism of integral affine manifolds f : M → M that identifies integral affine structures on them will be call an affinediffeomorphism . Acknowledgments.
The author would like to thank the staff of the Department ofDifferential Geometry and Applications of the MSU Faculty of Mechanics and Math-ematics, especially Prof. A. A. Oshemkov for his support in writing this paper. Thiswork was supported by the Russian Science Foundation grant (project № 17-11-01303).2
Integral affine 3-nilmanifolds
Theorem . Any compact integral affine -dimensional nilmanifold M is com-plete. Moreover, it is affinely diffeomorphic to a quotient of R by an action of one ofthe following groups Γ described in Tables 1 and 2. These tables contain matrices A,~b of affine transformations x Ax + ~b corresponding to generators g, h, t of the group Γ . In Series – ( described in Table 1) the manifold is diffeomorphic to the torus M ≈ T and the generators g, h, t commute. In other series (described in Table2) theysatisfy the relation [ g, t ] = [ h, t ] = e, [ g, h ] = t m , (1) where m ∈ N . In all series a , b , b , c ∈ Z \ { } , b ∗ ∈ Z , n ∈ N , x i , y i , z i ∈ R . Also, inall series except for the vector parts v ( g ) , v ( h ) , v ( t ) must be linearly independent. InSeries the following condition must be satisfied (cid:12)(cid:12)(cid:12)(cid:12) x y x − y x + m n x − ny y x − y x + mn y (cid:12)(cid:12)(cid:12)(cid:12) = 0 (2) g h t E, x y z E, x y z E, x y z b , y z E, x y E, x y a
00 1 00 0 1 , y z na , x nz z E, x a b ∗ c , z b , x b a z E, x Table 1: Integral affine 3-tori
Remark . From Theorem 2 one can obtain a classification of -dimensional in-tegral affine nilmanifolds by a scheme similar to the one described in [2] and [3] in thetwo-dimensional case. Manifolds from different series are affinely non-diffeomorphic(since they have different linear holonomy). In one series, one has to to identify themanifolds M that are obtained from each other by a change of integer affine frameof R or by a change of generators of the fundamental group π ( M ) . For exam-ple, two manifolds M and M ′ from Series are affinely diffeomorphic if and onlyif C (cid:16) x x x y y y z z z (cid:17) D = (cid:18) x ′ x ′ x ′ y ′ y ′ y ′ z ′ z ′ z ′ (cid:19) for some matrices C, D ∈ GL(3 , Z ) . Proof of theorem 2.
Completeness of integral affine nilmanifolds follows from the fol-lowing previously proved statement. 3 h t a
00 1 00 0 1 , y z E, x mx a E, x a b ∗ c , z E, x mx a E, x a
00 1 00 0 1 , y z na , mx a + nz z E, x c , x y − mx nc nc , x y my c E, x y a b ∗ c , z b , x mx + b z a E, x Table 2: Integral affine 3-nilmanifolds
Theorem . For a compact affine manifold M with nilpotent affine holonomy,the following conditions are equivalent:1. M is complete;2. M admits a parallel volume form;3. the linear holonomy is unipotent;4. M is a complete affine nilmanifold. Since the linear holonomy is unipotent, there exists a real basis of R in whichall matrices A of linear holonomy are upper unitriangular matrices. The followingstatement shows that this basis can be taken integer. Assertion . If all matrices from a certain subset of integer matrices M ⊂ GL n ( Z ) have a common eigenvector v with eigenvalue , then this vector can be chosen integer.Moreover, this vector v can be taken from some basis Z n .Proof of Assertion 1. Indeed, the vector v is given by linear equations ( A − E ) v = 0 , which must be satisfied for all A ∈ M . Since all matrices are integer, these linearequations have a common non-zero solution over R if and only if they have a commonnon-zero solution over Q . It remains to use the following simple statement. Assertion . An integer vector v ∈ Z n can be included in an integer basis Z n ifand only if its coordinates are mutually comprime. Thus, any vector from Q n is proportional to a vector from some basis Z n . Assertion 1is proved.After all the matrices A of the linear holonomy are simultaneously reduced to theupper unitriangular form, the problem is solved by brute force. Denote by UT n ( Z ) theset of upper unitriangular matrices with integer coefficients. We need to find all discrete4ubgroups Γ ⊂ UT ( Z ) ⋉ R n that act on R freely and properly discontinuously. It isconvenient to analyze the cases according to the rank of the subgroup of translations T in the group Γ .1. Let rk T = 3 . Then the quotient by the action of T is the torus T . The followingsimple statement shows that Γ = T , since there are no nontrivial elements offinite order in Γ /T . Assertion . If a group G acts freely and properly discontinuously on R n , thenany subgroup H ⊂ G , for which the space R n /H is compact, has finite index: [ G : H ] < ∞ . Obviously, in this case, we get Series from Table 1.2. Let rk T = 2 . In this case, it suffices to add one more element to T so that thequotient by the action of the generated group becomes compact. First of all, letus specify the form of the added element g ∈ Γ . Using Assertion 2, it is not hardto prove the following statement. Assertion . Assume that all eigenvalues of an element g ∈ AGL ( Z ) are equalto . Then there exists an integer affine frame in which g has one of the followingforms: a b ∗ c , z , b , yz or E, xyz (3) where a, b, b ∗ , c ∈ Z and a, b, c = 0 . Let us note that a change of integer affine frame corresponds to a conjugation byan element of
AGL ( Z ) .Then, for convenience of verification, let us give a formula for the commutator oftwo elements g i = (cid:16)(cid:16) a i b i c i (cid:17) , (cid:16) x i y i z i (cid:17)(cid:17) , i = 1 , : g g g − g − = (cid:12)(cid:12)(cid:12)(cid:12) a c a c (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) a y a y (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) b z b z (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) a c a c (cid:12)(cid:12)(cid:12)(cid:12) ( z + z ) (cid:12)(cid:12)(cid:12)(cid:12) c z c z (cid:12)(cid:12)(cid:12)(cid:12) (4) In particular a conjugation of a translation is again a translation: g = (cid:16) A,~b (cid:17) , t = (
E, ~v ) ⇒ gtg − = ( E, A~v ) (5)It immediately follows from (5) that the group of translations T ⊂ Γ generates atwo-dimensional subspace that is invariant with respect to L (Γ) . Then it is easyto prove the following assertion. Assertion . Consider a group Γ ⊂ AGL ( Z ) generated by three affine trans-formations g = a b c , x y z , t = E, x y z , t = E, x y z here g is one of the transformations from (3) and translations t and t arelinearly independent. The action of Γ on R is free and properly discontinuousif and only if the vectors (cid:16) x y z (cid:17) and (cid:16) x y z (cid:17) span an L ( g ) - invariant subspace and det (cid:16) x x x y y y z z z (cid:17) = 0 . Further, note that we can change a basis of T so that the relations (1) will holdfor h = t , t = t (there is also a commutative case, when m = 0 ). Indeed,according to (5) the subgroup of translations T is L ( g ) -invariant. An operator L ( g ) is unipotent, therefore it also acts on T as an unipotent operator. It remainsto use the following fact, which follows from Assertion 1. Assertion . For any unipotent operator A ∈ SL ( Z ) there exists an integerbasis of Z in which the matrix A has the form (cid:18) m (cid:19) , where m ∈ Z , m ≥ .The number m is uniquely defined and is equal to the GCD of elements of A . It is easy to check that after a suitable choice of basis in T for the groups fromAssertion 5 we get Series 2 from Table 1 and Series 5 and 6 from Table 2. Notethat these series are distinguished by the Jordan normal forms of L ( g ) in R andwhen acting on T .Assertion 5 guarantees that all these series define nilmanifolds. It remains toshow that we have described all the series with rk T = 2 . Assertion . Let rk T = 2 . Then there exists g ∈ Γ such that the group Γ isgenerated by g and T .Proof of Assertion 7. It suffices to take an element g , for which numbers b inSeries 2 and a in Series 5 and 6 are the smallest possible natural numbers.Indeed, if the number a (or b ) is the smallest possible, then by multiplying anyother element of g ′ ∈ Γ on g k we can always make the corresponding coefficient a ′ (or b ′ ) equal to zero. But according to Assertion 3 some degree of the element g ′ g k ∈ Γ must belong to the subgroup generated by g, t and t . This is possibleonly if g ′ g k ∈ T . Assertion 7 is proved.3. Let rk T = 1 . In this case, from (5) it follows that a vector of translation ~v is a common eigenvector for all operators L ( g ) , where g ∈ Γ . This case can beconsidered analogously to the case rk T = 2 , but now we have to add two elements g and h to a translation t ∈ T . Let us consider several cases.(a) Suppose that there exists an element g ∈ Γ such that L ( g ) = (cid:16) a b c (cid:17) ,where a , c = 0 . By changing coordinates we can assume that g = (cid:16)(cid:16) a b c (cid:17) , (cid:16) z (cid:17)(cid:17) . Since L ( g ) has only one eigenvector t = (cid:16) E, (cid:16) x (cid:17)(cid:17) .Further, without loss of generality, we can assume that a is positiveand the smallest possible and that the second element has the form h = (cid:16)(cid:16) b c (cid:17) , (cid:16) x y z (cid:17)(cid:17) . First consider the case c = 0 .6 ssertion . Consider a group Γ ⊂ AGL ( Z ) generated by three affinetransformations g = a b c ! , z !! , h = b ! , x y z !! , t = E, x !! where a , b , c , x = 0 and rk T = 1 . The action of Γ on R is free andproperly discontinuous if and only if z = 0 , z y x = 0 and a y − b z x ∈ Q .The quotient space R / Γ will then be a compact nilmanifold.Proof of Assertion 8. Indeed, according to formula (4) [ g, h ] = E, a y + b z − b z c z . (6)Since rk T = 1 , the vectors [ g, h ] and t must be commensurable, hence z = 0 and a y − b z x ∈ Q . Obviously, if z y x = 0 , then the action is not free. Itcan be explicitly checked that under the specified conditions the action isfree and properly discontinuous. It is easy to see that one of the fundamentaldomains is contained in the ≤ x ≤ x , ≤ y ≤ y , ≤ z ≤ z and therefore R / Γ is compact. Assertion 8 is proved.Let us now show that the case c = 0 is impossible. Assertion . A group Γ ⊂ AGL ( Z ) generated by three affine transforma-tions g = a b c ! , z !! , h = b c ! , x y z !! , t = E, x !! where a , c , c , x = 0 , and rk T = 1 , cannot act on R freely and discon-tinuously.Proof of Assertion 9. According to the formula (4) the linear part of thecommutator L ([ g, h ]) = (cid:16) a c (cid:17) . If the group Γ generated by g, [ g, h ] and t acts freely and properly discontinuously, then according to the Assertion 8the quotient R / Γ will be compact. But then by Assertion 3 some degreeof h has to belong to Γ , which is impossible. Assertion 9 is proved.It is easy to see that if in Assertion 8 we take t as a generator of the subgroupof translations T , then we get Series 4 from Table 1 (if [ g, h ] = e ) or Series9 from Table 2.(b) Assume that linear parts of all elements g ∈ Γ have the form L ( g ) = (cid:16) a g b g (cid:17) . First of all, let us find out how we can reduce the matrix X = (cid:16) a g b g a h b h (cid:17) for a given pair of elements g, h ∈ Γ . Since we can changethe elements g, h and take other elements of the basis e , e , the matrix X is defined up to the left-right GL(2 , Z ) × GL(2 , Z ) action. The followingstatement describes the orbits of this action on Mat × ( Z ) .7 ssertion . For any integer matrix ( a bc d ) ∈ Mat × ( Z ) there exist C, D ∈ GL(2 , Z ) such that C (cid:18) a bc d (cid:19) D = (cid:18) n kn (cid:19) , where n = GCD ( a, b, c, d ) and k = | ad − bc | n . Here we formally assume that k = n = 0 if a = b = c = d = 0 .Proof of Assertion 10. Among points (cid:0) a ′ b ′ c ′ d ′ (cid:1) of the orbits of the originalmatrix let us choose the one such that a ′ is the smallest possible naturalnumber. We claim that a ′ = n . Indeed, using the Euclidean algorithm, onecan always find matrices C and D such that C (cid:0) a ′ c ′ (cid:1) = (cid:0) GCD ( a ′ ,c ′ )0 (cid:1) and ( a ′ b ′ ) D = ( GCD ( a ′ ,b ′ ) 0 ) . Therefore, since a ′ is the smallest possible, we canassume that b ′ = c ′ = 0 . Then, for the same reasoning a ′ (cid:12)(cid:12) d ′ in the matrix (cid:0) a ′ d ′ (cid:1) .Then a ′ is the GCD of the elements of the final matrix. But the left-rightaction does not change the GCD (since it obviously does not decrease un-der multiplication on C and D , and since these matrices are invertible).Therefore, in the final matrix a ′ = n . It remains to note that we can assume d ′ = kn , since under the left-right action, the determinant is preserved upto sign. Assertion 10 is proved.Thus, we can assume that the matrix (cid:16) a g b g a h b h (cid:17) has the form (cid:0) a na (cid:1) . Bychanging the origin, we can always make the corresponding components ofthe translation equal to zero x = y = 0 . The following statement is provedanalogously to Assertion 8. Assertion . Consider a group Γ ⊂ AGL ( Z ) generated by three affinetransformations g = a
00 1 00 0 1 ! , y z !! , h = na ! , y z !! , t = E, x !! where a , x = 0 and rk T = 1 . The action of Γ on R is free and properlydiscontinuous if and only if the vector parts of the elements v ( g ) , v ( h ) and v ( t ) are linearly independent and a y − na z x ∈ Q . Then, the quotient space R / Γ is a compact nilmanifold. Thus, we obtain Series 3 from Table 1 (if [ g, h ] = e ) or Series 7 from Table 2.(c) It remains to consider the case when the linear parts of all elements havethe form L ( g ) = (cid:16) b g c g (cid:17) . Similarly to the previous cases, we can reducethe matrix (cid:0) b g b h c g c h (cid:1) to (cid:0) nc c (cid:1) and after that bring the elements to the formfrom Series from Table 2. It remains to show that the action is free andproperly discontinuous, and the quotient is compact exactly when (2) holds.For that it is convenient to switch from AGL ( Z ) to the group of all affinetransfromations Aff ( R ) and take a vector of translation as the first basisvector. 8 ssertion . Consider a group Γ ⊂ Aff ( R ) generated by three affinetransformations g = b c ! , x y z !! , h = b c ! , x y z !! , t = E, x !! where b i , c i ∈ R , ( c , c ) = (0 , , ( z , z ) = (0 , , x = 0 and rk T = 1 .The action of Γ on R is free and properly discontinuous if and only if c z = c z , b z − b z x ∈ Q and (cid:12)(cid:12)(cid:12)(cid:12) c y − c z c y − c z (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (7) The quotient space R / Γ is then a compact nilmanifold.Proof of Assertion 12 . Indeed, according to (4) [ g, h ] = E, b z − b z c z − c z . (8)Since rk T = 1 the vectors [ g, h ] and t must be commensurable and we get c z = c z and b z − b z x ∈ Q . Since ( c , c ) = (0 , , there exists u ∈ R such that z = c u, z = c u . Moreover, u = 0 , since ( z , z ) = (0 , . Theremaining condition (7) is equivalent to the fact that the matrices g = (cid:18)(cid:18) c (cid:19) , (cid:18) y c u (cid:19)(cid:19) , h = (cid:18)(cid:18) c (cid:19) , (cid:18) y c u (cid:19)(cid:19) , generates a discrete subgroup of rank in the commutative two-dimensionalgroup (cid:18)(cid:18) c (cid:19) , (cid:18) c u + tcu (cid:19)(cid:19) , c, t ∈ R which acts freely and transitively on the plane. It is precisely the case whenthe action is free and properly discontinuous and the quotient space R / Γ is compact. Assertion 12 is proved.It can be easily seen that condition (7) for Series 8 from Table 2 can berewritten as (2), which had to be shown.4. Let rk T = 0 . It remains to prove the following assertion. Assertion . Affine holonomy of a compact integral -dimensional affine nil-manifold contains at least one nontrivial translation. In other words, for the corre-sponding subgroup Γ ⊂ AGL ( Z ) the translation subgroup is nontrivial: rk T = 0 .Proof of Assertion 13 . On the contrary, assume that there is a group Γ , that actsfreely and properly discontinuously on R so that R / Γ is compact and rk T = 0 .Without loss of generality, we can assume that the generators of Γ have the form g = a b c ! , z !! , h = b c ! , x y z !! , t = b ! , x y z !! a , c , c , b , z = 0 and components a , c , b are the smallest possible nat-ural numbers. It is easy to check that if we take out any of the elements g, h, t ,then the quotient R / Γ cannot be compact. Since rk T = 0 we have [ h, t ] = e .Therefore, using formula (4) (or (8)) we get − c z = 0 , b z − b z = 0 ⇒ z = z = 0 Analogously [ g, t ] = e and thus a y = b z . Again, using the formula (4) we get [ g, h ] = a c a y + b z − b z − a c ( z + z ) c z − c z Since b is the smallest possible [ g, h ] = t m holds. But then a c = mb , − c z = my = m b z a . Thus a c z = − a c z , but a c z = 0 , and we get a contradiction. Assertion 13is proved.Theorem 2 is proved. Let us describe the remaining complete integral affine three-dimensional manifolds thatare not covered by nilmanifolds. All of them contain hyperbolic elements in the group Γ and generalize the following simple example. Example . For any matrix A = ( a bc d ) ∈ SL ( Z ) the quotient of R by the group Γ generated by elements g = a b c d , , t = , t = , is an integral affine -dimensional manifold M . Moreover, M is diffeomorphic to a T -bundle over S with monodromy matrix A . Remark . Note that there are naturally defined subsets in affine manifolds withtransformation maps from GL n ( Z ) ⋉ Z n (that is, all their elements are integers). Forexample, these are points that have integer coordinates in any chart (or rational coor-dinates from k Z n ). Remark . It is easy to see that two manifolds from Example 1 are affinelydiffeomorphic if and only if the corresponding matrices A and A lie in the sameconjugacy class in the group SL ( Z ) . Description of such conjugacy classes is a well-studied question (see, for example, [7]).If a matrix A ∈ SL ( Z ) is not unipotent, then in some basis it has the form (cid:0) e s e − s (cid:1) .The groups from Example 1 with non-unipotent matrix A correspond to the next seriesof affine crystallographic groups from [5]. 10 xample . For any λ ∈ R the group I λ = λe s u λe − s t e s
00 0 e − s s + λuttu , s, t, u ∈ R (9)acts freely and transitively on R . As a consequence, for any discrete subgroup Γ ⊂ I λ the quotient R / Γ is an affine manifold. It is easy to check that if the quotient R / Γ iscompact, then the group Γ possesses generators of the form g = λe s z λe − s y e s
00 0 e − s , s + λz y y z ,h = λz λy , λy z y z h = λz λy , λy z y z , (10)where det ( y z y z ) = 0 and (cid:0) e s e − s (cid:1) ( y z y z ) = ( y z y z ) A for some matrix A ∈ SL ( Z ) .If λ = 0 , then the group Γ from Example 2 is generated by three elements, two ofwhich are translastions. Using Assertion 1, it is easy to check that any such subgroup Γ ⊂ AGL ( Z ) is conjugate in AGL ( Z ) to one of the following subgroups. Assertion . Consider a group Γ ⊂ AGL ( Z ) , generated by three affine trans-formations g = p q a b c d , x , t = E, x y z , t = E, x y z , where a + d > , the vector parts v ( g ) , v ( t ) and v ( t ) are linearly independent and (cid:16) p q a b c d (cid:17) (cid:16) x x y y z z (cid:17) = (cid:16) x x y y z z (cid:17) A for some matrix A ∈ SL ( Z ) .Then the quotient R / Γ isan integral affine -dimensional manifold M that is diffeomorphic to a T -bundle over S with monodromy matrix A . It remains to analyze the case when Γ ⊂ AGL ( Z ) is conjugate in Aff ( R ) to adiscrete subgroup in I λ and λ = 0 . Using Assertions 1 and 10 we can simplify thelinear parts of generators of Γ . Moreover, by choosing a suitable origin, we can replacea conjugation by an element from Aff ( R ) with conjugation by an element from GL ( R ) .Thus, we obtain the following statement. Assertion . Assume that a subgroup Γ ⊂ AGL ( Z ) is conjugate in Aff ( R ) tosome discrete subgroup in the group I λ (given by (9) ), where λ = 0 . Then there is aninteger affine frame in which Γ is generated by elements of the form g = (cid:16)(cid:16) p q a b c d (cid:17) , (cid:16) x y z (cid:17)(cid:17) , h = (cid:16)(cid:16) a
00 1 00 0 1 (cid:17) , (cid:16) x y z (cid:17)(cid:17) , h = (cid:16)(cid:16) na (cid:17) , (cid:16) x y z (cid:17)(cid:17) , where a , n ∈ N and (cid:0) a na (cid:1) ( a bc d ) = A (cid:0) a na (cid:1) for some matrix A ∈ SL ( Z ) . In thiscase, generators g, h , h have the form (10) when conjugated by some matrix ( α u P ) ,where α ∈ R ∗ , u ∈ Mat × ( R ) and P ∈ GL ( R ) . emark . The condition
P AP − = (cid:0) e s e − s (cid:1) defines the matrix P up to multi-plication by a constant. If we fix the matrix P , then for any α ∈ R ∗ , u ∈ Mat × ( R ) the condition of conjugacy of g, h , h to the elements (10) provides restrictions on thecoefficients of the vector parts x i , y i , z i .Now let us formulate the main result of this paper. It is convenient to divide integralaffine -dimensional manifolds into three classes according to their type of geometry(for more information about geometries on -dimensional manifolds, see [8] or [9]). Theorem . Any compact integral affine -dimensional manifold M has a geo-metric structure modeled on E , Nil or Sol . The geometric structure on M is completelydetermined by π ( M ) .1. If π ( M ) is virtually Abelian, then M is a Euclidean manifold. Moreover, M isfinitely covered by one of the manifolds from Theorem 2 described in Table 1.2. If π ( M ) is virtually nilpotent, but not virtually Abelian, then M is a Nil -manifold. Moreover, M is finitely covered by one of the manifolds from The-orem 2, described in Table 2.3. If π ( M ) is solvable, but not virtually nilpotent, then M is a Sol -manifold. More-over, M is finitely covered by an integral affine manifold R / Γ for one of thegroups Γ , described in the Assertions 14 or 15.Proof of Theorem 4. According to [5] any compact -dimensional affine manifold isfinitely covered by a solvmanifold. By taking a finite cover we can assume that thealgebraic closure of ¯Γ is connected, and therefore, by the Lie–Kolchin theorem, thereexists a basis (over C ), in which all matrices from L (Γ) are upper triangular. Moreover,up to a finite cover, we can assume that all eigenvalues are real and positive.Thus, we can assume that M ≈ R / Γ are solvmanifolds, and that the linear partsof all elements L ( g ) are either unipotent or have Jordan normal form (cid:16) e s
00 0 e − s (cid:17) . If all L ( g ) are unipotent, then M is a nilmanifold, and this case is analyzed in Theorem 2.Otherwise, according to [5], the group Γ is conjugate (in Aff ( R ) ) to a discrete subgroupof the group I λ (see Example 2). And, therefore, Γ is conjugate in AGL ( Z ) to one ofthe groups from Assertions 14 or 15. Theorem 4 is proved. References [1] J. J. Duistermaat, “On global action–angle coordinates”,