aa r X i v : . [ m a t h . G R ] S e p On the coverings of Hantzsche-Wendt manifold
G. Chelnokov ∗ National Research University Higher School of Economics, Moscow, RussiaLaboratory of Combinatorial and Geometric Structures,Moscow Institute of Physics and Technology, Moscow, Russia [email protected]
A. Mednykh † Sobolev Institute of Mathematics, Novosibirsk, RussiaNovosibirsk State University, Novosibirsk, Russia [email protected]
Abstract
There are only 10 Euclidean forms, that is flat closed three dimensional man-ifolds: six are orientable G , . . . , G and four are non-orientable B , . . . , B . In thepresent paper we investigate the manifold G , also known as Hantzsche-Wendtmanifold; this is the unique Euclidean -form with finite first homology group H ( G ) = Z .The aim of this paper is to describe all types of n -fold coverings over G and cal-culate the numbers of non-equivalent coverings of each type. We classify subgroupsin the fundamental group π ( G ) up to isomorphism. Given index n , we calculatethe numbers of subgroups and the numbers of conjugacy classes of subgroups foreach isomorphism type and provide the Dirichlet generating series for the abovesequences.Key words: Euclidean form, platycosm, flat 3-manifold, non-equivalent cov-erings, crystallographic group, Dirichlet generating series, number of subgroups,number of conjugacy classes of subgroups. Introduction
Let M be a connected manifold with fundamental group G = π ( M ) . Two coverings p : M → M and p : M → M ∗ This work was supported by Ministry of Education and Science of the Russian Federation in theframework of MegaGrant no 075-15-2019-1926 † This work was supported by the Russian Foundation for Basic Research (grant 16-31-00138). h : M → M such that p = p ◦ h. According to the general theory of covering spaces, any n -fold covering isuniquely determined by a subgroup of index n in the group G . The equivalence classesof n -fold coverings of M are in one-to-one correspondence with the conjugacy classes ofsubgroups of index n in the fundamental group π ( M ) . See, for example, ([8], p. 67). Insuch a way the following natural problems arise: to describe the isomorphism classes ofsubgroups of finite index in the fundamental group of a given manifold and to enumeratethe finite index subgroups and their conjugacy classes with respect to isomorphism type.We use the following notations: let s G ( n ) denote the number of subgroups of index n in the group G , and let c G ( n ) be the number of conjugacy classes of such subgroups.Similarly, by s H,G ( n ) denote the number of subgroups of index n in the group G , whichare isomorphic to H , and by c H,G ( n ) the number of conjugacy classes of such subgroups.So, c G ( n ) coincides with the number of nonequivalent n -fold coverings over a manifold M with fundamental group π ( M ) ∼ = G , and c H,G ( n ) coincides with the number ofnonequivalent n -fold coverings p : N → M , where π ( N ) ∼ = H and π ( M ) ∼ = G . Thenumbers s G ( n ) and c G ( n ) , where G is the fundamental group of closed orientable or non-orientable surface, were found in ([14], [15], [16]). In the paper [17], a general method forcalculating the number c G ( n ) of conjugacy classes of subgroups in an arbitrary finitelygenerated group G was given. Asymptotic formulas for s G ( n ) in many important caseswere obtained in [13].The values of s G ( n ) for the wide class of 3-dimensional Seifert manifolds were calcu-lated in [11] and [12]. The present paper is a part of the series of our papers devotedto enumeration of finite-sheeted coverings over closed Euclidean 3-manifolds. Thesemanifolds are also known as flat 3-dimensional manifolds or Euclidean 3-forms.The class of such manifolds is closely related to the notion of Bieberbach group.Recall that a subgroup of isometries of R is called Bieberbach group if it is discrete,cocompact and torsion free. Each -form can be represented as a quotient R /G where G is a Bieberbach group. In this case, G is isomorphic to the fundamental group of themanifold, that is G ∼ = π ( R /G ) . Classification of three dimensional Euclidean formsup to homeomorphism was obtained by W. Nowacki [18] and W. Hantzsche and H.Wendt [7]. There are only 10 Euclidean forms: six are orientable G , . . . , G and four arenon-orientable B , . . . , B See monograph [25] for more details.In our previous paper [3] we describe isomorphism types of finite index subgroups H in the fundamental group G of manifolds B and B . Further, we calculate the respectivenumbers s H,G ( n ) and c H,G ( n ) for each isomorphism type H . In subsequent articles [4],[5] and [6] similar questions were solved for manifolds G , G , G , G , B and B .The aim of the present paper is to solve the same questions for the Hantzsche–Wendtmanifold G , undoubtedly the most weird among Euclidean -manifolds. This is theunique Euclidean -form with finite first homology group H ( G ) = Z . In contrast withmanifolds G − G , it has no fundamental set in the form of cube of hexagonal prism. Itsfundamental set consists of two cubes described below. The Hantzsche–Wendt manifoldis also known as Fibonacci manifold M . The Fibonacci manifold M n , n ≥ is a closedorientable three-dimensional manifold whose fundamental group is the Fibonacci group F (2 , n ) = h x , . . . , x n : x i x i +1 = x i +2 , i mod 2 n i . These manifolds were discovered by2. Helling, A.C. Kim and J. Mennicke [9]. It was shown by H.M. Hilden, M.T. Lozanoand J.M. Montesinos [10] that M n is the n -fold cyclic covering of the three-dimensionalsphere S branched over the figure-eight knot.Also, by A.Yu. Vesnin and A.D. Mednykh [24], M is the two fold covering of S branched over the Borromean rings. The outer automorphism group of the Hantzsche–Wendt manifold was calculated by B. Zimmermann [26]. Its high dimensional analogueswere investigated by A. Szczepa´nski [23].The description of G through Bieberbach group is the following: the group is gen-erated by isometries S : ( x, y, z ) ( x + 1 , − y, − z + 1) ,S : ( x, y, z ) ( − x + 1 , y + 1 , − z ) ,S : ( x, y, z ) ( − x, − y + 1 , z + 1) . In more geometric terms G can be described in the following way. We take the unionof two cubes [0 , ∪ [ − , as the fundamental domain of G , we call this cubes positiveand negative respectively. Now we have to provide six isometries to align each face ofthe negative cube with the respective face of the positive cube, we glue faces by thisalignment. • we align faces z = 0 and z = − with faces z = 1 and z = 0 through ( x, y, z ) ( x + 1 , − y, − z + 1) and ( x, y, z ) ( x + 1 , − y, − z − respectively (these areisometries S and S − S respectively); • align faces x = 0 and x = − with faces x = 1 and x = 0 through ( x, y, z ) ( − x + 1 , y + 1 , − z ) and ( x, y, z ) ( − x − , y + 1 , − z ) respectively (these areisometries S and S − S respectively); • finally, align faces y = 0 and y = − with faces z = 1 and z = 0 through ( x, y, z ) ( − x, − y + 1 , z + 1) and ( x, y, z ) ( − x, − y − , z + 1) respectively(these are isometries S and S − S respectively).In the present paper, we classify finite index subgroups in the fundamental group π ( G ) up to isomorphism. Given index n , we calculate the numbers of subgroups andthe numbers of conjugacy classes of subgroups for each isomorphism type. Also, weprovide the Dirichlet generating functions for all the above sequences.Numerical methods to solve these and similar problems for the three-dimensionalcrystallographic groups were developed by the Bilbao group [2]. The convenience oflanguage of Dirichlet generating series for this kind of problems was demonstrated in [22].The first homologies of all the three-dimensional crystallographic groups are determinedin [20]. Notations
Let G be a group, u , v are elements and H , F are subgroups in G . We use u v instead of vuv − and [ u, z ] instead of uvu − v − for the sake of brevity. By H v denote the subgroup3 u v | u ∈ H } . By H F denote the family of subgroups H v , v ∈ F . By Ad v : G → G denote the automorphism given by u → u v .By s H,G ( n ) we denote the number of subgroups of index n in the group G isomorphicto the group H ; by c H,G ( n ) the number of conjugacy classes of subgroups of index n in the group G isomorphic to the group H . Through this paper usually G and H arefundamental groups of manifolds G i , in this case we omit π in indices.Also we will need the following number-theoretic functions. Given a fixed n we widelyuse summation over all representations of n as a product of two or three positive integerfactors X ab = n and X abc = n . The order of factors is important. We assume this sum vanishesif n is not integer.To start with, this is the natural language to express the function σ ( n ) – the numberof representations of number n as a product of two factors σ ( n ) = X ab = n . We will alsoneed the following generalizations of σ : σ ( n ) = X ab = n a, σ ( n ) = X ab = n σ ( a ) = X abc = n a,d ( n ) = X ab = n σ ( a ) = X abc = n , ω ( n ) = X ab = n aσ ( a ) = X abc = n a b. The main goal of this paper is to prove the following two theorems.
Theorem 1.
Every subgroup ∆ of finite index n in π ( G ) is isomorphic to either π ( G ) ,or π ( G ) , or Z . The respective numbers of subgroups are ( i ) s G , G ( n ) = ω ( n , ( ii ) s G , G ( n ) = 3 ω ( n − ω ( n , ( iii ) s G , G ( n ) = n ( d ( n ) − d ( n d ( n − d ( n . Theorem 2.
Let
N → G be an n -fold covering over G . Then N is homeomorphic toone of G , G or G . The corresponding numbers of nonequivalent coverings are given bythe following formulas: ( i ) c G , G ( n ) = 14 ω (cid:0) n (cid:1) + 34 σ (cid:0) n (cid:1) + 94 σ (cid:0) n (cid:1) , ( ii ) c G , G ( n ) = 32 (cid:16) σ ( n σ ( n − σ ( n d ( n − d ( n − d ( n d ( n
16 ) − d ( n
32 ) (cid:17) , ( iii ) c G , G ( n ) = d ( n ) − d ( n d ( n − d ( n . emark. If n is odd then N ∼ = G . If n ≡ then N ∼ = G . Finally, if | n then N ∼ = G or N ∼ = G .Dirichlet generating series for the sequences provided by Theorems 1 and 2 are givenin Table 2 in Appendix. In this section we have collected some known statements that will be used later.
Proposition 1. (i) The sublattices of index n in the -dimensional lattice Z are inone-to-one correspondence with the matrices (cid:18) b c a (cid:19) , where a, b > , ab = n , ≤ c < b . Consequently, the number of such sublattices is σ ( n ) .(ii) The sublattices of index n in the -dimensional lattice Z are in one-to-one cor-respondence with the integer matrices c e f b d a , where a, b, c > , abc = n , ≤ d < b and ≤ f, e < c . Consequently, the number of such sublattices is ω ( n ) . For the proof see, for example, ([4], Proposition 1).
Corollary 1.
Let ℓ : Z Z be an automorphism of Z , given by ℓ ( a, b ) = ( u, − v ) .The sublattices ∆ of index n in the -dimensional lattice Z such that ℓ (∆) = ∆ are inone-to-one correspondence with the matrices union of two families of integer matrices, (cid:18) b a (cid:19) , where a, b > , ab = n , and (cid:18) b a/ a (cid:19) , where a, b > , ab = n and a is even.Consequently, the number of such sublattices is σ ( n ) + σ ( n ) . Corollary 2.
Let ℓ : Z Z be an automorphism of Z , given by ℓ ( u, v, w ) =( u, v, − w ) . Then the number of subgroups ∆ of index n in Z such that ℓ (∆) = ∆ isequal to σ ( n ) + 3 σ ( n ) . Proof in ([6], Corollary 3).In the next two propositions we enumerate the subgroups ∆ of index n in π ( G ) with ∆ ∼ = π ( G ) and conjugacy classes of such subgroups. This statements correspondto ([4], Proposition 3). Proposition 2.
The subgroups ∆ of index n in π ( G ) isomorphic to π ( G ) are inone-to-one correspondence with the triples ( k, H, h ) , where • k is an odd positive divisor of n , • H is a subgroup of index nk in Z , • h is a coset in Z /H . onsequently, the number of the above described subgroups is s G , G ( n ) = ω ( n ) − ω ( n ) . Proposition 3.
The conjugacy classes of subgroups ∆ of index n in π ( G ) isomorphicto π ( G ) are in one-to-one correspondence with the triples ( k, H, ¯ h ) , where • k is an odd positive divisor of n , • H is a subgroup of index nk in Z , • ¯ h is a coset in Z / h H, (2 , , (0 , i .Consequently, the number of conjugacy classes of the above described subgroups is c G , G ( n ) = σ ( n ) + 2 σ ( n ) − σ ( n ) . π ( G ) and π ( G ) The groups π ( G ) , π ( G ) and π ( G ) is given by generators and relations in the followingway π ( G ) = Z = h x, y, z : xyx − y − = xzx − z − = yzy − z − = 1 i ,π ( G ) = h x, y, z : xyx − y − = 1 , x z = x − , y z = y − i ,π ( G ) = h x, y, z : xy x − y = yx y − x = xyz = 1 i . (3.1)See [25] or [21]. Remark.
The above representation of the group π ( G ) is indeed symmetric withrespect to permutations of x , y and z . The relations xz x − z = yz y − z = zx z − x = zy z − y = 1 follow from given above.Next proposition provides the canonical form of an element in π ( G ) . Proposition 4. (i) Each element of π ( G ) can be represented in the canonical form g i x a y b z c , where g i ∈ { , x, y, z } and a, b, c are some integers.(ii) The subgroup h x , y , z i is normal in π ( G ) and isomorphic to Z .(iii) The following relations holds: x a y b z c · x = x · x a y − b z − c ,x a y b z c · y = y · x − a y b z − c ,x a y b z c · z = z · x − a y − b z c . (3.2) (iv) The product g i g j : g i , g j ∈ { , x, y, z } is given by Table 1. g i g j g j = 1 g j = x g j = y g j = zg i = 1 1 x y zg i = x x · x z · z − y · x − z g i = y y z · x y − · y x · x − g i = z z y · y − x · y z − · z Table 1 v) The representation in the canonical form w = g i x a y b z c for each element w ∈ π ( G ) is unique.Proof. Items (i–iv) follow routinely from the representation (3.1) of the group π ( G ) .To prove (v) consider the set G of all the expressions g i x a y b z c , where g i ∈ { , x, y, z } and a, b, c are some integers. Define the multiplication by concatenation and furtherreduction to the described above form through the relations (iii) and (iv). Direct veri-fication shows that G is a group with respect to this operation. Since the relations (iii)and (iv) are derived from the relations of the group π ( G ) , this group is a factor groupof the group G . By the other hand, one can verify that the relations of π ( G ) holds in G , thus G ∼ = π ( G ) . In particular, different canonical representations represent differentelements of π ( G ) . Notations.
Denote the subgroup h x , y , z i ✁ π ( G ) by Λ . Also denote the naturalby homomorphism, of factorization π ( G ) → π ( G ) / Λ by φ . Indeed, one can show that Λ coincide with the commutant of π ( G ) , see [26]. Definition 1.
Let g be an element of π ( G ) . In case g = x a y b z c we say that g haveexponents a , b , c at x , y , z respectively. In case g = zx a y b z c we say the respectiveexponents are a , b , c + 1 . Similarly in cases g = x · x a y b z c and g = yx a y b z c . Wedenote the exponents of g at x , y , z by exp x ( g ) , exp y ( g ) , exp z ( g ) respectively. We widely use the following statement, too trivial to be a lemma. Let g, h be someelements and exp y ( g ) , exp z ( g ) , exp y ( h ) , exp z ( h ) are even. Than exp x ( gh ) = exp x ( g ) + exp x ( h ) .Note that π ( G ) / Λ ∼ = Z , therefore there are only three possible isomorphism typesof a subgroup in Z , it is either trivial, or Z , or Z . Definition 2.
Let ∆ be a subgroup of finite index in π ( G ) . In case φ (∆) = 1 by X ∆ , Y ∆ , Z ∆ we refer to an arbitrary triple of generators of ∆ . If φ (∆) = { , x } by Z ∆ denote an arbitrary element of ∆ with the minimal positive odd exponent at x , andby X ∆ , Y ∆ denote an arbitrary pair of generators of ∆ T h x , y i . Similarly, in case φ (∆) = { , y } and φ (∆) = { , z } ( Z ∆ denote an element with the minimal positive oddexponent at y and z respectively). Finely, in case φ (∆) = { , x, y, z } by X ∆ , Y ∆ , Z ∆ denote an arbitrary element with minimal positive odd exponent at x , y , z respectively. Proposition 5.
Let ∆ be a subgroup of finite index in π ( G ) . Then ∆ have one of thefollowing three isomorphism types, defined by | φ (∆) | . Subgroup ∆ is isomorphic to Z , π ( G ) and π ( G ) in case | φ (∆) | = 1 , | φ (∆) | = 2 and | φ (∆) | = 4 respectively. In allcases ∆ is generated by elements X ∆ , Y ∆ , Z ∆ .Proof. In case | φ (∆) | = 1 the triple X ∆ , Y ∆ , Z ∆ generates ∆ by definition. Also, ∆ is asubgroup of Λ ∼ = Z , thus ∆ is a free abelian group. Since ∆ have finite index in Λ , wehave ∆ ∼ = Z .In case | φ (∆) | = 2 without loss of generality assume that φ (∆) = { φ (1) , φ ( x ) } .Denote the exponent of Z ∆ at x by m . Then for each g ∈ ∆ its exponent at x is amultiple of m . Otherwise multiplying either g or g − by the convenient power of Z ∆ we7et an element of ∆ with an odd exponent at x strictly between and m , which is thecontradiction with the definition of Z ∆ .To prove that X ∆ , Y ∆ , Z ∆ generate ∆ consider an arbitrary element g ∈ ∆ . Since itsexponent at x is divisible by m , gZ k ∆ ∈ ∆ T h y , z i for some k . Further, ∆ T h y , z i isgenerated by X ∆ , Y ∆ by virtue of their definition. We claim that different expressionsof the form X a ∆ Y b ∆ Z c ∆ represent different elements g ∈ ∆ . Indeed, the exponent of g at x uniquely determines c , and ∆ T h y , z i ∼ = Z , so different pairs ( a, b ) provide differentelements X a ∆ Y b ∆ .Note that the elements X ∆ , Y ∆ , Z ∆ yield the relations of the group π ( G ) hold for x, y, z , so we build the isomorphism π ( G ) → ∆ , given by x X ∆ , y Y ∆ , z Z ∆ .In case | φ (∆) | = 4 we set: X ∆ = x m y r z s , Y ∆ = y k x t z u and Z ∆ = z ℓ x v y w . Here m, k, ℓ, r, s, t, u, v, w are integers, moreover m, k, ℓ are odd positives.To prove that ∆ is generated by X ∆ , Y ∆ , Z ∆ do the following. Lemma 1.
If for some element g ∈ ∆ the numbers exp y ( g ) and exp z ( g ) are even, then m | exp x ( g ) . The proof is similar to the case | φ (∆) | = 2 . Analogous statements holds for anypermutation of x, y, z .Resume to the prove of Proposition 5. Note that X = x m , Y = y k and Z = x ℓ .By Lemma 1 if for some g ∈ ∆ all three numbers exp x ( g ) , exp y ( g ) , exp z ( g ) are even,then they are divisible by m , k and ℓ respectively; thus g can be expressed through X , Y , Z . If g ∈ ∆ has one exponent odd, without loss of generality exp x ( g ) is odd,then gX ∆ have all exponents even.If g ∈ ∆ has one exponent odd, without loss of generality we assume that exp x ( g ) isodd, then gX ∆ has all exponents even.To prove the isomorphism part note that the element X ∆ Y ∆ Z ∆ = x m − − t +2 v y − k +2 w − r z ℓ − u − s has all three exponents even, thus Lemma 1 implies m | m − − y +2 v , k | − k +2 w − r and ℓ | ℓ − u − s . So, by replacing X ∆ X ∆ Y i ∆ for some integer i and doingsimilar replacements for permuted generators, one can achieve that X ∆ Y ∆ Z ∆ = 1 andthe property of X ∆ , Y ∆ , Z ∆ given in Definition 2 holds.Now note that the elements X ∆ , Y ∆ , Z ∆ yield defining relations of the group π ( G ) .Then the mapping x X ∆ , y Y ∆ , z Z ∆ spawns the epimorphism ψ : π ( G ) → ∆ .We are going to prove that this epimorphism is indeed an isomorphism.Each element g of ∆ can be represented in the form g = g i X a ∆ Y b ∆ Z c ∆ , where g i ∈ { , X ∆ , Y ∆ , Z ∆ } ; whence such representation is possible in π ( G ) . So it is suf-ficient to prove that the above representation is unique for each g ∈ ∆ . Assume thecontrary, for some element g there are two different representations g = g i X a ∆ Y b ∆ Z c ∆ = g ′ i X a ′ ∆ Y b ′ ∆ Z c ′ ∆ .Note that for arbitrary g ∈ ∆ hold ( x ) g = x ± , ( y ) g = y ± and ( z ) g = z ± ,and the triple of signs in the exponents is solely determined by g i : (+ , + , +) , x (+ , − , − ) , y ( − , + , − ) and z ( − , − , +) . Thus g i X a ∆ Y b ∆ Z c ∆ = g ′ i X a ′ ∆ Y b ′ ∆ Z c ′ ∆ implies g i = g ′ i . Then X a ∆ Y b ∆ Z c ∆ = X a ′ ∆ Y b ′ ∆ Z c ′ ∆ , or x m ( a − a ′ ) y k ( b − b ′ ) z ℓ ( c − c ′ ) = 1 , whichis a contradiction with Proposition 4 (iv). 8 Proof of Theorem 1 and Theorem 2
The isomorphism types of finite index subgroups are already provided by Proposition 5.So we will consider isomorphism types separately in order to prove respective items ofboth theorems. ∆ ∼ = Z Recall that
Λ = h x , y , z i . By Proposition 5 each subgroup ∆ of index n in π ( G ) with ∆ ∼ = Z have φ (∆) = { } , that is ∆ Λ . Since | π ( G ) : Λ | = 4 , get | Λ : ∆ | = n .Applying Proposition 1 one gets s G , G ( n ) = ω ( n . Now we proceed to enumeration of the conjugacy classes of subgroups. Since Λ isabelian, the group π ( G ) acts by conjugation on subgroups of Λ as π ( G ) / Λ ∼ = Z .Thus each conjugacy class consists of one, two or four subgroups. Definition 3. By M denote the family of all normal subgroups ∆ , by M and M denote the families of subgroups ∆ , which belong to conjugacy classes, containing twoand four subgroups respectively. Also, by M x denote the family of subgroups ∆ such that ∆ x = ∆ . M y and M z are defined similar way. Each ∆ ∈ M belongs to all three of M x , M y , M z ; while each ∆ ∈ M belongs toexactly one of M x , M y , M z . Needless to say that ∆ ∈ M does not belongs to any of M x , M y , M z . So |M | + |M | = |M x | + |M y | + |M z | . Thus c G , G ( n ) = |M | + |M | |M | |M | + |M | + |M | |M | + |M | s G , G ( n )4 + |M x | + |M y | + |M z | . (4.3)Since in a suitable basis each of Ad x , Ad y , Ad z takes the form ( a, b, c ) ( − a, b, c ) ,Corollary 2 claims |M x | = |M y | = |M z | = σ ( n ) + 3 σ ( n ) . Thus c G ,G ( n ) = 14 ω (cid:0) n (cid:1) + 34 σ (cid:0) n (cid:1) + 94 σ (cid:0) n (cid:1) . By definition, |M | is the number of normal subgroups of index n in π ( G ) isomorphicto Z . So it is interesting in itself. It is explicitly calculated in Appendix 2. ∆ ∼ = π ( G ) By Proposition 5 each subgroup ∆ of index n in π ( G ) with ∆ ∼ = π ( G ) have | φ (∆) | = 2 .In other words, holds one of the inclusions ∆ h x, y , z i = Γ x , ∆ h y, x , z i = Γ y , ∆ h z, x , y i = Γ z . Since the above groups have index 2 in π ( G ) , subgroup ∆ has9ndex n in the respective subgroup. Further, ∆ belongs to just one of Γ x , Γ y , Γ z hencethe intersection of each two of them is the abelian group Λ = h x , y , x i .Also, subgroups Γ x , Γ y , Γ z are permutable by some outer automorphism of π ( G ) ,which permutes x, y, z . Thus it is sufficient to enumerate the subgroups of Γ = Γ x .Further during this subsection ∆ denotes a subgroup of index n in Γ isomorphic to π ( G ) . Pay attention, in spite of ∆ ∼ = Γ , ∆ is a non-trivial subgroup in Γ .Since Γ ∼ = π ( G ) , the number of the above subgroups ∆ in Γ is provided by Propo-sition 2, thus s G , G ( n ) = 3 s G , G ( n ω ( n − ω ( n . To enumerate conjugacy classes we need one more definition.
Definition 4.
Consider a subgroup ∆ . The set of subgroups { ∆ γ | γ ∈ Γ } we call a partial conjugacy class ∆ Γ . An enumeration of partial conjugacy classes of subgroups ∆ is given by Proposition 3.To enumerate conjugacy classes note that π ( G ) = Γ S y Γ , so Γ have index 2 in π ( G ) .Consequently Γ is normal in π ( G ) . Thus for each ∆ its conjugacy class consists ofone or two partial conjugacy classes ∆ Γ and (∆ Γ ) y depending upon whether partialconjugacy classes ∆ Γ and (∆ Γ ) y coincide or not. Notation. By K denote the set of partial conjugacy classes ∆ Γ , such that theequality ∆ Γ = (∆ Γ ) y holds. By K denote the set of partial conjugacy classes ∆ Γ with ∆ Γ = (∆ Γ ) y .In the introduced notation c G , G ( n ) = 3( |K | + |K | |K | + |K | |K | . (4.4)Proposition 3 implies |K | + |K | = σ ( n ) + 2 σ ( n ) − σ ( n ) . All that’s left is to calculate |K | , this is done in Lemma 3. First we need the following auxiliary statement. Lemma 2.
The following identity holds d ( n ) − d ( n d ( n − d ( n ( d ( n ) if n is odd if n is even . Proof.
Consider all factorizations n = abc and use inclusion-exclusion formula for alltriples of parities of a, b, c . Proof.
Recall that by definition d ( n ) is the number of ordered positive integer factoriza-tions abc = n . Then in case of an odd n equality d ( n ) − d ( n ) + 3 d ( n ) − d ( n ) = d ( n ) holds because terms d ( n ) , d ( n ) , d ( n ) vanish.Assume n is even. Note that positive integer factorizations abc = n with even a areenumerated by d ( n ) . Indeed, they bijectively correspond to factorizations a bc = n .Same holds for factorizations abc = n with even b , and factorizations abc = n with even c . Similarly, factorizations abc = n with even a and b simultaneously are enumerated10y d ( n ) . The same holds for permuted a, b, c . Finally, factorizations abc = n witheven a, b, c are enumerated by d ( n ) . Applying inclusion-exclusion formula we get that d ( n ) − d ( n ) + 3 d ( n ) − d ( n ) enumerates factorizations abc = n , where all three a, b, c are odd. Since n is even such factorization is impossible, the above expression vanishes.Next lemma finally calculates |K | . Lemma 3. |K | = d ( n/ − d ( n/ − d ( n/
8) + 5 d ( n/ − d ( n/ . Remark.
The above formula for |K | looks horribly; actually it means the following.Let n = 2 q r where ∤ s . Then • if q = 0 then |K | = 0 ; • if q = 1 then |K | = d ( r ) ; • if q = 2 then |K | = 2 d ( r ) ; • if q > then |K | = 0 . Proof.
Let ∆ be a subgroup of even index n in π ( G ) , such that ∆ ∼ = π ( G ) and (∆ Λ ) y = ∆ Λ .Let Z ∆ = x k y s z t , where k is an odd positive, and k | n . We set H ∆ = ∆ T h y , z i .Further we identify h y , z i with Z , that is we address an element y a z b as ( a, b ) .Proposition 3 implies that the condition (∆ Λ ) y = ∆ Λ means that the groups ∆ and ∆ y have the same triples ( k, H, ¯ h ) of invariants. Consider two conditions: (i) the groups ∆ and ∆ y share the same invariant H , (ii) the groups ∆ and ∆ y share the same invariant ¯ h . Condition (i) means that ∆ y T h y , z i = ∆ T h y , z i , i.e. Ad y ( H ) = H . Since theaction of Ad y on h y , z i is given by ( u, v ) ( u, − v ) , Corollary 1 claims that either H = h ( a, , (0 , b ) i or H = h ( a, , ( a/ , b ) i , where a, b > and ab = n k ; additionally a is evenin the second case. We say that a subgroups H is of the first type if H = h ( a, , (0 , b ) i ,likewise H is of the second type if H = h ( a, , ( a/ , b ) i .Consider condition (ii). Note that ( Z ∆ ) y = ( x k y s z t ) y = x − k y s − z − t − . Thus theelement ( Z y ∆ ) − ∈ ∆ y satisfies the definition of the element Z ∆ y . So the correspondingvalue of ¯ h is ( s − , − t − . Then the condition (ii) is reformulated as ( s, t ) ∈ ( s − , − t −
1) + h H, (2 , , (0 , i , or equivalently (1 , ∈ h H, (2 , , (0 , i .In case H = h ( a, , (0 , b ) i this implies a and b are odd, thus n = kab is odd ( k is odddue to Proposition 2). Vice versa, in case n is odd, an arbitrary positive factorization n = kab spawns the unique group H = h ( a, , (0 , b ) i , that defines the unique coset Z / h H, (2 , , (0 , i , so we get the unique partial conjugacy class ∆ Λ with ∆ Λ = (∆ Λ ) y corresponding to each factorization n = kab . Thus there are d ( n ) conjugacy classes ofthe first type if n is odd. Then by Lemma 2 the first type provides d ( n ) − d ( n ) +3 d ( n ) − d ( n ) partial conjugacy classes in K .11n case H = h ( a, , ( a/ , b ) i condition (1 , ∈ h H, (2 , , (0 , i implies b and a are odd. Then | Z / h ( a, , ( a/ , b ) , (2 , , (0 , i| = 2 . Vice versa, if n is even but notdivisible by 4, then each factorization n = kab , where k, b are odd provides the uniquesubgroup H of the second type. In turn, each subgroup H provides two cosets ¯ h because | Z / h H, (2 , , (0 , i| = 2 . Thus if n is odd there are d ( n ) partial conjugacy classesof the second type in K , again by Lemma 2 this amount is equal to d ( n ) − d ( n ) +6 d ( n ) − d ( n ) .Summing up one gets |K | = d ( n/ − d ( n/ − d ( n/
8) + 5 d ( n/ − d ( n/ .Substituting the result of Lemma 3 into equation (4.4) we get c G , G ( n ) = 3( |K | + |K | |K | (cid:16) σ ( n σ ( n − σ ( n d ( n − d ( n − d ( n d ( n
16 ) − d ( n
32 ) (cid:17) . ∆ ∼ = π ( G ) We claim that the following two propositions holds.
Notation.
Given integers m, n with n > , by [ m ] n denote the integer number,defined by ≤ [ m ] n < n and m ≡ [ m ] n mod n . Proposition 6.
The subgroups ∆ of index n in π ( G ) isomorphic to π ( G ) are in one-to-one correspondence with the -plets ( k, ℓ, m, u, v, w ) , ≤ v < m , ≤ u < ℓ , ≤ w The conjugacy classes of subgroups ∆ of index n in π ( G ) isomorphicto π ( G ) are in one-to-one correspondence with the triples ( k, ℓ, m ) , where k, ℓ, m areodd positive integers and kℓm = n .Proof. By Proposition 6, a subgroup ∆ of the above type is defined by the -plet ( m, k, ℓ, u, v, w ) . Note that the conjugation with the elements x , y , z acts on theabove -plets in the following way: Ad x : ( k, ℓ, m, u, v, w ) ( k, ℓ, m, u, [ v − m , w ) ,Ad y : ( k, ℓ, m, u, v, w ) ( k, ℓ, m, u, v, [ w − k ) ,Ad z : ( k, ℓ, m, u, v, w ) ( k, ℓ, m, [ u − ℓ , v, w ) . ( k, ℓ, m ) are conjugated by a suitableelement of h x , y , z i . Obviously the conjugation with any element can not change thetriple ( k, ℓ, m ) . Corollary 3. s G , G ( n ) = n ( d ( n ) − d ( n d ( n − d ( n ,c G , G ( n ) = d ( n ) − d ( n d ( n − d ( n . Proof. To get the second formula use Lemma 2. To proceed to the first formula notethat for any triple ( m, k, ℓ ) there are m choices of v , k choices of w and ℓ choices of u .By the second formula there exist d ( n ) − d ( n ) + 3 d ( n ) − d ( n ) triples ( m, k, ℓ ) , eachcorresponds to exactly n different -plets ( m, k, ℓ, u, v, w ) . Appendix Given a sequence { f ( n ) } ∞ n =1 , the formal power series b f ( s ) = ∞ X n =1 f ( n ) n s is called a Dirichlet generating function for { f ( n ) } ∞ n =1 . To reconstruct the sequence f ( n ) from b f ( s ) one can use Perron’s formula ([1], Th. 11.17). Given sequences f ( n ) and g ( n ) we call their convolution ( f ∗ g )( n ) = P k | n f ( k ) g ( nk ) . In terms of Dirichlet generatingseries the convolution of sequences corresponds to the multiplication of generating series [ f ∗ g ( s ) = b f ( s ) b g ( s ) . For the above facts see, for example, ([1], Ch. 11–12).Here we present the Dirichlet generating functions for the sequences s H,G ( n ) and c H,G ( n ) . Since theorems 1–4 provide the explicit formulas, the remainder is done bydirect calculations.Consider the Riemann zeta function ζ ( s ) = ∞ X n =1 n s . Following [1] note that b σ ( s ) = ζ ( s ) , b σ ( s ) = ζ ( s ) ζ ( s − , b d ( s ) = ζ ( s ) , b ω ( s ) = ζ ( s ) ζ ( s − ζ ( s − . Table 2. Dirichlet generating functions for the sequences s H, G ( n ) and c H, G ( n ) . H s H, G c H, G π ( G ) 4 − s ζ ( s ) ζ ( s − ζ ( s − 2) 4 − s − ζ ( s ) ζ ( s − (cid:0) ζ ( s − 2) + 3(1 + 3 · − s ) ζ ( s ) (cid:1) π ( G ) 2 − s (cid:0) − − s (cid:1) ζ ( s ) ζ ( s − ζ ( s − 2) 3 · − s − (1 − − s ) ζ ( s ) (cid:0) (1 + 3 · − s ) ζ ( s − 1) +(1 − − s ) (1 + 2 − s +1 ) ζ ( s ) (cid:1) π ( G ) (cid:0) − − s +1 (cid:1) ζ ( s − (cid:0) − − s (cid:1) ζ ( s ) Appendix 2 The purpose of this section is to enumerate the normal subgroups ∆ of index n in π ( G ) ,such that ∆ ∼ = Z . In the notations of Section 4.1 the following holds. Proposition 8. The number of normal subgroups of index n in π ( G ) , isomorphic to Z is given by the formula |M | = d ( n/ 4) + 4 d ( n/ 8) + d ( n/ 16) + 2 d ( n/ . Proof. As it was shown above, a subgroup ∆ of described type is a subgroup of index n in Λ , so we use Proposition 1. The matrix a d f b e c determines a normal subgroup if andonly if (2 d, , ∈ h ( a, , i and (2 f, e, ∈ h ( a, , , ( d, b, i . Thus we have to findthe number of integer matrixes among the following eight: a b 00 0 c , a a/ b 00 0 c , a a/ b 00 0 c , a a/ a/ b 00 0 c , a b b/ 20 0 c , a a/ b b/ 20 0 c , a a/ a/ b b/ 20 0 c , a a/ a/ b b/ 20 0 c .The first matrix is always integer, that is appears d ( n ) times, once in each factorizationof the type abc = n . The next three matrices are integer if a is even, that is they appearin d ( n ) factorizations abc = n . Analogously the fifth matrix is integer if b is even, sothis matrix is counted d ( n ) times. The sixth matrix is integer if a and b are both even,it is counted d ( n ) times. The seventh and eighth matrices are integer if | a and b iseven, they are counted d ( n ) times. So |M | = d ( n/ 4) + 4 d ( n/ 8) + d ( n/ 16) + 2 d ( n/ . Remark. The respective Dirichlet generating function is − s (cid:0) · − s + 2 − s +2 · − s (cid:1) ζ ( s ) . Proposition 9. The number of normal subgroups ∆ of index n in π ( G ) isomorphic to π ( G ) equals to in case n is of the form m + 2 ; in case n is of the form m + 8 ; in all other cases.Proof. In this proof we follow notations and overall ideas of Section 4.2 (see first twoparagraphs). So it is sufficient to enumerate the subgroups ∆ in Γ x of the type consideredin Proposition 9. Proposition 2 claims that a subgroup ∆ of the above type is uniquelydefined by a triple ( k, H, h ) , while Proposition 3 describes the transformations of suchtriple under conjugation of the group ∆ with an element g ∈ Γ x . Similarly, the proofof Lemma 3 describes the transformation of triple ( k, H, h ) induced by the conjugationof the group ∆ by an element y . Since π ( G ) = Γ x S y Γ x , each conjugation can beachieved as a composition of described above.14ummarizing, we get that the invariant h is preserved by any conjugation if and only if (2 , , (0 , , (1 , ∈ H . This holds for just two subgroups H having index and in h y , z i ∼ = Z respectively. Both subgroups are normal in G , thus in both cases H isautomatically preserved by any conjugation. So, case k = n , where k is odd, providesone subgroup of Γ x which is normal in π ( G ) . Case k = n , where k is odd, providestwo subgroups of Γ x which are normal in π ( G ) . No other values of k provides normalsubgroups. Proposition 10. Any normal subgroup ∆ ✂ π ( G ) isomorphic to π ( G ) coincide withwhole π ( G ) . Actually this is shown in the proof of Proposition 7. References [1] T. Apostol, Introduction to Analytic Number Theory, Springer Science+BusinessMedia, New York, 1976.[2] M. I. Aroyo, A. Kirov, C. Capillas, J.M. Perez-Mato, H. Wondratschek,Bilbao Crystallographic Server II: Representations of crystallographic pointgroups and space groups, Acta Crystallogr., Sect. A62 (2006) 115–128, doi:10.1107/S0108767305040286.[3] G. Chelnokov, M. Deryagina, A. Mednykh, On the coverings of Euclidean manifolds B and B , Comm. Algebra 45 (4) (2017) 1558–1576.[4] G. Chelnokov, A. Mednykh, On the coverings of Euclidean manifolds G and G ,Comm. Algebra 48 (7) (2020) 2725–2739, doi:10.1080/00927872.2019.1705468.[5] G. Chelnokov, A. Mednykh, On the coverings of Euclidean manifolds G and G , J.Algebra