GROUP ALGEBRAS OF KLEINIAN TYPE AND GROUPS OF UNITS

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1 GROUP ALGEBRAS OF LEINIAN TYPE AND GROUPS OF UNITS GABRIELA OLTEANU AND ÁNGEL DEL RÍO Abstract. The algebras of leinian type are finite dimensional semisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of leinian groups. We classify the Schur algebras of leinian type and the group algebras of leinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such that the group of units of RG is virtually a direct product of free-by-free groups. The study of leinian groups goes back to the works of Poincaré [Poi] and Bianchi [Bia] and it has been an active field of research ever since. During the last decades, leinian groups have been strongly related to the Geometrization Program of Thurston for the classification of 3-manifolds [EGM, MR, Mas, Thu]. The use of the methods of leinian groups to the study of the groups of units of group rings was started in [PRR] and led to the notions of algebras of leinian type and finite groups of leinian type. Let be a number field, A a central simple -algebra and R an order in A. By order we mean always a Z-order. Let R 1 denote the group of units of R of reduced norm 1. Every embedding σ : C induces an embedding σ : A M d C, where d is the degree of A. Furthermore, σr 1 SL d C. We say that A is of leinian type if either A = or A is a quaternion algebra over and σr 1 is a discrete subgroup of SL C for some embedding σ of in C. More generally, an algebra of leinian type [PRR] is, by definition, a direct sum of simple algebras of leinian type. A finite group G is of leinian type if and only if the rational group algebra G is of leinian type. The finite groups of leinian type have been classified in [JPRRZ] where it has been also proved that a finite group G is of leinian type if and only if the group of units ZG of its integral group ring ZG is commensurable with a direct product of free-by-free groups. Recently, Alan Reid asked us in a private communication about the consequences of replacing the ring of rational integers by another ring of integers. This leads to the following two problems: Problem 1. Classify the group algebras of leinian type of finite groups over number fields. Problem. Given a group algebra of leinian type G, describe the structure of the group of units of the group ring RG for R an order in. The simple factors of G are Schur algebras over their centers. So, in order to solve Problem 1, it is natural to start classifying the Schur algebras of leinian type. This is obtained in Section 1. Using this classification and that of finite groups of leinian type given in [JPRRZ] we obtain the classification of the group algebras of leinian type in Section. In Section 3 we obtain a partial solution for Problem. 1. Schur algebras Throughout is a number field. We refer to the field homomorphisms R as real embeddings of. By abuse of notation, a pair of complex embeddings of is, by definition, a pair of conjugate field homomorphisms C whose images are not embedded in R. Recall that the infinite places of correspond to the real embeddings of and the pairs of complex embeddings of. Furthermore, the finite places of correspond to the prime ideals of the ring of integers of. A Schur algebra over is a central simple -algebra A which is generated over by a finite subgroup of the group of units A of A. Equivalently, a Schur algebra over is a simple factor, with center, of a group algebra of a finite group. If L/ is a finite cyclic extension of degree n, with GalL/ = σ and a, then L/, σ, a denotes the cyclic algebra L[u uxu 1 = σx, u n = a]. Sometimes we abbreviate L/, σ, a by writing L/, a, if 000 Mathematics Subject Classification. 16S34, 0C05, 16A6, 16U60, 11R7. ey words and phrases. Group algebras, leinian groups, groups of units. Research supported by M.E.C. of Romania CEEX-ET 47/006, D.G.I. of Spain and Fundación Séneca of Murcia. 1

2 GABRIELA OLTEANU AND ÁNGEL DEL RÍO the generator σ is either clear from the context or not relevant. Recall that L/, a is split if and only if a is a norm of the extension L/ [Rei, Theorem 30.4]. A cyclic cyclotomic algebra is a cyclic algebra L/, a with L/ a cyclotomic extension and a a root of unity. A cyclic cyclotomic algebra L/, a is a Schur algebra because it is generated over by the finite metacyclic group u, ζ, where ζ is a root of unity of L such that L = ζ. Conversely, every algebra generated by a finite metacyclic group is cyclic cyclotomic. uaternion algebras are cyclic algebras of degree and take the form a,b = [i, j i = a, j = b, ji = ij], for a, b. We abbreviate H = 1, 1. If A = a,b and σ is a real embedding of then A is said to ramify at σ if R σ A HR, or equivalently, if σa, σb < 0. Recall that a totally definite quaternion algebra is a quaternion algebra A over a totally real field which is ramified at every real embedding of its center. Let G be a finite group. A strong Shoda pair of G is a pair M, L of subgroups of G such that L M G, M/L is cyclic and M/L is maximal abelian in N G L/L. The definition in [ORS] is more general but for our purposes we do not need such a generality. If M = L and hence M = G, then let εm, M = M = 1 M m M m M; otherwise, let εm, L = L Ŝ, where S runs on the minimal subgroups of M containing L properly. Finally, let eg, M, L denote the sum of the different G-conjugates of εm, L. For every positive integer n, ζ n denotes a complex primitive n-th root of unity. We quote the following result from [ORS]. Proposition 1.1. Let G be a finite group and M, L a strong Shoda pair of G. Let N = N G L, k = [M : L] and n = [G : N]. Then e = eg, M, L is a primitive central idempotent of G and Ge is isomorphic with M n ζ k σ τ N/M, an n n-matrix algebra over a crossed product of N/M over the cyclotomic field ζ k, with defining action and twisting given as follows: Let x be a generator of M/L and let γ : N/M N/L be a left inverse of the natural epimorphism N/L N/M. Then, for every a, b N/M, one has ζ σa k = ζ i k, if x γa = x i and τa, b = ζ j k, if γab 1 γaγb = x j. In [ORS] it was proved that if G is a finite metabelian group, then every primitive central idempotent of G is of the form eg, M, L for some strong Shoda pair M, L of G. This can be used to compute the Wedderburn decomposition of G for G metabelian. A method to compute the Wedderburn decomposition of G for G an arbitrary finite group is given in [Olt]. It has been implemented in the GAP package wedderga [BOOR]. We will make use several times of this method at different places of the paper. We quote the following proposition from [JPRRZ]. Proposition 1.. The following statements are equivalent for a central simple algebra A over a number field. 1 A is of leinian type. A is either a number field or a quaternion algebra which is not ramified at at most one infinite place. 3 One of the following conditions holds: a A =. b A is a totally definite quaternion algebra. c A M. d A M d, for d a square-free negative integer. e A is a quaternion division algebra, is totally real and A ramifies at all but one real embeddings of. f A is a quaternion division algebra, has exactly one pair of complex non-real embeddings and A ramifies at all real embeddings of. We need the following lemmas. Lemma 1.3. If = d with d a square-free negative integer then 1 H is a division algebra if and only if d 1 mod 8. is a division algebra if and only if d 1 mod 3. Proof. 1 Writing H as ζ 4 /, 1, one has that H is a division algebra if and only if 1 is a sum of two squares in. It is well known that this is equivalent to d 1 mod 8 [FGS].

3 GROUP ALGEBRAS OF LEINIAN TYPE AND GROUPS OF UNITS 3 Assume first that A = is not split. Then A is ramified at at least two finite places p1 and p. Writing A as ζ 3 /, 1 and using [Rei, Theorem 14.1], one deduces that p 1 and p are divisors of 3. Thus 3 is totally ramified in and this implies that D 3 = 1, where D is the discriminant of [BS, Theorem 3.8.1]. Since D = d or D = 4d and p 3 p mod 3, for each rational prime p, one has d = d 3 = D 3 1 mod 3. Conversely, assume that d 1 mod 3. Then 3 is totally ramified in. Let p be a prime divisor of 3 in. Then the residue field of p has order 3 and p ζ 4 / p is the unique unramified extension of degree of p [Rei, Theorem 5.8]. Since v p 3 = 1, we deduce from [Rei, Theorem 14.1] that 3 is not a norm of the extension p ζ 4 / p. Thus p A = p ζ 4 / p, 3 is a division algebra, hence so is A. Lemma 1.4. Let D be a division quaternion Schur algebra over a number field. Then D is generated over by a metabelian subgroup of D. Proof. By means of contradiction we assume that D is not generated over by a metabelian group. Using Amitsur s classification of the finite subgroups of division rings see [Ami] or [SW] we deduce that D is generated by a group G which is isomorphic to one of the following three groups: O, the binary octahedral group of order 48; SL, 5, the binary icosahedral group of order 10; or SL, 3 M, where M is a metacyclic group. Recall that O = x, y, a, b x 4 = x y = x b = a 3 = 1, a b = a 1, x y = x 1, x b = y, x a = x 1 y, y a = x 1. We may assume without loss of generality that G is one of these three groups. Let D 1 be the rational subalgebra of D generated by G. It is enough to show that D 1 is generated over by a metabelian group. So we may assume that D is generated over by G, and so D is one of the factors of the Wedderburn decomposition of G. Computing the Wedderburn decomposition of O and SL, 5 and having in mind that D has degree we obtain that D ζ 8 /, 1, if G = O, and D ζ 5 / 5, 1, if G = SL, 5. In both cases D is generated over its center by a finite metacyclic group. Finally, assume that G = SL, 3 M, with M metacyclic. Then D is a simple factor of A 1 A, where A 1 is a simple epimorphic image of SL, 3 and A is a simple epimorphic image of M. Since A is generated by a metacyclic group, it is enough to show that so is A 1. This is clear if A 1 is commutative. Assume otherwise that A 1 is not commutative. Having in mind that D is a division quaternion algebra, one deduces that so is A 1 and, computing the Wedderburn decomposition of SL, 3, one obtains that A 1 is isomorphic to H. This finishes the proof because H is generated over by a quaternion group of order 8. For a positive integer n we set η n = ζ n + ζ 1 n and λ n = ζ n ζ 1 n. Observe that ηn λ n = 4 and hence ηn = λ n. Furthermore, if n 3, then λ n is totally negative because ζn i + ζn i 1, for every i Z prime with n. Therefore, if λ λ n then ramifies at every real embedding of. We are ready to classify the Schur algebras of leinian type. Theorem 1.5. Let be a number field and let A be a non-commutative central simple -algebra. Then A is a Schur algebra of leinian type if and only if A is isomorphic to one of the following algebras: 1 M, with = or d for d a square-free negative integer. H d, for d a square-free negative integer, such that d 1 mod 8. 3, for d a square-free negative integer, such that d 1 mod 3. 4, where n 3, η n and has at least one real embedding and at most one pair of 3, 1 d λ n, 1 complex non-real embeddings. Proof. That the algebras listed are of leinian type follows at once from Proposition 1.. Let be a field. Then M is an epimorphic image of D 8 and if λ λ n n then the algebra, 1 is an epimorphic ζ 4 image of n. This shows that the algebras listed are Schur algebras because H =, 1 and ζ =. 6, 1 n, 1

4 4 GABRIELA OLTEANU AND ÁNGEL DEL RÍO Now we prove that if A is a Schur algebra of leinian type then one of the cases 1-4 holds. If A is not a division algebra, Proposition 1. implies that A = M for = or an imaginary quadratic extension of, so 1 holds. In the remainder of the proof we assume that A is a division Schur algebra of leinian type. By Lemma 1.4, A is generated over by a finite metabelian group G. Then A = L B, where B is a simple epimorphic image of G with center L and, by Proposition 1.1, B is a cyclic cyclotomic algebra ζ n /L, ζn a of degree. Since A is of leinian type, so is B. Now we prove that L is totally real. Otherwise, since L is a Galois extension of, L is totally complex and therefore is also totally complex. By Proposition 1., both L and are imaginary quadratic extensions of and so L = and ϕn = 4, where ϕ is the Euler function. Then either a n = 8 and = ζ 4 or = ; or b n = 1 and = ζ 4 or = ζ 3. If n = 8, then B is generated over by a group of order 16 containing an element of order 8. Since B is a division algebra, G = 16 and so B = H, a contradiction. Thus n = 1 and hence B = ζ 1 /ζ d, ζd a, where d = 6 or 4. Since ζ 6 is a norm of the extension ζ 1 /ζ 6, necessarily d = 4. So A = B = ζ 1 /ζ 4, ζ a 4 = n, 1 L ζ a 4, 3 ζ 4. Since X = 1 + ζ 4, Y = ζ 4 is a solution of the equation ζ 4 X 3Y = 1, ζ 4 is a norm of the extension ζ 1 /ζ 4, and hence so is ζ4 a, yielding a contradiction. So L is a totally real field of index in ζ n. Then L = η n and necessarily B is isomorphic to λ λ ζ n /η n, 1. This implies that A. If has some embedding in R, then 4 holds. Otherwise = d, for some square-free negative integer d. This implies that L =. Then n = 3, 4 or 6 and so A is isomorphic to either H or. Since A is a division algebra, Lemma 1.3 implies that, in the first case, d 1 mod 8 and condition holds, and, in the second case, d 1 mod 3 and condition 3 holds. n, 1. Group algebras In this section we classify the group algebras of leinian type, that is the number fields and finite groups G such that G is of leinian type. The classification for = was given in [JPRRZ]. We start with some notation. The cyclic group of order n is usually denoted by C n. To emphasize that a C n is a generator of the group, we write C n either as a or a n. Recall that a group G is metabelian if G has an abelian normal subgroup N such that A = G/N is abelian. We simply denote this information as G = N : A. To give a concrete presentation of G we will write N and A as direct products of cyclic groups and give the necessary extra information on the relations between the generators. By x we denote the coset xn. For example, the dihedral group of order n and the quaternion group of order 4n can be given by D n = a n : b, b = 1, a b = a 1. 4n = a n : b, a b = a 1, b = a n. If N has a complement in G then A can be identify with this complement and we write G = N A. For example, the dihedral group can be also given by D n = a n b with a b = a 1 and the semidihedral groups of order n+ can be described as D + n+ = a n+1 b, a b = a n +1. D n+ = a n+1 b, a b = a n 1. Following the notation in [JPRRZ], for a finite group G, we denote by CG the set of isomorphism classes of noncommutative simple quotients of G. We generalize this notation and, for a semisimple group algebra G, we denote by CG the set of isomorphism classes of noncommutative simple quotients of G. For simplicity, we represent CG by listing a set of representatives of its elements. For example, using the isomorphisms D 16 = 4 M M and D + 16 = 4 i M i one deduces that CD + 16 = {M i} and CD 16 = {M, M }. The following groups play an important role in the classification of groups of leinian type.

5 GROUP ALGEBRAS OF LEINIAN TYPE AND GROUPS OF UNITS 5 W = t x y : x y, with t = y, x and ZW = x, y, t. W 1n = t i n y i x 4, with t i = y i, x and ZW 1n = t 1,..., t n, x. i=1 i=1 W n = y i 4 x 4, with t i = y i, x = yi and ZW n = t 1,..., t n, x. i=1 V = t x 4 y 4 : x y, with t = y, x and ZW = x, y, t. V 1n = t i n y i 4 x 8, with t i = y i, x and ZV 1n = t 1,..., t n, y1,..., yn, x. i=1 i=1 V n = y i 8 x 8, with t i = y i, x = yi 4 and ZV n = t i, x. i=1 U 1 = t ij 3 3 yk : y k, with t ij = y j, y i and 1 i<j 3 k=1 k=1 ZU 1 = t 1, t 13, t 3, y1, y, y3. U = 3 t 3 y1 y 4 y3 4 : y k, with t ij = y j, y i, y 4 = t 1, y3 4 = t 13 and k=1 ZU = t 1, t 13, t 3, y1, y, y3. T = t 4 y 8 : x, with t = y, x and x = t = x, t. T 1n = t i 4 n y i 4 x 8, with t i = y i, x, t i, x = t i and ZT 1n = t 1,..., t n, x. i=1 i=1 T n = y i 8 x 4, with t i = y i, x = y i and ZT n = t 1,..., t n, x. i=1 T 3n = y1t 1 y 1 8 n y i 4 : x, with t i = y i, x, t i, x = t i, x = t 1, i= ZT 3n = t 1, y,..., yn, x and, if i then t i = yi, S n,p, = C3 n P = C3 n : x, with a subgroup of index in P and z x = z 1 for each z C3 n. We collect the following lemmas from [JPRRZ]. Lemma.1. Let G be a finite group and A an abelian subgroup of G such that every subgroup of A is normal in G. Let H = {H H is a subgroup of A with A/H cyclic and G H}. Then CG = H H CG/H. Lemma.. Let A be a finite abelian group of exponent d and G an arbitrary group. 1 If d then CA G { = CG. } If d 4 and CG M, H,, M ζ 4 then CA G CG {M ζ 4 }. { } 3 If d 6 and CG M, H,, M ζ 3 then CA G CG {M ζ 3 }. Lemma.3. 1 CW 1n = {M }. CW = CW n = {M, H}. 3 CV, CV 1n, CV n, CU 1, CU, CT 1n {M, H, M ζ 4 }. 4 CT, CT n, CT 3n {M, H, M ζ 4, H, M }. 5 Let G = S n,p,. { } a If P = x is cyclic of order n then CG = CG/ x ζ n 1, 3 ζ n 1. In particular, if { } P = C then CG = {M }, if P = C 4 then CG = M, and if P = C 8 then { } CG = M,, M ζ 4. { } b If P = W 1n and = y 1,..., y n, t 1,..., t n, x then CG = M,, M ζ 3. c If P = W 1 and = y1, x then CG = {M, H 3, M ζ 4, M ζ 3 }. We are ready to present our classification of the group algebras of leinian type. Theorem.4. Let be a number field and G a finite group. Then G is of leinian type if and only if G is either abelian or an epimorphic image of A H, for A an abelian group, and one of the following conditions holds: 1 = and one of the following conditions holds.

6 6 GABRIELA OLTEANU AND ÁNGEL DEL RÍO a A has exponent 6 and H is either W, W 1n or W n, for some n, or H = S m,w1n, with = y 1,..., y m, t 1,..., t m, x, for some n and m. b A has exponent 4 and H is either U 1, U, V, V 1n, V n or S n,c8,c 4, for some n. c A has exponent and H is either T, T 1n, T n, T 3n or S n,w1, with = y 1, x, for some n. and has at most one pair of complex non-real embeddings, A has exponent and H = 8. 3 is an imaginary quadratic extension of, A has exponent and H is either W, W 1n, W n or S n,c4,c, for some n. 4 = ζ 3, A has exponent 6 and H is either W, W 1n or W n, for some n, or H = S m,w1n, with = y 1,..., y m, t 1,..., t m, x, for some n and m. 5 = ζ 4, A has exponent 4 and H is either U 1, U, V, V 1n, V n, T 1n or S n,c8,c 4, for some n. 6 =, A has exponent and H is either D 16 or T n, for some n. Proof. To avoid trivialities, we assume that G is non-abelian. The main theorem of [JPRRZ] states that G is of leinian type if and only if G is an epimorphic image of A H for A abelian and A and H satisfy one of the conditions a-c from 1. So, in the remainder of the proof, we assume that. First we prove that if one of the conditions -6 holds, then G is of leinian type. For that we compute CG and check that it is formed by algebras satisfying one of the conditions of Theorem 1.5. Since CG CG, for G an epimorphic image of G, it is enough to compute CG for G = A H and, A and H satisfying one of the conditions -6. We use repeatedly Lemmas. and.3 which provide an approximation of CG and CG = {ZA ZA A : A CG}. If holds, then CG = {H}. If 3 holds, then CG {M, H, }, and so CG {M, H, }. Similarly, if 4 holds then CG CH {M ζ 3 } {M, H,, M ζ 3 }. Hence CG = {M ζ 3 }, by Lemma 1.3. Arguing similarly one deduces that if 5 holds then CG = {M ζ 4 }. Finally, assume that 6 holds. If H = D16 then CG = {M, M } and so CG = {M }. Otherwise H = T n for some n. In this case we show that CG = {M }. For that, we need a better approximation of CG that the one given in Lemma.3. Namely, we show that CG = {M, H, M }. Let L be a proper subgroup of H the derived subgroup of H such that H /L is cyclic. Using that y, xy = 1, for each y y 1,..., y n, one has that T n /L is an epimorphic image of T 1 C n 1. Then Lemmas.1 and. imply that CG = CT 1. So we may assume that G = T 1. Now take B = ZT 1 = t, x C and L a subgroup of B such that B/L is cyclic. If t L, then T 1 /L is an epimorphic image of W and therefore CT 1 /L {M, H}. Otherwise L = x or L = x t ; hence T 1 /L D16 and so CT 1/L {M, M }. Using Lemma.1, one deduces that CT n = {H, M, M } as claimed. Conversely, assume that G is of leinian type and G is non-abelian and. Then G is of leinian type, that is G is an epimorphic image of A H for A and H satisfying one of the conditions a-c from 1. Furthermore, has at most one pair of complex embeddings, by Theorem 1.5. We have to show that, A and H satisfy one of the conditions -6. We consider several cases. Case 1. Every element of CG is a division algebra. This implies that G is Hamiltonian and so G 8 E F with E an elementary abelian -group and F abelian of odd order [Rob, 5.3.7]. If F = 1, then holds. Otherwise, CG contains Hζ n, where n is the exponent of F. Therefore n = 3 and = ζ 3, by Theorem 1.5. Since 8 is an epimorphic image of W, condition 4 holds. In the remainder of the proof we assume that CG contains a non-division algebra B. Then B = M L for some field L and therefore M L CG. Since, L is an imaginary quadratic extension of and L. Let E be the center of an element of CG. Then E is the center of an element of CG. If E, then the two complex embeddings of extends to more than two complex embeddings of E, yielding a contradiction. This shows that contains the center of each element of CG. Case. The center of each element of CG is. Then Lemmas. and.3 imply that CG {M, H, }. Using this and the main theorem of [LR] one has that G = A H, where A is an elementary abelian -group and H is an epimorphic image of W, W 1n, W n or S n,c4,c, for some n. So G satisfies 3.

7 GROUP ALGEBRAS OF LEINIAN TYPE AND GROUPS OF UNITS 7 Case 3. At least one element of CG has center different from. Then the center of each element of CG is either or. Using Lemmas. and.3 one has: If A and H satisfy condition 1.a then = ζ 3, hence 4 holds. If either A and H satisfy 1.b or they satisfy 1.c with H = T 1n, for some n, then = ζ 4 and condition 5 holds. Otherwise, A has exponent and H is either T, T n, T 3n, or S n,w1, y1,x, for some n. Since A has exponent, one may assume that G = A H 1, for H 1 an epimorphic image of H and H 1 is not an epimorphic image of any of the groups considered above. We use the standard bar notation for the images of H in H 1. Assume first that H = T. Then M = y, t, L = ty is a strong Shoda pair of T and, by using Proposition 1.1, one deduces that if e = et, M, L = L 1 y4, then Ge H. Since H is not of leinian type, we have that e = 0, or equivalently y 4 L. Hence either y 4 = 1, t = 1 or t = y. So H 1 is an epimorphic image of either T / y 4, T / t or T / ty. In the first case H 1 is an epimorphic image of T 11 and in the second case H 1 is an epimorphic image of V. In both cases we obtain a contradiction with the hypothesis that H 1 is not an epimorphic image of the groups considered above. Thus H 1 is an epimorphic image of T / ty D 16. In fact H 1 = D 16, because every proper non-abelian quotient of D 16 is an epimorphic image of W. Then CG = CD 16 = {M, M } and so =. Hence condition 6 holds. Assume now that H = T n. By the first part of the proof we have CH = {H, M, M }. Since we are assuming that one element of CG has center different from, then M CH 1 and so =. Hence condition 6 holds. In the two remaining cases we are going to obtain some contradiction. Suppose that H = T 3n. We may assume that n is the minimal positive integer such that G is an epimorphic image of T 3n A, for A an elementary abelian -group. This implies that t 1, t,..., t n is elementary abelian of order n and hence y 1, y,..., y n C n 4. Let M = y 1, y,..., y n and L 1 = t 1 y 1, y, y 3,..., y n. Then M, L 1 is a strong Shoda pair of H. By using Proposition 1.1, we obtain that HeH, M, L 1 H. This implies that eh, M, L 1 = 0, or equivalently t 1 = y4 1 L 1. Since t 1 has order and t 1 t,..., t n one has t 1 = t 1y1 tα tαn n for some α 1,..., α n {0, 1}. Since, by assumption, H 1 is not an epimorphic image of T n, we have α i 0 for some i. After changing generators one may assume that α = 1 and α i = 0 for i 3. Thus t 1 = y1 t. Let now L = t 1 y1, y y1, y 3,..., y n. Then M, L is also a strong Shoda pair of H and GeH, M, L H. The same argument shows that y 4 1 = t 1 L = t 1 y1, y y1, y 3,..., y n. This yields a contradiction because t 1 y1 = y y1 y y1, y 3,..., y n and y 14 y y1, y 3,..., y n. Finally assume that H = S n,w1,, with = y1, x and set y = y 1. Since, by assumption, G does not satisfy 1.a, one has t = t 1 1. Moreover, as in the previous case, one may assume that n is minimal for G to be a quotient of H A, with A elementary abelian -group. Let M = C3 n, x, t and L = Z 1, tx, where Z 1 is a maximal subgroup of Z = C3 n. Then M, L is a strong Shoda pair of H and HeH, M, L H 3. Thus 0 = eh, M, L = L1 ẑ1 t, where z Z \ Z 1. Comparing coefficients and using the fact that t z, for each z Z, we have L1 ẑ = 0, that is L = Z and this contradicts the minimality of n. 3. Groups of units In this section we study the virtual structure of RG, for G a finite group and R an order in a number field. More precisely, we characterize the finite groups G and number fields for which RG is finite, virtually abelian, virtually a direct product of free groups or virtually a direct product of free-by-free groups. We say that a group virtually satisfies a group theoretical condition if it has a subgroup of finite index satisfying the given condition. Notice that the virtual structure of RG does not depend on the order R and, in fact, if S is any order in G, then S and RG are commensurable see e.g. [Seh, Lemma 4.6]. Recall that two subgroups of a given group are said to be commensurable if their intersection has finite index in both. It is easy to show that a group commensurable with a free group respectively a free-by-free group, a direct product of free groups, a direct product of free-by-free groups is also virtually free respectively a free-by-free group, a direct product of free groups, a direct product of free-by-free groups. One important tool is the following lemma.

8 8 GABRIELA OLTEANU AND ÁNGEL DEL RÍO Lemma 3.1. Let A = n i=1 A i be a finite dimensional rational algebra with A i simple for every i. Let S be an order in A and S i an order in A i. 1 S is finite if and only if for each i, A i is either, an imaginary quadratic extension of or a totally definite quaternion algebra over. S is virtually abelian if and only if for each i, A i is either a number field or a totally definite quaternion algebra. 3 S is virtually a direct product of free groups if and only if for each i, A i is either a number field, a totally definite quaternion algebra or M. 4 S is virtually a direct product of free-by-free groups if and only if for each i, S 1 i is virtually free-byfree. Proof. We are going to use the following facts: a S is commensurable with n i=1 S i and Si is commensurable with ZS i Si 1. This is because S and n i=1 S i are both orders in A. b Si 1 is finite if and only if A i is either a field or a totally definite quaternion algebra see [le1] or [Seh, Lemma 1.3]. c If A i is neither a field nor a totally definite quaternion algebra and Si 1 is commensurable with a direct product of groups G 1 and G then either G 1 or G is finite [R]. d Si 1 is infinite and virtually free if and only if A i M see e.g. [le, page 33]. 1 By a, S is finite if and only if Si is finite for each i if and only if ZS i and Si 1 are finite for each i. By the Dirichlet Unit Theorem, if A i is a number field, then Si is finite if and only if A i is either or an imaginary quadratic extension of. Using this and b, one deduces that, if A i is not a number field then ZS i and Si 1 are finite if and only if A i is a totally definite quaternion algebra over. By a, S is virtually abelian if and only if Si 1 is virtually abelian for each i. If A i is either a number field or a totally definitive quaternion then Si 1 is finite, by b. Conversely, assume that Si 1 is virtually abelian. We argue by contradiction to show that A i is either a number field or a totally definite quaternion algebra. Otherwise Si 1 is virtually infinite cyclic by b and c and the fact that Si 1 is finitely generated. Then d implies that A i = M and so Si 1 is commensurable with SL Z. This yields a contradiction because SL Z is not virtually cyclic. 3 By a and [JR, Lemma 3.1], S is virtually a direct product of free groups if and only if so is Si 1 for each i. As in the previous proof, if A i is neither commutative nor a totally definite quaternion algebra, then Si 1 is virtually a direct product of free groups if and only if it is virtually free if and only if A i M. 4 Is proved in [JPRRZ, Theorem.1]. The characterization of when RG is finite respectively virtually abelian, virtually a direct product of free groups is an easy generalization of the corresponding result for integral group rings. Theorem 3.. Let R be an order in a number field and G a finite group. Then RG is finite if and only if one of the following conditions holds: 1 = and G is either abelian of exponent dividing 4 or 6, or isomorphic to 8 A, for A an elementary abelian -group. is an imaginary quadratic extension of and G is an elementary abelian -group. 3 = ζ 3 and G is abelian of exponent 3 or 6. 4 = ζ 4 and G is abelian of exponent 4. Proof. If =, then R = Z and it is well known that ZG is finite if and only if G is abelian of exponent dividing 4 or 6 or it is isomorphic to 8 A, for A an elementary abelian -group [Seh]. If one of the conditions 1-4 holds, then G is a direct product of copies of, imaginary quadratic extensions of and H. Then RG is finite by Lemma 3.1. Conversely, assume that RG is finite and. Then ZG is finite and therefore G is either abelian of exponent dividing 4 or 6 or isomorphic to 8 A, for A an elementary abelian -group. Moreover, R is finite and so = d for d a square-free negative integer. If the exponent of G is then holds. If the exponent of G is 3 or 6, then one of the simple components of G is d, ζ 3, hence d = 3, and therefore 3 holds. If the exponent of G is 4 then one of the simple components of G is d, ζ 4 and therefore d = 1, that is 4 holds.

9 GROUP ALGEBRAS OF LEINIAN TYPE AND GROUPS OF UNITS 9 Theorem 3.3. Let R be an order in a number field and G a finite group. Then RG is virtually abelian if and only if either G is abelian or is totally real and G 8 A, for A an elementary abelian -group. Proof. As in the proof of Theorem 3., the sufficient condition is a direct consequence of Lemma 3.1. Conversely, assume that RG is virtually abelian. Then ZG is virtually abelian and therefore it does not contain a non-abelian free group. Then G is either abelian or isomorphic to G 8 A, for A an elementary abelian -group [HP]. In the latter case, one of the simple components of G is H and hence is totally real, by Lemma 3.1. Theorem 3.4. Let R be an order in a number field and G a finite group. Then RG is virtually a direct product of free groups if and only if either G is abelian or one of the following conditions holds: 1 = and G H A, for A an elementary abelian -group and H is either W, W 1n, W 1n / x, W n, W n / x, W n / x t 1, T 3n or H = S n,ct,c t, for some n and t = 1, or 4. is totally real and G 8 A, for A an elementary abelian -group. Proof. The finite groups G such that ZG is virtually a direct product of free groups were classified in [JR] and are the abelian groups and those satisfying condition 1. So, in the remainder of the proof, one may assume that R Z, or equivalently, and we have to show that RG is virtually a direct product of free groups if and only if either G is abelian or holds. If either G is abelian or holds then RG is virtually abelian, hence RG is virtually a direct product of free groups, because it is finitely generated. Conversely, assume that RG is virtually a direct product of free groups and G is non-abelian. Since, M is not a simple quotient of G, hence Lemma 3.1 implies that every simple quotient of G is either a number field or a totally definite quaternion algebra. In particular, G is Hamiltonian, that is G = 8 A F, where A is an elementary abelian -group and F is abelian of odd order. If n is the exponent of F then Hζ n is a simple quotient of G and this implies that n = 1 and is totally real. Now we prove the main result of this section which provides a characterization of when RG is virtually a direct product of free-by-free groups. Theorem 3.5. Let R be an order in a number field and G a finite group. Then RG is virtually a direct product of free-by-free groups if and only if either G is abelian or one of the following conditions holds: 1 G is an epimorphic image of A H with A abelian and, A and H satisfy one of the conditions 1, 4, 5 or 6 of Theorem.4. is totally real and G A 8, for A an elementary abelian -group. 3 = d, for d a square-free negative integer, SL Z[ d] is virtually free-by-free, and G A H where A is an elementary abelian -group and one of the following conditions holds: a H is either W 1n, W 1n / x, W n / x or S n,c,1, for some n. b H is either W, W n or W n / x t 1, for some n and d 1 mod 8. c H = S n,c4,c for some n and d 1 mod 3. Proof. To avoid trivialities we assume that G is not abelian. We first show that if and G satisfy one of the listed conditions then RG is virtually a direct product of free-by-free groups. By Lemma 3.1, this is equivalent to show that for every B CG and S an any order in B, we have that S 1 is virtually free-by-free. By using Lemmas. and.3 and the computation of CG in the proof of Theorem.4, it is easy to show that if and G satisfy one of the conditions 1 or, then every element of CG is either a totally definite quaternion algebra or isomorphic to M for = d, with d = 0, 1, or 3. In the first case S 1 is finite and in the second case S 1 is virtually free-by-free see Lemma 3.1 and [JPRRZ, Lemma 3.1] or alternatively [le1], [MR, page 137] and [WZ]. If and G satisfy condition 3 then Lemmas 1.3,. and.3 imply that CG = {M d}. Since S 1 and SL Z[ d] are commensurable and, by assumption, the latter is virtually free-by-free, we have that S 1 is virtually free-by-free. Conversely, assume that RG is virtually a direct product of free-by-free groups. Let B be a simple factor of G and S an order in B. By Lemma 3.1, S 1 is virtually free-by-free and hence the virtual cohomological dimension of S 1 is at most. Then B is of leinian type by [JPRRZ, Corollary 3.4]. This proves that G is of leinian type. Furthermore, B is of one of the types a-f from Proposition 1.. However, the virtual

10 10 GABRIELA OLTEANU AND ÁNGEL DEL RÍO cohomological dimension of S 1 is 0, if B is of type a or b; 1 if B is of type c; if it is of type d or e; and 3 if B is of type f [JPRRZ, Remark 3.5]. Thus B is not of type f. Since every simple factor of G contains, either is totally real or is an imaginary quadratic extension of and G is split. By Theorem.4, G is an epimorphic image of A H with A abelian and, A and H satisfying one of the conditions 1-6 of Theorem.4. If they satisfy one of the conditions 1, 4, 5 or 6 of Theorem.4, then condition 1 of Theorem 3.5 holds. So, we assume that, A and H satisfy either condition or 3 of Theorem.4. Since A is an elementary abelian -group, one may assume that G = A H 1 with H 1 an epimorphic image of H. Assume first that, A and H satisfy condition of Theorem.4. Then one of the simple quotient of G is isomorphic to H. If is totally real then condition holds. Otherwise = d for d a square-free negative integer and H is split. By Lemma 1.3, d 1 mod 8. Since 8 W 1 / x t 1, condition 3b holds. Secondly assume that, A and H satisfy condition 3 of Theorem.4 and set = d for d a square-free negative integer. Then CG { H, M, }, by Lemma.3. By the main theorem of [JR], H 1 is isomorphic to either W, W 1n, W n, W 1n / x 1, W n / x 1, W n / x 1t 1, S n,c,1 or S n,c4,c, for some n. If H is either W 1n, W 1n / x 1, W n / x 1 or S n,c,1, then 3a holds. If H is either W, W n or W n / x t 1 then CG = {M, H} and, arguing as in the previous paragraph, one deduces that d 1 mod 8. In this case condition 3b holds. Finally, if G = S n,c4,c then CG = {M, the second part of Lemma 1.3 one deduces that d 1 mod 3, and condition 3c holds. } and using The main theorem of [JPRRZ] states that a finite group G is of leinian type if and only if ZG is commensurable with a direct product of free-by-free groups. One implication is still true when Z is replaced by an arbitrary order in a number field. This is a consequence of Theorems.4 and 3.5. Corollary 3.6. Let R be an order in a number field and G a finite group. If RG is commensurable with a direct product of free-by-free groups then G is of leinian type. It also follows from Theorems.4 and 3.5 that the converse of Corollary 3.6 fails. The group algebras G of leinian type for which the group of units of an order in G is not virtually a direct product of free-by-free groups occur under the following circumstances, where G = A H for A an elementary abelian -group: 1 is a number field of index 3 over with exactly one pair of complex embeddings and at least one real embedding and H = 8. = d, for d a square-free negative integer with d 1 mod 8 and H = W, W n or W n / x t 1, for some n. 3 = d for d a square-free negative integer with d 1 mod 3 and H = S n,c4,c. 4 = d and d and H satisfy one of the conditions 3a-3c from Theorem 3.5, but SL Z[ d] is not virtually free-by-free. Resuming, if G is of leinian type, then we have a good description of the virtual structure of RG for R an order in, except in the four cases above. It has been conjectured that SL Z[ d] is virtually free-by-cyclic for every negative integer. This conjecture has been verified for d = 1,, 3, 7 and 11 see [MR] and [WZ]. Thus, maybe the last case does not occur and the hypothesis of SL Z[ d] being virtually free-by-free in Theorem 3.5 is superfluous. In order to obtain information on the virtual structure of RG in the four cases 1-4 above, one should investigate the groups S, for S an order in the following algebras: H, for a number field with exactly one pair of complex embeddings and H is not split, H d with d 1 mod 8, with d d 1 mod 3 and, of course, M d for d a square-free negative integer. Notice that = 7 and G = 8 = W 11 / x t 1 is an instance of cases above and, if R is an order in, then R 8 is commensurable with HR. A presentation of HR, for R the ring of integers of 7 has been computed in [CJLR]. References [Ami] S.A. Amitsur, Finite subgroups of division rings, Trans. Amer. Math. Soc , [Bia] L. Bianchi, Sui gruppi dei sustituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari, Math. Ann

11 GROUP ALGEBRAS OF LEINIAN TYPE AND GROUPS OF UNITS 11 [BS] Z.I. Borevich and I.R. Shafarevich, Number Theory, Academic Press, [BOOR] O. Broche Cristo, A. onovalov, A. Olivieri, G. Olteanu and Á. del Río, Wedderga Wedderburn Decomposition of Group Algebras, Version 4.0; [CJLR] C. Corrales, E. Jespers, G. Leal and Á. del Río, Presentations of the unit group of an order in a non-split quaternion algebra, Adv. Math , no., [EGM] J. Elstrodt, F. Grunewald and J. Mennicke, Groups Acting on Hyperbolic Space, Harmonic Annalysis and Number Theory, Springer-Verlag, Berlin, [FGS] B. Fein, B. Gordon and J.H. Smith, On the representation of 1 as a sum of two squares in an algebraic number field J. Number Theory , [HP] B. Hartley and P.F. Pickel, Free subgroups in the unit group of integral group rings, Canad. J. Math , [JPRRZ] E. Jespers, A. Pita, Á. del Río, M. Ruiz and P. Zalesski, Groups of units of integral group rings commensurable with direct products of free-by-free groups, Adv. Math. 007, doi: /j.aim [JR] E. Jespers and Á. del Río, A structure theorem for the unit group of the integral group ring of some finite groups, J. Reine Angew. Math , [le1] E. leinert, A theorem on units of integral group rings, J. Pure Appl. Algebra , no. 1-, [le] E. leinert, Units of classical orders: a survey, L Enseignement Mathématique , [R] E. leinert and Á. del Río, On the indecomposability of unit groups, Abh. Math. Sem. Univ. Hamburg , [LR] G. Leal and Á. del Río, Products of free groups in the unit group of integral group rings II, J. Algebra , [MR] C. Maclachan and A. W. Reid, The Arithmetic of Hyperbolic 3-Manifolds, Springer, New York, 00. [Mas] B. Maskit, leinian groups, Springer-Verlag, Berlin, [ORS] A. Olivieri, Á. del Río and J.J. Simón, On monomial characters and central idempotents of rational group algebras, Comm. Algebra 3 004, no. 4, [Olt] G. Olteanu, Computing the Wedderburn decomposition of group algebras by the Brauer-Witt theorem, Math. Comp , [PRR] A. Pita, Á. del Río and M. Ruiz, Groups of units of integral group rings of leinian type, Trans. Amer. Math. Soc , [Poi] H. Poincaré, Mémoires sur les groups kleinéens, Acta Math [Rei] I. Reiner, Maximal orders, Academic Press 1975, reprinted by LMS 003. [Rob] D.J.S. Robinson, A course in the theory of groups, Springer-Verlag, New York-Berlin, 198. [Seh] S.. Sehgal, Units in integral group rings, Longman Scientific and Technical Essex, [SW] M. Shirvani and B.A.F. Wehrfritz, Skew Linear Groups, Cambridge University Press, [Thu] W. Thurston, The geometry and topology of 3-manifolds, Lecture notes, Princeton, [WZ] J.S. Wilson and P.A. Zalesski, Conjugacy separability of certain Bianchi groups and HNN-extensions, Math. Proc. Cambridge Philos. Soc , 7 4. Department of Mathematics and Computer Science, North University of Baia Mare, Victoriei 76, 4301 Baia Mare, Romania. olteanu@math.ubbcluj.ro Current address: Department of Mathematics, University of Murcia, Murcia, Spain. golteanu@um.es Department of Mathematics, University of Murcia, Murcia, Spain. adelrio@um.es