We study the solvability of equations over groups within a given variety or another class of groups. The classes of nilpotent and solvable groups were considered as main classes to investigate from such point of view. The natural analogues of the famous Kervaire - Laudenbach and Levin conjectures were raised to the challenge. It was also noted that the "solvable" version of the known theorem by Brodskii is not true. In this paper, for each n Е N, n ^ 2, we prove that every regular equation over the free metabelian group M_{n} is solvable in the class M of all metabelian groups. Moreover, there is a metabelian group M_{n} that contains a solution of every unimodular equation over M_{n}. These results are extended to the class of rigid metabelian groups. Also, we give an example showing that there exists an equation over a locally indicable torsion-free metabelian group G that has no solution in any solvable overgroup of G. It follows that solvable versions of the Levin conjecture are not true. Another example presents an unimodular equation over a locally indicable torsion-free metabelian group G that has no solution in any metabelian overgroup of G. Hence, the Kervaire - Laudenbach conjecture is not valid for the variety of all metabelian groups. We prove that there is an unimodular equation over a finite metabelian group G that has no solutions in any finite metabelian overgroup of G. This means that analog of the famous theorem by Gerstenhaber and Rothaus (about solvability of each unimodular equation over a finite group G in some finite overgroup of G) is not valid for the class of finite metabelian groups.

On solvability of regular equations in the variety of metabelian groups.pdf Introduction The purpose of this note is to study solvability of equations over groups in varieties or other classes of groups, emphasizing the variety of metabelian groups. The process of solving equations is central to much of mathematics. In general setting, there are two questions to answer when presented with an equation. The first question is about existing of a solution, the second one is about finding a solution if it exists. Equations in groups are the old and well-established area of group theory. For instance, see the surveys [1-4]. An equation in k variables over a group G is an expression of the form (1) u(x1,..., xk) = 1, where u(x1,..., xk) = goж^1 g1... gn-1x-; Е F(X) * G. Here, each gj is a group element, each exponent Sj is 1 or -1, each Xj. is taken from an alphabet of variables X = {x1,..., xk,...}, and F(X) denotes the free group with basis X. xThis research was supported by Russian Science Foundation (project 16-11-10002). For brevity, we often use the word "equation" to refer a word u(x1,...,xk), i.e., the left-hand side of an equation of the form (1). Also, an equation can be written in the form u(x1,..., uk) = v(x1,..., xk), where u(x1,..., xk), v(x1,..., xk) E F(X) * G. This equation is equivalent to u(x1,..., xk)v(x1,..., xk)-1 = 1. By the preceding convention, the free product GX = F(X) * G is the space of all equations with variables in X and coefficients in G. We assume that n ^ 1 and if j = iJ+1 and + = 0, then gj = 1. Also, we assume that if g0 = 1 and i1 = in, then e1 + en = 0. If not, we can conjugate the both sides of (1) by x-£1 to get an equivalent equation with a shorter length. We exclude the case that u is conjugate to an element of G. The gj's are called the coefficients or constants of the equation and n is called the length of the equation. For every positive integer k, let Xk = {x1,..., xk} С X. The free product GXk = F(Xk) * G is the space of all equations over G in k variables. A solution of (1) in G is an assignment xj i-У hj E G such that #oh-! g1.. .gn-1h-;n = 1 Let u = u(x1,... ,xk) E GX and let U denote the normal closure of u in the group GX. We say that (the equation) u = 1 has a solution (equivalently, is solvable) over the group G if there is an overgroup H of G such that u = 1 has a solution in H. In this case, we say that u =1 has a solution or u =1 is solvable. Clearly, u =1 has a solution if and only if the canonical map from G to the group H = GX/U is an embedding. An one-variable equation of the form g0x£!g1... gn-1x£n = 1 of length n with the n exponent sum (exponent) e = E £ is called regular or non-singular if e = 0, unimodular if i=1 e =1, and singular if e = 0. Similarly, we call the underlined word u = g0x£!g1... gn-1x£n regular, unimodular or singular, respectively. Not every singular equation is solvable. If g,f E G and |g| = |f | (by |v| we denote the order of element v), then the equation gxfx-1 = 1 is obviously non-solvable. For any word w = w(x1,..., xk) E GX and for each j = 1,..., k, we denote the exponent sum of xj in w by expj (w). Let {u1 = 1,...,um = 1} (2) be a system of equations in variables x1,...,xk. Let = expj(u^) for i = 1,...,m; j = 1,..., k. The system (2) is called independent if the matrix (a^) has rank m. There are the following three major results on solvability of equations in groups. Theorem 1 (M. Gerstenhaber, O.S. Rothaus, 1962 [5]). Every independent system of equations over a compact connected Lie group has a solution. It follows that every independent system of equations over a finite group or, more generally, over a locally residually finite group [6] has a solution. In particular, every regular equation over a finite or, more generally, locally residually finite group has a solution. Theorem 2 (A. A. Klyachko, 1993 [7]). If G is a torsion-free group, then every unimodular equation in one variable over G is solvable. Theorem 3. (S. D. Brodskii, 1980 [8], 1984 [9]; J.Howie, 1982 [10]; S. M. Gersten, 1985 [11]; S. Krstic, 1985 [12]). If G is locally indicable, then every independent system of equations over G is solvable. Moreover, if G is locally p-indicable for some prime number p, then every p-independent system of equations over G is solvable. Recall that G is locally indicable if every finitely generated subgroup of G maps onto the infinite cyclic group C, and locally p-indicable (p is a prime) if every finitely generated subgroup of G maps onto the cyclic group Cp of order p. A system of m equations is called p-independent if the corresponding matrix (aio-) has rank m mod p. S. D. Brodskii proved the locally indicable statement in the case of a single equation. In fact, he proved that quotient G*H/ncl(w) of a free product G*H of two non-trivial locally indicable groups by a normal closure ncl(w) of element w G G * H, which is not conjugate to element of GUH, contains the natural images of the factors. Independently, J. Howie proved the locally indicable statement for arbitrary independent system. S. M. Gersten established the locally p-indicable version, and S. Krstic gave an alternative proof of the locally p-indicable statement. Not every singular equation is solvable. If g, f G G and |g| = |f |, then the equation gxfx-1 = 1 is obviously non-solvable. Since not every equation is solvable over an arbitrary group, some restrictions on group or equation are required. The best known restrictions are the Kervaire - Laudenbach conjecture and the Levin conjecture. Conjecture 1 (Kervaire - Laudenbach conjecture). Every regular equation in one variable over an arbitrary group is solvable. Conjecture 2 (Levin conjecture). Every equation over arbitrary torsion-free group is solvable. In general, these two conjectures are still open. At this point, we need to introduce the following notation: [g, f] means the commutator gfg-1f-1, gf stands for conjugate gfg-1, G' means the derived subgroup (commutant) of group G, Z is the ring of integers, N is the set of positive integers, Q denotes the field of rationals. By Z[G], we denote the group ring of group G. Recall that a group G is said to be metabelian if there exists a short exact sequence A м G ^ B of groups with abelian A and B. In particular, one can take A = G' and B = G/G'. Hence, metabelian groups are precisely the solvable groups of derived length at least two. We think of A (in particular, G') as a Z[B]-module via conjugation; that is, we define a о q = gag-1, where g м q under the standard homomorphism G ^ B. Since A is abelian, this is well-defined. 1. Solvability of equations over a group in a given class of groups In [4], a new direction was initiated to study solvability of equations over groups within a given variety or another class of groups. The classes of nilpotent and solvable groups were considered as main classes to investigate from such point of view. The natural analogues of the Kervaire - Laudenbach and Levin conjectures were raised to the challenge. When we study the solvability of equations over a group G in a given class C of groups, there is an important issue -what is the proper space of equations for a given group G? The free product GX seems suitable if the class C coincides with the class G of all groups. On the other hand, if C is the class A of abelian groups (and G is an abelian group), then it is natural to consider an equation g1x-1g2x2x1 = 1 as essentially the same as g1g2x2 = 1. More generally, it is natural to consider two equations as essentially the same if one can be transformed into the other by applying identities of the variety of abelian groups. So the natural space of equations in variables X over an abelian group G, when we restrict ourselves by the class A, is the direct product G x A(X) of G and a free abelian group A(X) with basis X. Similarly, if G is in a variety V, and we study solvability of equations over G in V, then the space of equations in variables X over G relative to V is the free product in the variety V of G and a V-free group with basis X. Following this approach to a metabelian group G, we define the space of metabelian equations as the set of elements of a free product GFm(X), where M is the variety of all metabelian groups, is the free product in M, and Fm(X) is a free metabelian group with basis X. Let Mn denote the free metabelian group of rank n. 2. Solvability of equations over a group in the variety M of metabelian groups: negative results We are interesting in solvability of equations over metabelian groups in the class M. We say that the equation u = 1 has a solution (equivalently, is solvable) over the metabelian group G in the class M if there is an overgroup H G M of G such that u = 1 has a solution in H. It has been noted in [4] that the "solvable" version of theorem 3 is not true. Indeed, an equation of the form xgx-1 = [g,hgh-1], (3) where g and h are elements of a torsion-free solvable group G, for which [g, hgh-1] = 1 (such elements exist in every non-abelian solvable group), has no solution in any solvable overgroup of G. It also follows that solvable versions of the Levin conjecture are not true. Hence, these conjectures are not true in the class M. Indeed, let the equation (3) has a solution in a solvable group H of the derived length r and H ^ G. Suppose g G H(k), where H(k) denotes the k-th member of the derived series of H. Then the right side of (3) belongs to the (k + 1)-th member H(fc+1) = [H(k), H(k)] of the derived series of H. By continuing in this way, we get g G H(r) = {1}. This contradicts the our assumption. The following example shows that there is not only regular even unimodular equation that is not solvable over some torsion-free metabelian group in the class M. Example 1. Let G be subgroup of group GL2(Q) generated by two matrices: A=(0 1) andB=(-02 2 Then the following unimodular equation is not solvable in every metabelian group H ^ G: x2Bx-1B-1 = [A,B ]. (4) ( 1 2 ) Proof. Note that [A, B] = ( 0 1 ) . Let x = h is a solution of the equation (4) in a metabelian group H ^ G. Then h = [A,B][h,B]-1 G H/. We consider H/ as a module over group ring Z[H]. Group H acts on H/ by conjugation. As H/ acts trivially, we get a module H/ over group ring Z[H/H/]. We denote the image of f G H in H/H/ by f. For each f G H/H/ and each u G H/, we write uf as a result of action of f on u. The equality corresponding to (4) can be written in the form h2-B = [A, B]. Then h(2-B)(2+B) = h4-B2 = 1, (5) because B2 = 4E (E is identical matrix). Hence, the action of 2 + B on the left-hand side of (5) gives trivial element as a result. But direct computation gives the following result of the action of 2 + B on the right-hand side of equality (4): [A,B]2+B =(1 1)=[A,B]. (6) In view of (5), the nontriviality of the right-hand side of (6) contradicts our assumption about solvability of (4) in H. ■ The group G in Example 1 is obviosly torsion-free. Moreover, it is locally indicable. The equation (4) is unimodular. Thus, the statement of Example 1 implies that both Theorems 2 and 3 are not true in the class M of metabelian groups. Moreover, it is easy to modify Example 1 considering subgroup Gp of GL2(Zp) generated by matrices A and B over the modular ring (finite prime field) Zp (p is an odd prime) and to prove that the equation (4) is unsolvable in the class Mfin of all finite metabelian groups. The argument is practically the same as in the proof above. This shows that Theorem 1 is wrong in Mfin. 3. Solvability of equations over a group in the variety M of metabelian groups: positive results Let Mn be the free metabelian group of rank n with basis |/1,...,/raj and An = = Мп/МП. Then An is a free abelian group with the basis {a1,..., an} corresponding to the basis {/1,..., /n}, that is, / м- aj under the standard homomorphism ^ : Mn ^ An, i = 1,..., n. For brevity, we denote g = ^(g) for each g Е Mn. Thus, a» = /, i = 1,..., n. Now, we apply the partial Fox derivatives and the Magnus embedding. For the basic notions and properties of these tools see [13]. The partial Fox derivatives on the group ring Z[Mn] may be defined as the mappings d/d/j : Z[Mra] м Z[Ara] for 1 ^ j ^ n, satisfying the following conditions for all а, в Е Z, u,v Е Z[Mn], and g,h Е Mn: d/j/d/j = ^jj (where is the Kronecker delta), S(au + ev)/d/j = adu/d/j + edv/S/j, d(gh)/d/j = dg/d/j + g ■ dh/d/j. For any n Е N, n ^ 2, the free metabelian group Mn can be treated via the Magnus embedding as a subgroup of the wreath product W = AraWrAra, where An is the free abelian group of rank n. In linear representation, the group W is the group of matrices of the form n a E ujtj j=1 01 where {t1,...,tn} is a basis of the free module T over Z[An], a Е An, uj Е Z[An] for j = 1,..., n. Hence, W = | An S Z[An] ■ ij The group Mn is embedded into W by the following map, where g Е Mn and ^ : Mn ^ An is the standard homomorphism, n , . »(g) E dg/dfj • tj v: g ^ I j=1 0 1 In particular, v(fi) = ^ ^ 1 ^ for i = 1,... ,n. The group ring Z[An] is the ring of Laurent polynomials Z[a±1,..., a^1]. This ring is an integral domain; so, we can consider the field of fractions Frac(Z[An]) of Z[An]. Recall that the fraction u/v denotes the equivalence class of pairs (u,v), where (u, v) is equivalent to (r, s) if and only if us = rv. The field of fractions Frac(Z[An]) is defined as the set of all such fractions u/v. The sum of u/v and r/s is defined as us + rv/vs, and the product of u/v and r/s is defined as ur/vs. The embedding of Z[An] into Frac(Z[An]) maps each u Е Z[An] to the fraction u/1. The wreath product W (thus, the free metabelian group Mn) can be considered as subgroup of the following metabelian group: n WV= | An E Frac(Z[An])ti 01 Now, we are ready to prove the following theorem. Theorem 4. For each n Е N, n ^ 2, every regular equation over the free metabelian group Mn is solvable in the class M of all metabelian groups. Moreover, there is a metabelian group Mn that contains a solution of every unimodular equation over Mn. Proof. Let go • x£1 • g1 • ... • gn-1 • x£" = 1 (7) be a regular equation over Mn with exponent e = 0. This equation induces equation of the form x£g = 1 over the free abelian group An = Mn/M'n. Here, g is the image of g = = gog1... gn-1 in An. Suppose that e = 1, i. e., the equation (7) is unimodular. In this case, we set x = g-1y, where y is a new variable. The new induced equation over An is expressed as y = 1. Then (7) is equivalent to an equation of the form go • (g-1y)£1 • g1 •... • gn-1 • (g-1y)£n = 1. (8) The equation (8) has a solution y in the group IV such that y belongs to the abelian i i tV / 1 E Frac(Z[An])ti \ T , , normal subgroup V1 =1 i=1 I. Indeed, we consider this abelian normal 01 subgroup as a module via conjugations over group ring Z[An] as we did before. Then (8) can be rewritten in the form yu = v, where u Е Z[An], v Е M'n. Since Mn ^ W1 and W1 is a vector space, it will be enough to prove that u = 0. We can fc+1 k take y = vu and succeed. Since (8) is unimodular, v = E fi - E hj for some k Е N i=1 j=1 and /i,hj € An. Let 5 : Z[An] м Z be а trivialization homomorphism that sends every group element a € to 1. Then 5(v) - 1 and v - 0. We proved that every unimodular equation (7) is solvable in metabelian group Mn. Suppose that e > 1, i.e., the equation (7) is regular, but not unimodular. We deal with Mn as a subgroup of other group ~ Mn with the basis {h1,..., hn} via embedding corresponding to map / м- hf for i - 1,..., n. This embedding is well-defined. Indeed, a subgroup of a free metabelian group is free if and only if it is generated by a set of elements, which are independent modulo the derived group [14]. Thus, a subgroup of Mn generated by some k independent modulo MП elements is isomorphic to Mk with the basis consisting of k given elements. Denote - Mra/M^ and Bn - . Then is a free abelian subgroup of Bn, ~ Bn, with the basis {b1,..., bn} corresponding to {h1,..., hn}. As above, (7) induces an equation of the form xfg - 1 over the free abelian group An, or, more generally, over Bn. There is a unique solution x - g of this induced equation. The rest of the proof is practically the same as the proof above in the case of unimodular equation. The group WW in this proof is constructed for bigger free metabelian group Mn. There is a k+f k small difference in the last step in these two proofs. Now, v - E / - E hj. Then 5(v) - e. i=1 j=1 But we have v - 0, and succeed again. ■ Remark 1. We see from the proof that in the case of regular and not unimodular equation (7), this equation is solvable in metabelian overgroup isomorphic to W. The difference with the unimodular case is in embedding of Mn into WW. This embedding depends on e. We can now give a generalization of Theorem 4 to the class of rigid metabelian groups. Recall that a group G is called m-rigid if it contains a finite series of normal subgroups G > G1 > ... > Gm+1 - 1, with abelian quotients Gi/Gi+1 (thus G is solvable) such that each quotient Gi/Gi+1 considered as a module over Z[G/G*] via conjugation has no module torsion, i.e., for each nontrivial g € Gi/Gi+1 and for every nontrivial v € Z[G/G*], it follows - 1. Such a series (if it exists) is uniquely defined by the group G that is solvable of the derived length exactly m [15]. A group G is called rigid if it is m-rigid for some m. If G is metabelian, then m - 2. In the class of all rigid groups, we distinguish divisible groups G with quotients Gi/Gi+1, whose elements are divisible by any nontrivial element of the respective group ring Z[G/G*], i.e., for each g € Gi/Gi+1 and for every nontrivial v € Z[G/G*], there exists h € Gi/Gi+1 such that hv - g. It has been shown in [16] that every m-rigid group can be embedded into divisible m-rigid group. In particular, every metabelian rigid group can be embedded into metabelian divisible rigid group. It has been proved in [16] that there exists a minimal divisible m-rigid completion G of the given m-rigid group G preserving linear independence, and G is uniquely defined up to G-isomorphism. Theorem 5. Every regular equation over a rigid metabelian group M is solvable in the class M of all metabelian groups. Moreover, every regular equation over a rigid metabelian group M has a solution in the divisible rigid completion M. A proof of Theorem 5 is practically such as the proof of Theorem 4.

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