On Quasivarities of Groups and Equations over Groups

We prove that the quasivariety of groups generated by finite and locally indicable groups does not contain the class of periodic groups. This result is related to (and inspired by) the solvability of equations over groups. The proof uses the Feit-Thompson theorem on the solvability of finite groups of odd order, Kostrikin-Zelmanov results on the restricted Burnside problem and applies techninal details of a recent construction of weakly finitely presented periodic groups. Let X = {x1, x2, . . . } be a countably infinite alphabet and U1, . . . , Un, V be words in X = X ∪ X (called X-words). A quasiidentity is an expression of the form (U1 = 1 ∧ · · · ∧ Um = 1) ⇒ V = 1, where ∧ and ⇒ are the signs of conjunction and implication, respectively. A quasiidentity holds in a group G if it is a true formula for any substitution gi → xi, where gi ∈ G, i = 1, 2 . . . . A quasivariety of groups is the class of groups defined by a set of quasiidentities; that is, the class of all groups in which every quasiidentity of a given set holds (see [23] for more details). For example, the quasiidentity x = 1 → x = 1 defines the class of groups without involutions. Recall that an equation over a group G with an unknown y is an expression of the form w(y,G) = yg1 . . . y gl = 1, (1) where ε1, . . . , εl ∈ {±1} and g1, . . . , gl ∈ G. The elements g1, . . . , gl of G are called the coefficients and l is the length of equation (1). The equation (1) is called solvable if there is a group H that contains G as its subgroup and there is an element h ∈ H so that the substitution h → y in (1) results in the equality w(h,G) = 1 in H . Clearly, the equation (1) is solvable if and only if G naturally embeds in the quotient of the free product 〈y〉∞ ∗G by relation (1), where 〈y〉∞ is the infinite cyclic group generated by y. The equation (1) is called nonsingular if ε1 + · · ·+ εl 6= 0 (otherwise, it is singular). It is a well-known conjecture of Kervaire and Laudenbach that every nonsingular equation is solvable over any group G. We will say that the equation (1) is strongly solvable over a group G if it is solvable for an arbitrary l-tuple (g1, . . . , gl) of elements of G. Clearly, the Kervaire-Laudenbach conjecture is equivalent to its strong version, that is, the Kervaire-Laudenbach conjecture holds if and only if every nonsingular equation is strongly solvable over any group G. It is also easy to see that if the equation (1) is strongly solvable over a nontrivial group G then this equation is nonsingular. 2000 Mathematics Subject Classification. Primary 20E06, 20F05, 20F06, 20F50. Supported in part by NSF grants DMS 98-01500, DMS 00-99612.