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Abstract. The aim of the present note is to show that free Burnside groups of sufficiently large odd exponent are non–amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating sets. This result is applied to the solution of a strong version of the von Neumann – Day problem concerning amenability of groups without non–abelian free subgroups. As another consequence, we obtain that the above–mentioned groups are of uniform exponential growth. This answers a question of de la Harpe [12]. 1.

. The Burnside problem about periodic groups asks whether any finitely generated group with the law x n j 1 is necessarily finite. This is proven only for n 4 and n = 6. A negative solution to the Burnside problem for odd n AE 1 was given by Novikov and Adian. The article presents a discussion of a recent solution of the Burnside problem for even exponents n AE 1 and related results. 1991 Mathematics Subject Classification: Primary 20F05, 20F06, 20F10, 20F50 Recall that the notorious Burnside problem about periodic groups (posed in 1902, see [B]) asks whether any finitely generated group that satisfies the law x n j 1 (n is a fixed positive integer called the exponent of G) is necessarily finite. A positive solution to this problem is obtained only for n 4 and n = 6. Note the case n 2 is obvious, the case n = 3 is due to Burnside [B], n = 4 is due to Sanov [S], and n = 6 to M. Hall [Hl] (see also [MKS]). A negative solution to the Burnside problem for odd exponents was g...

We study existentially closed CSA-groups. We prove that existentially closed CSA-groups without involutions are simple and divisible, and that their maximal abelian subgroups are conjugate. We also prove that every countable CSA-group without involutions embeds into a finitely generated one having the same maximal abelian subgroups, except maybe the infinite cyclic ones. We deduce from this that there exist 2 ℵ0 countable existentially closed CSA-groups without involutions and that their firstorder theories have 2 ℵ0 types over ∅. 1

Abstract. To solve a number of problems on varieties of groups, stated by Kleiman, Kuznetsov, Ol’shanskii, Shmel’kin in the 1970’s and 1980’s, we construct continuously many varieties of groups in which all periodic groups are abelian and whose pairwise intersections are the variety of all abelian groups. 1.

Abstract. 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 ±1 = X ∪ X −1 (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