On abelian subgroups of relatively free groups

Infinite cyclic verbal subgroups of relatively free groups

“…He repeated this question later in [3]. It is interesting to note that the construction described in [5] shows that there exists a word u(x, y) and a relatively free group K such that u(K) is infinite cyclic and all the elements of u(K) are values of the word u(x, y) in K. In this paper we give the negative answer to the question raised by S. Ivanov.…”

“…. , introduced in [5], namely w 1 (x, y). Therefore, when studying the group w(F 2 ) and groups associated with it, we can use lemmas proved in [5].…”

“…, introduced in [5], namely w 1 (x, y). Therefore, when studying the group w(F 2 ) and groups associated with it, we can use lemmas proved in [5]. In particular, by Lemma 17 of [5], all the nontrivial values of w(x, y) in G = F 2 /[w(F 2 ), F 2 ] form a basis of the free abelian group w(G).…”

“…Therefore, when studying the group w(F 2 ) and groups associated with it, we can use lemmas proved in [5]. In particular, by Lemma 17 of [5], all the nontrivial values of w(x, y) in G = F 2 /[w(F 2 ), F 2 ] form a basis of the free abelian group w(G). Now let V be the subgroup of G generated by all the elements of the form w(X, Y ) and [5] and Lemma 15 of [5], the fact that w(S 1 , T 1 ) = w(S 2 , T 2 ) in G yields that S 1 is conjugate to S 2 and T 1 is conjugate to T 2 in the group F 2 /w(F 2 ).…”

A group variety defined by a semigroup law

“…While [3] is the main source of information for references, in this paper we also use a few results obtained in [5]. LEMMA Since the words X and Y are not commutative in rank i, the inequality | / | < 100?"'…”