A combinatorial problem in infinite groups

Abstract

Let $w$ be a word in the free group of rank $n\in \mathbb{N}$ and let $\mathcal{V}(w)$ be the variety of groups defined by the law $w=1$. Define $\mathcal{V}(w^{\ast })$ to be the class of all groups $G$ in which for any infinite subsets $X_1,...,X_n$ there exist $x_i\in X_i$, $1\le i\le n$, such that $w(x_1,...,x_n)=1$. Clearly, $\mathcal{V}(w)\cup \mathcal{F}\subseteq \mathcal{V}(w^{\ast })$; $\mathcal{F}$ being the class of finite groups. In this paper, we investigate some words $w$ and some certain classes $\mathcal{P}$ of groups for which the equality $(\mathcal{V}(w)\cup \mathcal{F})\cap \mathcal{P}=\mathcal{P}\cap \mathcal{V}(w^{\ast })$ holds.

8 pages, to appear in Bull. Malaysian Math. Soc