A combinatorial problem in infinite groups

TL;DR

This paper explores a combinatorial property of words in free groups and characterizes classes of groups where certain algebraic and combinatorial conditions coincide, extending understanding of group varieties.

Contribution

It investigates conditions under which the union of a variety defined by a word and finite groups equals a class defined by a combinatorial property within specific group classes.

Findings

Identifies classes of groups where algebraic and combinatorial properties align.

Provides conditions for the equality of group classes involving words and finite groups.

Enhances understanding of the structure of group varieties and combinatorial group theory.

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 $\mathcal{V}(w^*)$ to be the class of all groups $G$ in which for any infinite subsets $X_1, ..., X_n$ there exist $x_i \in X_i$, $1\leq i\leq n$, such that $w(x_1, ..., x_n)=1$. Clearly, $\mathcal{V}(w) \cup \mathcal{F} \subseteq \mathcal{V}(w^*)$; $\mathcal{F}$ 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 $(\mathcal{V}(w) \cup \mathcal{F})\cap \mathcal{P}= \mathcal{P} \cap \mathcal{V}(w^*)$ holds.