[Paper Review] A combinatorial problem in infinite groups

Aliréza Abdollahi|ArXiv.org|Dec 2, 2002

Finite Group Theory Research42 references3 citations

TL;DR

This paper investigates the relationship between varieties of groups defined by a word $ w = 1 $ and a stronger combinatorial condition $ w^* $, where every infinite tuple of subsets contains a solution to $ w = 1 $. It proves that under certain finiteness and structural conditions—such as local finiteness, local solubility, or finite rank—groups satisfying $ w^* $ must actually lie in the variety $ \mathcal{V}(w) $, thereby establishing equality $ \mathcal{V}(w) \cup \mathcal{F} = \mathcal{V}(w^*) $ in broad classes of 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 $\mathcal{V}(w^*)$ 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.

Motivation & Objective

• To resolve the conjecture that $ \mathcal{V}(w) \cup \mathcal{F} = \mathcal{V}(w^*) $, where $ \mathcal{V}(w^*) $ consists of groups in which every infinite tuple of subsets contains a solution to $ w = 1 $.
• To identify structural conditions on groups (e.g., local finiteness, local solubility, finite rank) under which $ \mathcal{V}(w^*) $-groups must lie in $ \mathcal{V}(w) $.
• To generalize prior results on Engel groups, nilpotent groups, and metabelian groups by extending the equality to broader classes of groups.
• To establish reductions for the equality problem by analyzing the absence of certain infinite simple or locally finite sections in $ \mathcal{V}(w^*) $-groups.

Proposed method

• Introduces the class $ \mathcal{V}(w^*) $ as the set of groups where, for any $ n $ infinite subsets, there exist elements from each subset satisfying $ w(x_1,\dots,x_n) = 1 $.
• Applies structural group theory, particularly results on minimax groups, finite residual structure, and finite rank soluble groups.
• Uses Kropholler’s theorem to show that finitely generated soluble groups without sections isomorphic to $ C_p \text{wr} C_\infty $ are minimax.
• Applies results on residually finite and locally graded groups, especially leveraging the fact that infinite locally graded or finite-rank groups in $ \mathcal{V}(w^*) $ must lie in $ \mathcal{V}(w) $.
• Analyzes the role of specific groups like $ M(\alpha,p) $ and $ C_q \text{wr} C_\infty $ as obstructions: if such groups are not in $ \mathcal{V}(w) $, then $ \mathcal{V}(w^*) $-groups must be in $ \mathcal{V}(w) $.
• Employs contradiction arguments via section theory: if a group in $ \mathcal{V}(w^*) $ is not in $ \mathcal{V}(w) $, it must have an infinite simple section, which is ruled out under the given conditions.

Experimental results

Research questions

• RQ1 Under what conditions on a word $ w $ and a class of groups $ \mathcal{P} $ does $ \mathcal{V}(w^*) \cap \mathcal{P} = (\mathcal{V}(w) \cup \mathcal{F}) \cap \mathcal{P} $ hold?
• RQ2 Can the equality $ \mathcal{V}(w) \cup \mathcal{F} = \mathcal{V}(w^*) $ be established for infinite locally soluble or finite-rank groups?
• RQ3 What role do specific groups like $ C_p \text{wr} C_\infty $ and $ M(\alpha,p) $ play in obstructing or enabling the equality $ \mathcal{V}(w^*) = \mathcal{V}(w) \cup \mathcal{F} $?
• RQ4 Does the absence of infinite linear simple locally finite groups in $ \mathcal{V}(w^*) $ imply that $ \mathcal{V}(w^*) $-groups are in $ \mathcal{V}(w) $?
• RQ5 How do properties like local finiteness, local solubility, and residual finiteness interact with the $ w^* $-condition to force membership in $ \mathcal{V}(w) $?

Key findings

• Every infinite finitely generated locally graded restrained group in $ \mathcal{V}(w^*) $ lies in $ \mathcal{V}(w) $, provided all finitely generated residually finite groups in $ \mathcal{V}(w) $ are polycyclic-by-finite.
• Every infinite residually [(locally $ \mathcal{P} $)-by-finite] group in $ \mathcal{V}(w^*) $ belongs to $ \mathcal{V}(w) $, provided $ \mathcal{P} $ is closed under subgroups, consists of soluble groups, and the equality holds for $ (\mathcal{P} \text{ by finite}) $-groups.
• Every infinite locally soluble group in $ \mathcal{V}(w^*) $ lies in $ \mathcal{V}(w) $, provided $ C_q \text{wr} C_\infty \not\in \mathcal{V}(w) $ for all primes $ q $, or $ M(\alpha,p) \not\in \mathcal{V}(w) $ for all $ p \geq 0 $.
• Non-linear simple locally finite groups do not belong to $ \mathcal{V}(w^*) $, which helps eliminate counterexamples to the equality $ \mathcal{V}(w) \cup \mathcal{F} = \mathcal{V}(w^*) $.
• For soluble words $ w = w_d $, an infinite locally finite $ \mathcal{V}(w^*) $-group lies in $ \mathcal{V}(w) $ if and only if it has no infinite linear simple locally finite section.
• Infinite locally soluble groups of finite rank in $ \mathcal{V}(w^*) $ belong to $ \mathcal{V}(w) $, as such groups are minimax and hence satisfy the required finiteness conditions.