[Paper Review] Acyclicity in finite groups and groupoids

Martin Otto|arXiv (Cornell University)|Jun 22, 2018

semigroups and automata theory12 references3 citations

TL;DR

This paper presents a novel, generic construction of finite groups and groupoids whose Cayley graphs satisfy strong N-acyclicity conditions by controlling cycles formed by cosets of subgroups rather than just individual generators. By generalizing Biggs' method to second-order structures (cosets), the authors achieve highly symmetric, locally tree-like graphs with controlled acyclicity, correcting a critical flaw in prior work and enabling applications in finite hypergraph coverings and model theory.

ABSTRACT

We expound a concise construction of finite groups and groupoids whose Cayley graphs satisfy graded acyclicity requirements. Our acyclicity criteria concern cyclic patterns formed by coset-like configurations w.r.t. subsets of the generator set rather than just by individual generators. The proposed constructions correspondingly yield finite groups and groupoids whose Cayley graphs satisfy much stronger acyclicity conditions than large girth. We thus obtain generic and canonical constructions of highly homogeneous graph structures with strong acyclicity properties, which support known applications in finite graph and hypergraph coverings that locally unfold cyclic configurations.

Motivation & Objective

• To develop a generic, canonical construction of finite groups and groupoids with strong acyclicity properties beyond large girth.
• To address the limitation of prior methods that only control cycles formed by individual generators, by instead controlling cycles formed by cosets of subgroups.
• To correct a serious error in the construction of N-acyclic groupoids previously sketched in [15], which had implications for the Henckell–Rhodes conjecture in semigroup theory.
• To unify the treatment of finite groups and groupoids under a common framework of N-acyclicity based on coset configurations.
• To support applications in finite graph and hypergraph coverings, particularly in contexts requiring symmetry preservation and local unfolding of cyclic structures.

Proposed method

• Introduces E-graphs and E-groups as foundational structures to model generator and coset interactions over a base set I.
• Defines I-coset cycles as sequences of cosets linked by compatibility conditions over shared generator sets, generalizing generator cycles to second-order structures.
• Applies free amalgamation of E-graphs along common substructures to build larger acyclic configurations while preserving N-acyclicity.
• Uses compatibility and homomorphism conditions to ensure consistency across amalgamated components, especially in the context of I-skeletons and constraint patterns.
• Establishes connectivity and separation conditions for I-coset cycles to enforce acyclicity, ensuring no chords or unintended cycles form.
• Extends the group construction to groupoids via I-groupoids, allowing for branched coverings in hypergraph settings while maintaining acyclicity and symmetry.

Experimental results

Research questions

• RQ1 How can finite groups be constructed such that their Cayley graphs satisfy strong acyclicity conditions based on coset cycles rather than just generator cycles?
• RQ2 What is the correct and generic construction of N-acyclic groupoids that avoids the flaw present in the earlier sketch from [15]?
• RQ3 How can the acyclicity criteria for groups be generalized to groupoids to support branched coverings in finite hypergraph theory?
• RQ4 What conditions ensure that coset cycles do not form chords or unintended cycles in the Cayley graph of the constructed structures?
• RQ5 How can symmetry and compatibility be preserved in the amalgamation of E-graphs to maintain N-acyclicity in the final group or groupoid?

Key findings

• The paper corrects a fundamental flaw in the construction of N-acyclic groupoids previously proposed in [15], which had been used in attempts to resolve the Henckell–Rhodes conjecture.
• A new, rigorous construction of N-acyclic E-groups is provided via free amalgamation of E-graphs over I-skeletons, ensuring strong acyclicity at the coset level.
• The method achieves N-acyclicity by enforcing both connectivity and separation conditions on I-coset cycles, preventing the formation of chords and cycles of length ≤ N.
• The construction generalizes Biggs’ method to second-order structures (cosets), yielding Cayley graphs that are not only of large girth but also locally tree-like in a stronger, more structured sense.
• The framework successfully supports applications in finite hypergraph coverings and finite model theory, particularly in the guarded fragment, by preserving local symmetries and enabling unfoldings.
• The transfer of acyclicity from groups to groupoids is formally established, showing that N-acyclic groups can be used to construct N-acyclic I-groupoids with preserved symmetries and compatibility.