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Acyclicity in finite groups and groupoids

Martin Otto

TL;DR

This work develops a generic, symmetry-preserving framework to construct finite groups and groupoids with graded coset-acyclicity, extending Biggs’ large-girth paradigm beyond generator cycles to coset configurations. The key technique is a conservative unfolding approach using free amalgams of E-graphs and their Cayley graphs, ensuring that acyclicity at higher levels does not disrupt lower-level structure or symmetries. The authors supply inductive constructions of $N$-acyclic $ extsf{E}$-groups and, via constraint patterns, translate these results to $\mathbb{I}$-groupoids, with a transfer principle from groups to groupoids that preserves acyclicity, compatibility, and symmetries. The resulting finite, highly homogeneous structures enable robust finite coverings of graphs and hypergraphs, providing a unified route to local-to-global unfolding of cyclic configurations with broad applicability in discrete geometry and combinatorial topology. The work also clarifies and corrects prior flawed groupoid constructions, reinforcing the methodological bridge between algebraic and graph-theoretic perspectives for controlled acyclicity.

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.

Acyclicity in finite groups and groupoids

TL;DR

This work develops a generic, symmetry-preserving framework to construct finite groups and groupoids with graded coset-acyclicity, extending Biggs’ large-girth paradigm beyond generator cycles to coset configurations. The key technique is a conservative unfolding approach using free amalgams of E-graphs and their Cayley graphs, ensuring that acyclicity at higher levels does not disrupt lower-level structure or symmetries. The authors supply inductive constructions of -acyclic -groups and, via constraint patterns, translate these results to -groupoids, with a transfer principle from groups to groupoids that preserves acyclicity, compatibility, and symmetries. The resulting finite, highly homogeneous structures enable robust finite coverings of graphs and hypergraphs, providing a unified route to local-to-global unfolding of cyclic configurations with broad applicability in discrete geometry and combinatorial topology. The work also clarifies and corrects prior flawed groupoid constructions, reinforcing the methodological bridge between algebraic and graph-theoretic perspectives for controlled acyclicity.

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.

Paper Structure

This paper contains 21 sections, 28 theorems, 92 equations, 7 figures.

Table of Contents

  1. Introduction
  2. General patterns
  3. Cayley & Biggs: the basic construction
  4. E-graphs and E-groups
  5. Compatibility and homomorphisms
  6. Coset cycles and acyclicity criteria
  7. Free amalgams of E-graphs
  8. Amalgamation chains
  9. Amalgamation clusters
  10. Construction of N-acyclic E-groups
  11. Constraints on generator sequences
  12. Small coset amalgams over I-skeletons
  13. Construction of N-acyclic E-groups over I
  14. From groups to groupoids
  15. Constraint patterns for groupoids
  16. ...and 6 more sections

Key Result

Lemma 2.10

Let $\hat{\mathbb{G}} \succcurlyeq \mathbb{G}$ be $\mathsf{E}$-groups, $\hat{\mathbb{G}} = \mathrm{sym}(\mathbb{H})$ for an $\mathsf{E}$-graph $\mathbb{H}$. In this situation, the subgroups $\mathbb{G}[\alpha]$ and $\hat{\mathbb{G}}[\alpha]$ generated by $\alpha \subseteq E$ are isomorphic as $\alph

Figures (7)

  • Figure 1: Free amalgamation.
  • Figure 2: Pattern for identifications in $\mathsf{CE}(\mathbb{H}[\alpha],\mathbb{G},\alpha)$, see $(\dagger)$ in Definition \ref{['smallcosetamalgCEdef']}: for $i=1,2$, $g_{i0} = [w_{i0}]_\mathbb{G}$ links $v_0$ to $v_i$ in their $\alpha_i'$-component of $\mathbb{H}$.
  • Figure 3: Pattern for two consecutive identifications.
  • Figure 4: Bridge-freeness.
  • Figure 5: Walks in relevant $\alpha_i$-components of $\mathsf{CE}(\mathbb{H}[\alpha],\mathbb{G},\alpha)$ and $\mathsf{CE}(\hat{\mathbb{H}}[\alpha],\hat{\mathbb{G}},\alpha)$.
  • ...and 2 more figures

Theorems & Definitions (94)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.10
  • proof
  • Definition 2.11
  • ...and 84 more