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Beyond graph products and cactus groups: quandle products of groups

Anthony Genevois

TL;DR

The paper introduces quandle products of groups as a unifying framework for graph products, cactus groups, wreath products, trickle groups and related constructions, using quasi-median geometry to extract structural and algorithmic results. It develops a comprehensive theory: defining quandle systems with oposets and holonomy, solving the word problem via ranked braids, and proving the Cayley graphs are quasi-median with controlled cubical dimension. This geometry yields decomposition theorems (iterated semidirect and parabolic subgroups), stability results for properties like solvable word problems, torsion, orderability, and a-T-menability, and it clarifies the role of holonomy in the tractability of these products. It also situates trickle groups within this framework (as quandle products of cyclic groups with trivial holonomy) and provides a broad collection of examples (twisted graph products, wreath products, cactus products, trickle groups) along with open questions about residual finiteness, CAT(0)ness, subgroups, and algorithmic problems.

Abstract

In this paper, we introduce and initiate the study of quandle products of groups, a family of groups that includes graph products of groups, cactus groups, wreath products, and the recently introduced trickle groups. Our approach is geometric: we show that quandle products admit quasi-median Cayley graphs; and, then, we exploit this geometry to deduce various valuable information about quandle products.

Beyond graph products and cactus groups: quandle products of groups

TL;DR

The paper introduces quandle products of groups as a unifying framework for graph products, cactus groups, wreath products, trickle groups and related constructions, using quasi-median geometry to extract structural and algorithmic results. It develops a comprehensive theory: defining quandle systems with oposets and holonomy, solving the word problem via ranked braids, and proving the Cayley graphs are quasi-median with controlled cubical dimension. This geometry yields decomposition theorems (iterated semidirect and parabolic subgroups), stability results for properties like solvable word problems, torsion, orderability, and a-T-menability, and it clarifies the role of holonomy in the tractability of these products. It also situates trickle groups within this framework (as quandle products of cyclic groups with trivial holonomy) and provides a broad collection of examples (twisted graph products, wreath products, cactus products, trickle groups) along with open questions about residual finiteness, CAT(0)ness, subgroups, and algorithmic problems.

Abstract

In this paper, we introduce and initiate the study of quandle products of groups, a family of groups that includes graph products of groups, cactus groups, wreath products, and the recently introduced trickle groups. Our approach is geometric: we show that quandle products admit quasi-median Cayley graphs; and, then, we exploit this geometry to deduce various valuable information about quandle products.
Paper Structure (39 sections, 49 theorems, 80 equations, 2 figures)

This paper contains 39 sections, 49 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Step 1.
  3. Step 2.
  4. Step 3.
  5. A word about trickle groups.
  6. Basics on quandle products
  7. Oposets
  8. Quandle systems and groups
  9. Holonomy
  10. Parabolic subgroups
  11. Word problem
  12. Quasi-median geometry
  13. Preliminary
  14. Alternative characterisations.
  15. Paths and geodesics.
  16. ...and 24 more sections

Key Result

Theorem 1.1

Cayley graphs of quandle products with respect to the union of their factors is a quasi-median graph.

Figures (2)

  • Figure 1: From left to right: the no-$K_4^-$ condition, the house condition, and the $3$-cube condition.
  • Figure 2: Hyperplanes in a quasi-median graph.

Theorems & Definitions (131)

  • Theorem 1.1: Theorem \ref{['thm:QuandleQM']}
  • Theorem 1.2: Corollary \ref{['cor:DecompositionQuandle']}
  • Theorem 1.3
  • Theorem 1.4: Proposition \ref{['prop:QuandleVsTrickle']}
  • Theorem 1.5: Corollary \ref{['cor:Trickle']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Definition 2.4
  • ...and 121 more