{
  "id": "2412.04932",
  "version": "v1",
  "published": "2024-12-06T10:41:11.000Z",
  "updated": "2024-12-06T10:41:11.000Z",
  "title": "Trickle groups",
  "authors": [
    "Paolo Bellingeri",
    "Eddy Godelle",
    "Luis Paris"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "A new family of groups, called trickle groups, is presented. These groups generalize right-angled Artin and Coxeter groups, as well as cactus groups. A trickle group is defined by a presentation with relations of the form $xy = zx$ and $x^\\mu = 1$, that are governed by a simplicial graph, called a trickle graph, endowed with a partial ordering on the vertices, a vertex labeling, and an automorphism of the star of each vertex. We show several examples of trickle groups, including extended cactus groups, certain finite-index subgroups of virtual cactus groups, Thompson group F, and ordered quandle groups. A terminating and confluent rewriting system is established for trickle groups, enabling the definition of normal forms and a solution to the word problem. An alternative solution to the word problem is also presented, offering a simpler formulation akin to Tits' approach for Coxeter groups and Green's for graph products of cyclic groups. A natural notion of a parabolic subgraph of a trickle graph is introduced. The subgroup generated by the vertices of such a subgraph is called a standard parabolic subgroup and it is shown to be the trickle group associated with the subgraph itself. The intersection of two standard parabolic subgroups is also proven to be a standard parabolic subgroup. If only relations of the form $xy = zx$ are retained in the definition of a trickle group, then the resulting group is called a preGarside trickle group. Such a group is proved to be a preGarside group, a torsion-free group, and a Garside group if and only if its associated trickle graph is finite and complete.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-06T10:41:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "standard parabolic subgroup",
      "trickle graph",
      "coxeter groups",
      "word problem",
      "pregarside trickle group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}