Trickle groups

TL;DR

This paper introduces trickle groups, a new family generalizing several known groups, with a focus on their structure, normal forms, and solutions to the word problem, expanding the understanding of complex algebraic groups.

Contribution

The paper defines trickle groups via a novel presentation, establishes rewriting systems for them, and explores their subgroup structures and Garside properties, unifying various group classes.

Findings

Established a confluent rewriting system for trickle groups.

Proved standard parabolic subgroups are themselves trickle groups.

Identified conditions under which trickle groups are Garside groups.

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.