Trickle groups
Paolo Bellingeri (LMNO), Eddy Godelle (LMNO), Luis Paris (IMB)

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 and , 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…
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Taxonomy
TopicsParkinson's Disease and Spinal Disorders
