A conjugation-free geometric presentation of fundamental groups of arrangements II: Expansion and some properties

TL;DR

This paper extends the class of arrangements with conjugation-free geometric presentations of their fundamental groups to include those with disjoint union of cycle-tree graphs, and explores properties like completeness and solvability of the word problem.

Contribution

It generalizes previous results by including arrangements with cycle-tree graphs and analyzes key properties of their fundamental group presentations.

Findings

Fundamental groups of arrangements with cycle-tree graphs have conjugation-free geometric presentations.

These presentations satisfy Dehornoy's completeness property when the graph has no edges.

The properties lead to implications like left-cancellativity and solvability of the word problem.

Abstract

A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators $x_1, ..., x_n$ and the cyclic relations: $x_{i_k}x_{i_{k-1}} ... x_{i_1} = x_{i_{k-1}} ... x_{i_1} x_{i_k} = ... = x_{i_1} x_{i_k} ... x_{i_2}$ with no conjugations on the generators. We have already proved that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs. Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement has no edges. The completeness property is a powerful property which leads to many nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group).