Forest-skein groups II: construction from homogeneously presented monoids

TL;DR

This paper introduces a class of forest-skein groups constructed from homogeneous monoid presentations, classifies them via wreath products, and shows they inherit several important algebraic and topological properties.

Contribution

It extends the construction of forest-skein groups to those derived from homogeneous monoids and provides a classification and inheritance results for their properties.

Findings

Groups decomposed as wreath products.

Classification of groups up to isomorphism.

Inheritance of properties like Haagerup property and orderability.

Abstract

Inspired by the reconstruction program of conformal field theories of Vaughan Jones we recently introduced a vast class of so called forest-skein groups. They are built from a skein presentation: a set of colours and a set of pairs of coloured trees. Each nice skein presentation produces four groups similar to Richard Thompson's group F,T,V and the braided version BV of Brin and Dehornoy. In this article, we consider forest-skein groups obtained from one-dimensional skein presentations; the data of a homogeneous monoid presentation. We decompose these groups as wreath products. This permits to classify them up to isomorphisms. Moreover, we prove that a number of properties of the fraction group of the monoid pass through the forest-skein groups such as the Haagerup property, homological and topological finiteness properties, and orderability.