Forest-skein groups II: Construction from homogeneously presented monoids

Abstract

Inspired by the reconstruction program of conformal field theories of Vaughan Jones we recently introduced a vast class of the so-called forest-skein groups. They are built from a skein presentation: a set of colors and a set of pairs of colored trees. Each nice skein presentation produces four groups similar to Richard Thompson’s group [Formula: see text] and the braided version [Formula: see text] of Brin and Dehornoy. In this paper, 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.