Invariants of inversive 2-structures on groups of labels

Abstract

For a finite set D of nodes let E₂(D)={(x, y)[mid ]x, y∈D, x≠y}. We define an inversive Δ2-structure g as a function g[ratio ]E₂(D)→Δ into a given group Δ satisfying the property g(x, y)=g(y, x)⁻¹ for all (x, y)∈E₂(D). For each function (selector) σ[ratio ]D→Δ there is a corresponding inversive Δ2-structure g^σ defined by g^σ(x, y)=σ(x)·g(x, y)·σ(y)⁻¹. A function η mapping each g into the group Δ is called an invariant if η(g^σ)=η(g) for all g and σ. We study the group of free invariants η of inversive Δ2-structures, where η is defined by a word from the free monoid with involution generated by the set E₂(D). In particular, if Δ is abelian, the group of free invariants is generated by triangle words of the form (x₀, x₁)(x₁, x₂)(x₂, x₀).