The R∞-property for right-angled Artin groups

Abstract

Given a group G and an automorphism φ of G, two elements x,y∈G are said to be φ-conjugate if x=gyφ(g)−1 for some g∈G. The number of equivalence classes is the Reidemeister number R(φ) of φ, and if R(φ)=∞ for all automorphisms of G, then G is said to have the R∞-property.A finite simple graph Γ gives rise to the right-angled Artin group AΓ, which has as generators the vertices of Γ and as relations vw=wv if and only if v and w are joined by an edge in Γ. We conjecture that all non-abelian right-angled Artin groups have the R∞-property and prove this conjecture for several subclasses of right-angled Artin groups.