The $R_{\infty }$-property for right-angled Artin groups

Abstract

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

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