$R_{\infty}$-property for groups commensurable to nilpotent quotients of RAAGs

Abstract

Let $G$ be a group and $\varphi$ an automorphism of $G$. Two elements $x,y \in G$ are said to be $\varphi$-conjugate if there exists a third element $z \in G$ such that $z x \varphi(z)^{-1} = y$. Being $\varphi$-conjugate defines an equivalence relation on $G$. The group $G$ is said to have the $R_{\infty}$-property if all its automorphisms $\varphi$ have infinitely many $\varphi$-conjugacy classes. For finitely generated torsion-free nilpotent groups, the so-called Mal'cev completion of the group is a useful tool in studying this property. Two groups have isomorphic Mal'cev completions if and only if they are abstractly commensurable. This raises the question whether the $R_{\infty}$-property is invariant under abstract commensurability within the class of finitely generated torsion-free nilpotent groups. We show that the answer to this question is negative and provide counterexamples within a class of 2-step nilpotent groups associated to edge-weighted graphs. These groups are commensurable to 2-step nilpotent quotients of right-angled Artin groups.

Figures (5)

• Figure 1: Graphs $\Gamma(k)$ and $\overline\Gamma$ with introduced notation
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Theorems & Definitions (49)

• Proposition 1.1
• Theorem 1.2
• Lemma 2.1: roma11-1,dg14-1
• Remark 2.2
• proof : Proof of Proposition \ref{['prop:RinftyAndCommensurability']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3