Given a group G and an automorphism φ of G, two elements $x,y\in G$ are said to be φ-conjugate if $x=gy\phi {(g)}^{-1}$ for some $g\in G$. The number of equivalence classes is the Reidemeister number $R(\phi )$ of φ, and if $R(\phi )=\infty$ for all automorphisms of G, then G is said to have the $R_{\infty }$-property.

A finite simple graph Γ gives rise to the right-angled Artin group $A_{\mathrm{\Gamma }}$, 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 $R_{\infty }$-property and prove this conjecture for several subclasses of right-angled Artin groups.

中文翻译：

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在[R _∞ -特性为直角阿廷组

给定的一组G ^和构φ的ģ，两个元件$X，ÿ\in G$被称为φ-共轭$X=Gÿ\phi {（G）}^{-\mathrm{1个}}$ 对于一些 $G\in G$。等价类的数量是Reidemeister数$\mathrm{[R}（\phi ）$的φ，如果$\mathrm{[R}（\phi ）=\infty$对于G的所有自同构，则称G具有${\mathrm{[R}}_{\infty }$-属性。

有限简单图Γ引起直角Artin组 ${\mathrm{一种}}_{\mathrm{\Gamma }}$，它具有Γ的顶点和关系 $vw=wv$当且仅当v和w通过Γ中的边连接时。我们推测所有非阿贝尔直角阿丁族群都有${\mathrm{[R}}_{\infty }$属性，并证明对直角Artin组的几个子类的猜想。