Mathematics – Group Theory
Scientific paper
2009-11-17
Geom. Dedicata (2011) 151:233-243
Mathematics
Group Theory
13 pages
Scientific paper
A group $G$ is said to have the property $R_\infty$ if every automorphism $\phi \in {\rm Aut}(G)$ has an infinite number of $\phi$-twisted conjugacy classes. Recent work of Gon\c{c}alves and Kochloukova uses the $\Sigma^n$ (Bieri-Neumann-Strebel-Renz) invariants to show the $R_{\infty}$ property for a certain class of groups, including the generalized Thompson's groups $F_{n,0}$. In this paper, we make use of the $\Omega^n$ invariants, analogous to $\Sigma^n$, to show $R_{\infty}$ for certain finitely generated groups. In particular, we give an alternate and simpler proof of the $R_{\infty}$ property for BS(1,n). Moreover, we give examples for which the $\Omega^n$ invariants can be used to determine the $R_{\infty}$ property while the $\Sigma^n$ invariants techniques cannot.
Koban Nic
Wong Peter
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