  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.
