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Sigma theory and twisted conjugacy-II: Houghton groups and pure symmetric automorphism groups

We say that $x,y\in Γ$ are in the same $ϕ$-twisted conjugacy class and write $x\sim_ϕy$ if there exists an element $γ\in Γ$ such that $y=γxϕ(γ^{-1})$. This is an equivalence relation on $Γ$ called the $ϕ$-twisted conjugacy. Let $R(ϕ)$ denote the number of $ϕ$-twisted conjugacy classes in $Γ$. If $R(ϕ)$ is infinite for all $ϕ\in Aut(Γ)$, we say that $Γ$ has the $R_\infty$-property. The purpose of this note is to show that the symmetric group $S_\infty$, the Houghton groups and the pure symmetric automorphism groups have the $R_\infty$-property. We show, also, that the Richard Thompson group $T$ has the $R_\infty$-property. We obtain a general result establishing the $R_\infty$-property of finite direct product of finitely generated groups.