We consider twisted conjugacy classes of continuous automorphisms \(\varvec{\varphi }\) of a Lie group \(\varvec{G}\) . We obtain a necessary and sufficient condition on \(\varvec{\varphi }\) for its Reidemeister number, the number of twisted conjugacy classes, to be infinite when \(\varvec{G}\) is connected and solvable or compactly generated and nilpotent. We also show for a general connected Lie group \(\varvec{G}\) that the number of conjugacy classes is infinite. We prove that for a connected non-nilpotent Lie group \(\varvec{G}\) , there exists \(\varvec{n\in }\mathbb {N}\) such that the Reidemeister number of \(\varvec{\varphi ^n}\) is infinite for every \(\varvec{\varphi }\) . We say that \(\varvec{G}\) has topological \(\varvec{R_\infty }\) -property if the Reidemeister number of every \(\varvec{\varphi }\) is infinite. We obtain conditions on a connected solvable Lie group under which it has topological \(\varvec{R_\infty }\) -property; which, in particular, enables us to prove that the group of invertible \(\varvec{n}\times \varvec{n}\) upper triangular real matrices and its quotient group modulo its center have topological \(\varvec{R_\infty }\) -property for every \(\varvec{n}\ge \varvec{2}\) . We also prove that the Walnut group also has this property. We show that \(\mathbf {SL(2,}\,\mathbb {R} {\textbf {)}}\) and \({\textbf {GL(2,}}\,\mathbb {R} {\textbf {)}}\) have topological \(\varvec{R_\infty }\) -property, and construct many examples of Lie groups with this property.
