By Prof. Riddhi Shah from JNU

Abstract 

Twisted conjugacy classes (Reidemeister classes) of an automorphism ϕ of a group G are defined as {gxϕ(g−1) | g ∈ G}, x ∈ G. We consider continuous auto-morphisms ϕ of a Lie group G and study when its Reidemeister number R(ϕ), that is, the number of its twisted conjugacy classes is finite or infinite. When G is connected and solvable, we obtain equivalent conditions on ϕ for R(ϕ) to be infinite, and show that some structural conditions on G ensures that R(ϕ) is infinite for every ϕ, that is, G has topological R∞-property. This enables us to show that the group of invertible n × n upper triangular real matrices has this property, n ≥ 2. We also prove that the Walnut group also has this property. While for a general connected Lie group G, the number of conjugacy classes is infinite, in case G is not nilpotent there exists n ∈ N such that R(ϕn) is infinite for every ϕ. We show that SL(2, R) and GL(2, R) have topological R∞-property, and construct many examples of solvable and non-solvable Lie groups with this property. We will also briefly discuss this for general semisimple Lie groups and com-pact (non-Lie) groups. (Joint work with Ravi Prakash, Transformation Groups (2026). https://doi.org/10.1007/s00031-026-09967-x)