González Prieto, José ÁngelLogares Jiménez, Marina LucíaMuñoz Velázquez, Vicente2023-06-172023-06-172021González Prieto, Á., Logares Jiménez, M. L. & Muñoz Velázquez, V. «Representation Variety for the Rank One Affine Group». Mathematical Analysis in Interdisciplinary Research, editado por Ioannis N. Parasidis et al., vol. 179, Springer International Publishing, 2021, pp. 381-416. DOI.org (Crossref), https://doi.org/10.1007/978-3-030-84721-0_18.978-3-030-84720-310.1007/978-3-030-84721-0_18https://hdl.handle.net/20.500.14352/8864The aim of this chapter is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group.engbook parthttps://doi.org/10.1007/978-3-030-84721-0_18open access512.7TQFTModuli spacesE-polynomialRepresentation varietiesGeometria algebraica1201.01 Geometría Algebraica