Motivated by rigidity results for quotients spaces coming from topology, functional analysis, and set-theory, Kanovei and Reeken showed that if \(N\) and \(M\) are countable dense subgroups of the additive group of the reals \(\mathbb{R}\), then every Borel–definable homomorphism from \(\mathbb{R}/N\) to \(\mathbb{R}/M\) is of a certain "trivial" form. In the same paper, they asked whether quotients of the \(p\)–adic groups satisfy similar rigidity phenomena.

I will present my joint work with J. Bergfalk and M. Lupini, in which we answer Kanovei-Reeken's question in a broader context. I will also illustrate how these results inform the "Definable Algebraic Topology" research program, which enriches classical invariants from homological algebra and algebraic topology with descriptive set-theoretic data, yielding definable invariants that offer stronger tools for classification.