$PS{L}_2$ tropicalization and lines on surfaces

The paper is based on a talk given by the first author at the Gökova Geometry & Topology conference in May 2024. The subject is an interplay between the ideas of tropical geometry and two-by-two matrices with an intention to explore new types of geometries. More concretely, the article gives a preliminary account for a non-abelian version of phase tropicalization for subvarieties of $PS{L}_2\mathrm{.}$