Extended positive representations

We introduce the notion of extended positive representation of a Fuchsian group into a semisimple Lie group with a $\mathrm{\Theta }$-positive (in the sense of Guichard--Wienhard) structure. Examples of such representation include Fock-Goncharov's positive framed representations, maximal representations and classical $\mathrm{\Theta }$-positive representations. We show that extended positive representations are divergent and extended geometrically finite (in the sense of Weisman).