Teichmüller space has several avatars, for example it can be identified with a connected component of the character variety Hom(H,PSL(2,R))/PSL(2,R) that consists entirely of discrete and faithful representations, where H is the fundamental group of a closed surface. Surprisingly components consisting of discrete and faithful representations exist also for the character varieties Hom(H,G)/G of other groups G, e.g. if G is split or Hermitian. By the analogy those components are called „higher rank Teichmueller spaces“. Guichard-Wienhard developed a conjectural framework on the existence of higher rank Teichmueller spaces showing that PO(p,q) should admit such components which are filled by „positive representations“.

In this talk we want to discuss properties of those positive representations into PO(p,q), in particular we study similarities of those to hyperbolizations, and we discuss a proof of why positive representations

constitute higher rank Teichmüller spaces. Based on joint work with Beatrice Pozzetti.

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