Degenerations of k-positive surface group representations

Abstract

We introduce k-positive representations, a large class of {1, . . . , k}-Anosov surface group representations into PGL(E) that share many features with Hitchin representations, and we study their degenerations: unless they are Hitchin, they can be deformed to non-discrete representations, but any limit is at least (k − 3)-positive and irreducible limits are (k − 1)-positive. A major ingredient, of independent interest, is a general limit theorem for positively ratioed representations. JONAS BEYRER AND BEATRICE POZZETTI 6.2. k-positive representations 27 6.3. Transversality properties of k-positive representations 28 6.4. k-tridirect representations 30 7. Degenerations of k-positive representations 31 7.1. Anosov limits of k-positive representations 31 7.2. Limits of k-positive representations 32 7.3. Coherence of the boundary map in the limit 33 8. Examples and open questions 36 8.1. All transversality could get lost in the limit 37 8.2. The set of positively ratioed representations is not open 37 8.3. Limits of k-positive representations 38 References 39