Degenerations of k-positive surface group representations

TL;DR

This paper introduces k-positive surface group representations, explores their degenerations, and establishes a limit theorem for positively ratioed representations, revealing how these representations behave under deformation and their positivity properties.

Contribution

It defines k-positive representations, analyzes their degenerations, and proves a new limit theorem for positively ratioed representations, advancing understanding of their structure and limits.

Findings

Hitchin representations are a special case of k-positive representations.

Degenerations of non-Hitchin k-positive representations can become non-discrete.

Irreducible limits of these representations are at least (k-1)-positive.

Abstract

We introduce \emph{k-positive representations}, a large class of $\{1,\ldots,k\}$--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.