  This article concerns the locus of all locally constant
$\mathrm{SL}(2,\mathbb{R})$-valued cocycles that are uniformly hyperbolic,
called the hyperbolic locus. Using the theory of semigroups of M\"obius
transformations we introduce a new locus in $\mathrm{SL}(2,\mathbb{R})^N$ which
allows us to study the complement of the hyperbolic locus. Our results answer a
question of Avila, Bochi and Yoccoz, and Jacques and Short, while motivating a
new line of investigation on the subject.
