In this talk we discuss locally constant cocycles taking
values in the special linear group.
Such a cocycle is *uniformly hyperbolic* if the norms of matrix products
in the image of the cocycle
grow exponentially with respect to the length of the product.
Using techniques from the theory of Möbius semigroups, we study the
locus of all uniformly hyperbolic cocycles
which was introduced by Avila, Bochi and Yoccoz.
Our goal is to answer a question posed by these three authors and
modified by Jacques and Short.