  The term degenerate is used to describe abelian varieties whose Hodge rings
contain exceptional cycles -- Hodge cycles that are not generated by divisor
classes. We can see the effect of the exceptional cycles on the structure of an
abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate
group. In this article we examine degeneracy through these different but
related lenses. We specialize to a family of abelian varieties of Fermat type,
namely Jacobians of hyperelliptic curves of the form $y^2=x^m-1$. We prove that
the Jacobian of the curve is degenerate whenever $m$ is an odd, composite
integer. We explore the various forms of degeneracy for several examples, each
illustrating different phenomena that can occur.
