  This is a study of orbifold-quotients of quantum groups (quantum orbifolds
$\Theta \rightrightarrows G_q$). These structures have been studied extensively
in the case of the quantum $SU_2$ group. I will introduce a generalized
mechanism which allows one to construct quantum orbifolds from any compact
simple and simply connected quantum group. Associated with a quantum orbifold
there is an invariant subalgebra as well as a crossed product algebra. For each
spin quantum orbifold, there is a unitary equivalence class of Dirac spectral
triples over the invariant subalgebra, and for each effective spin quantum
orbifold associated with a finite group action, there is a unitary equivalence
class of Dirac spectral triples over the crossed product algebra. As an
application I will study a Hopf-equivariant Fredholm index problem.
