Abstract:The L-space Conjecture asserts the equivalence, for prime 3-manifolds, of three properties: not being an L-space (NLS), having left-orderable fundamental group (LO), and supporting a co-orientable taut foliation (CTF). We study these properties for toroidal 3-manifolds. For example, Eftekhary and Hanselman-Rasmussen-Watson have shown that toroidal homology spheres are NLS; we show that they are LO. We also show that the cyclic branched covers of a prime satellite knot are NLS and LO, and CTF if the companion is fibered. A partial extension to links allows us to show that a prime quasi-alternating link is either a (2,q) torus link or hyperbolic, generalizing Menasco's classical result for non-split alternating links. This is joint work with Steve Boyer and Ying Hu.