Slope detection, toroidal 3-manifolds, and the L-space conjecture

We will describe an approach to the L-space conjecture for toroidal 3-manifolds based on notions of slope detection corresponding to the three properties in the conjecture. We will give some applications to toroidal manifolds with small first homology, cyclic branched covers of satellite links, and Dehn surgeries on satellite knots. This is joint work with Steve Boyer and Ying Hu.