Slope detection and toroidal 3-manifolds
The $L$-space conjecture asserts the equivalence, for prime 3-manifolds, of three properties: not being an $L$-space, having a left-orderable fundamental group, and admitting a co-oriented taut foliation. We investigate these properties for toroidal $3$-manifolds using various notions of slope detection. Our main technical result gives sufficient conditions for certain slopes on the boundaries of rational homology solid tori to be detected by left-orders, foliations, and Heegaard Floer homology, using Thurston's universal circle actions, Li's result on laminar branched surfaces, and Rasmussen-Rasmussen's result on L-space intervals, respectively. This leads to a proof that toroidal integer homology spheres have left-orderable fundamental groups, as predicted by the $L$-space conjecture. It also allows us to show that the cyclic branched covers of prime satellite knots are not $L$-spaces and have left-orderable fundamental groups, as conjectured by Gordon and Lidman. Similarly we show that a cyclic branched cover of a satellite knot admits a co-oriented taut foliation when it has a fibred companion. A partial extension of these results to toroidal links leads to a proof that prime quasi-alternating links are either hyperbolic or $(2, m)$-torus links, which generalises Menasco's classical theorem that non-split alternating links are either hyperbolic or $(2, m)$-torus links.