Heegaard Floer homology groups:

\[cor:fiberedgrowth\] Let $K\subset Y$ be a fibered knot, and let $\Sigma_n(K)$ denote its $n$-fold branched cyclic cover. Then $$\dim_{{\mathbb F}}HF^{red}(\Sigma_n(K)) \ge n\cdot|\tau(\phi_K)|-1.$$ In particular, if $K$ has right- (or left-)veering monodromy then all cyclic branched covers over $K$ with sufficiently large order are not L-spaces.

Note that work of Kazez and Roberts could be used to show that high order branched cyclic covers of fibered knots with right -or left-veering monodromy are not L-spaces, but that their work wouldn’t produce the quantitative growth rate of the corollary. We expect that for most knots the rank of the reduced Floer homology of branched cyclic covers will grow at least linearly in the order of the cover, but there are examples (for example, the figure eight knot [@BaldwinGenusOne]) for which all branched cyclic covers are L-spaces.

Coming back to Question \[question2\], we should remark that there are many useful refinements of $N_Y$ made possible by taking into account further structure on the Floer groups (see the remarks after the proof of Theorem \[basicthm\]), and these refinements should be accounted for in making our question precise. For example, since our argument depends only on one [[ ]{}]{}structure at a 