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Taut foliations in branched cyclic covers and left-orderable groups. (English) Zbl 1445.57017
The $$L$$-space Conjecture is that for a closed, connected, irreducible, orientable $$3$$-manifold $$M$$, the following statements are equivalent: (1) $$M$$ is not a Heegaard Floer $$L$$-space, (2) $$M$$ admits a co-orientable taut foliation, (3) $$\pi_1 (M)$$ is left-orderable.
The conjecture is known to be true in many cases, for example when $$M$$ has positive first Betti number, or is a non-hyperbolic $$3$$-manifold, or is a graph manifold. Gordon and Lidman call $$M$$ excellent, if $$M$$ satisfies conditions (2) and (3). Since (2) and (3) imply (1), the conjecture holds for excellent $$3$$-manifolds.
The authors consider hyperbolic fibered knots $$K$$ with monodromy $$h$$ in an oriented integer homology 3-sphere and show that the $$n$$-fold cyclic branched covering $$\Sigma_n (K)$$ is excellent for $$n |c(h)| \geq 1$$, where $$c(h)$$ is the fractional Dehn twist coefficient of $$h$$. In particular, if $$c(h)\neq 0$$ and $$g$$ is the genus of $$K$$, then $$\Sigma_n (K)$$ is excellent for $$n \geq 2(2g-1)$$. Considering closed braids, the authors complete the proof of the $$L$$-space conjecture for closed, connected, orientable, irreducible $$3$$-manifolds containing a genus one fibred knot. They also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibered knot in an integer homology $$3$$-sphere is excellent, even if the surgered manifold is not, and that the same holds for many branched covers of satellite knots with braided patterns.
The paper is well written by providing in the first six chapters (after the introduction) all the necessary definitions and facts about cyclic branched coverings, mapping class groups and braids, fractional Dehn twist coefficients, Euler classes of representations and oriented circle bundles, and the universal circle representation associated to a rational homology 3- sphere endowed with a co-oriented taut foliation. In particular they give a detailed proof of the fact, due to Thurston, that the Euler class of the universal circle representation coincides with that of the associated foliation’s tangent bundle. No proof of this result has previously appeared in the literature.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57R30 Foliations in differential topology; geometric theory 20F60 Ordered groups (group-theoretic aspects) 57M99 General low-dimensional topology 20F36 Braid groups; Artin groups
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