Explicit rank bounds for cyclic covers (Q2630667)

Let \(M\) be a closed, orientable hyperbolic \(3\)-manifold, and \(S\subset M\) be a connected, two-sided \textit{incompressible} surface, namely an embedded surface (not equal to a 2-sphere) with a \(\pi_1\)-injective inclusion map on fundamental groups. By cutting \(M\) along \(S\), one obtains a new \(3\)-manifold \(M'\), and a graph \(\mathcal G\) (associated to the \textit{graph of spaces decomposition} of \(M\)), with a single edge, corresponding to \(S\), and one or two vertices corresponding to the connected components of \(M'\). By Bass-Serre's theory, these data provide also an action of the fundamental group \(\pi_1 M\) on a tree \(T_S\), such that \(\mathcal G = T_S / \pi_1 M\). On the other hand, a general action of a group \(G\) on a tree \(T\) is said to be \textit{\(k\)-acylindrical} if no element of the group (except the identity) fixes a segment of \(T\) of length \(> k\), whereas a surface \(S\) of the \(3\)-manifold \(M\) is called a \textit{semifiber} if it separates \(M\) into a disjoint union of twisted \(I\)-bundles over the surface double covered by \(S\). Somehow, any homotopy between two closed curves in \(S\) can be decomposed into several particular homotopies of \(M'\), and there is actually a connection between the number of these homotopies and the action of \(\pi_1 M\) on \(T_S\) (as outlined in \textit{Z. Sela}'s paper [Invent. Math. 129, No. 3, 527--565 (1997; Zbl 0887.20017)]). Roughly speaking, the aim of the present paper is to specify this last observation, to give a detailed proof of it, to provide an upper bound for the number of such homotopies in some special cases, and to apply all these results for bounding the rank of the fundamental groups of cyclic covers of certain \(3\)-manifolds. The main theorem (Theorem 4.1) is a little bit technical, but it basically shows that if \(M\) is a closed, orientable hyperbolic 3-manifold and \(S\subset M\) is a connected, two-sided incompressible surface \(S\) of genus \(g\) that is not a fiber nor a semifiber, then certain homotopies in \((M,S)\) have bounded length (or, more precisely, the length of a non-degenerate reduced homotopy in \((M,S)\) is less than or equal to \(14g -12\)). In the theorem above, a \textit{homotopy in \((M,S)\)} is a homotopy \(H\) from some topological space \(K\) to \(M\) such that \(H(K \times \partial I) \subset S\). The homotopy \(H\) is called \textit{non-degenerate} if \(H_{\ast} (\pi_1 K) \neq {1}\), and it is \textit{reduced of length k} if it can be gotten from \(k\) homotopies in \((M',\partial M')\), satisfying several technical conditions. As a consequence of Theorem 4.1, the author shows that, for a closed, orientable hyperbolic \(3\)-manifold \(M\) with a connected, two-sided incompressible surface \(S\subset M\) of genus \(g\), that is not a fiber or a semifiber, the action of the fundamental group \(\pi_1 M\) on the tree \(T_S\) is exactly \((14g-12)\)-acylindrical. (Note that if \(S\) is a fiber, then any element of its fundamental group fixes the whole \(T_S\); whereas, whenever \(S\) is a semifiber, it can be lifted to a fiber of a fibration of the two-fold cover of \(M\).) Finally, the author applies the above-mentioned results to cyclic covers. Let \(M\) be a closed, orientable hyperbolic \(3\)-manifold, and \(\phi : \pi_1 M \to \mathbb Z\) an onto homomorphism which is not induced by a fibration \(M\to S^1\). Then the minimal number of generators of the subgroups \(\phi ^{-1} (n \mathbb Z)\) is bounded below, linearly in \(n\) (an explicit formula is provided). The general idea being that, given such a \(\phi\), there exists an associated surface \(S\subset M\) that is in some sense dual to \(\phi\), and hence a graph of spaces decomposition of \(M\) and its associated \(\pi_1 M\) action on the tree \(T_S\).