#### Volume 16, issue 3 (2016)

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Explicit rank bounds for cyclic covers

### Jason DeBlois

Algebraic & Geometric Topology 16 (2016) 1343–1371
##### Abstract

For a closed, orientable hyperbolic $3$–manifold $M$ and an onto homomorphism $\varphi :{\pi }_{1}\left(M\right)\to ℤ$ that is not induced by a fibration $M\to {S}^{1}\phantom{\rule{0.3em}{0ex}}$, we bound the ranks of the subgroups ${\varphi }^{-1}\left(nℤ\right)$ for $n\in ℕ$, below, linearly in $n$. The key new ingredient is the following result: if $M$ is a closed, orientable hyperbolic $3$–manifold and $S$ is a connected, two-sided incompressible surface of genus $g$ that is not a fiber or semifiber, then a reduced homotopy in $\left(M,S\right)$ has length at most $14g-12$.

##### Keywords
rank, rank gradient, JSJ decomposition
##### Mathematical Subject Classification 2010
Primary: 20F05, 57M10
Secondary: 20E06
##### Publication
Received: 4 November 2013
Revised: 19 October 2015
Accepted: 4 November 2015
Published: 1 July 2016
##### Authors
 Jason DeBlois Department of Mathematics University of Pittsburgh 301 Thackeray Hall Pittsburgh, PA 15260 United States http://www.pitt.edu/~jdeblois