Explicit rank bounds for cyclic covers

DeBlois, Jason

doi:10.2140/agt.2016.16.1343

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Explicit rank bounds for cyclic covers

Jason DeBlois

Algebraic & Geometric Topology 16 (2016) 1343–1371

DOI: 10.2140/agt.2016.16.1343

Abstract

For a closed, orientable hyperbolic $3$ –manifold $M$ and an onto homomorphism $ϕ : π_{1} (M) \to ℤ$ that is not induced by a fibration $M \to S^{1}$ , we bound the ranks of the subgroups $ϕ^{- 1} (n ℤ)$ 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 $(M, S)$ has length at most $14 g - 12$ .

Keywords

rank, rank gradient, JSJ decomposition

Mathematical Subject Classification 2010

Primary: 20F05, 57M10

Secondary: 20E06

References

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

Volume 16, issue 3 (2016)