Volume 16, issue 3 (2016)

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Explicit rank bounds for cyclic covers

Jason DeBlois

Algebraic & Geometric Topology 16 (2016) 1343–1371

For a closed, orientable hyperbolic 3–manifold M and an onto homomorphism ϕ: π1(M) that is not induced by a fibration M S1, we bound the ranks of the subgroups ϕ1(n) for 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 14g 12.

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rank, rank gradient, JSJ decomposition
Mathematical Subject Classification 2010
Primary: 20F05, 57M10
Secondary: 20E06
Received: 4 November 2013
Revised: 19 October 2015
Accepted: 4 November 2015
Published: 1 July 2016
Jason DeBlois
Department of Mathematics
University of Pittsburgh
301 Thackeray Hall
Pittsburgh, PA 15260
United States