Algebraic & Geometric Topology

Explicit rank bounds for cyclic covers

Jason DeBlois

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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.

Article information

Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1343-1371.

Received: 4 November 2013
Revised: 19 October 2015
Accepted: 4 November 2015
First available in Project Euclid: 28 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F05: Generators, relations, and presentations 57M10: Covering spaces
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations

rank rank gradient JSJ decomposition


DeBlois, Jason. Explicit rank bounds for cyclic covers. Algebr. Geom. Topol. 16 (2016), no. 3, 1343--1371. doi:10.2140/agt.2016.16.1343.

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