## Algebraic & Geometric Topology

### Explicit rank bounds for cyclic covers

Jason DeBlois

#### Abstract

For a closed, orientable hyperbolic $3$–manifold $M$ and an onto homomorphism $ϕ: π1(M) → ℤ$ that is not induced by a fibration $M → S1$, we bound the ranks of the subgroups $ϕ−1(nℤ)$ for $n ∈ ℕ$, 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

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

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

https://projecteuclid.org/euclid.agt/1511895848

Digital Object Identifier
doi:10.2140/agt.2016.16.1343

Mathematical Reviews number (MathSciNet)
MR3523041

Zentralblatt MATH identifier
1354.57008

#### Citation

DeBlois, Jason. Explicit rank bounds for cyclic covers. Algebr. Geom. Topol. 16 (2016), no. 3, 1343--1371. doi:10.2140/agt.2016.16.1343. https://projecteuclid.org/euclid.agt/1511895848

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