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Alexander varieties and largeness of finitely presented groups. (English) Zbl 1316.57004

A 2-complex \(X\) is said to be large if there is a finite cover of \(X\) whose fundamental group surjects on a free group of rank 2. The main result of this paper is:
{ Theorem 1.1} Let \(X\) be a finite CW-complex. The following are equivalent:
(1)
\(X\) is large.
(2)
There is a finite cover \(Y\to X\) such that for each \(n\in\mathbb N\) there is a finite abelian cover \(Y_n\to Y\) with \(b_1(Y_n)\geq n\).
(3)
There is a finite cover \(Y\to X\) such that the Alexander variety of \(Y\) contains infinitely many torsion points.
As a first corollary the author recovers a special case of a result of Cooper, Long and Reid: A noncompact, finite volume, fibered, hyperbolic 3-manifold is large. Then:
{ Corollary 1.3} Let \(S\) be an orientable surface with finite negative Euler-characteristic and let \(\psi \) be a nontrivial element of the mapping class group of \(S\).
(1)
If \(\psi \) is not pseudo-Anosov then the mapping torus \(T_\psi \) is large.
(2)
If there is no lift \(\tilde{\psi }\) to a finite cover such that the spectral radius of the action of \(\tilde{\psi }\) on \(H_1(\tilde{S}, \mathbb C)\) is greater than 1, then \(T_\psi \) is large.
The last corollary states that if a group \(G\) has a finite presentation with at least 2 more generators than relators, then \(G\) is large.

MSC:

57M20 Two-dimensional complexes (manifolds) (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions
20F65 Geometric group theory
20M07 Varieties and pseudovarieties of semigroups
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References:

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