Abstract

Abstract Let G be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from G to ℤ have finitely generated kernel, and we compute the rank of the kernel. We thus describe all possible ways of expressing G as the mapping torus of a free group automorphism. This is similar to the case for 3-manifold groups, and different from the case of mapping tori of exponentially growing free group automorphisms. The proof uses a hierarchical decomposition of G and requires determining the Bieri–Neumann–Strebel invariant of the fundamental group of certain graphs of groups.

Highlights

  • Given an automorphismof a group F, one may form its mapping torus G D F Ì Z D hF; t j t 1f t D.f /i and obtain an exact sequence 1 ! F ! G ! Z ! 1

  • Any fibration over the circle leads to such an exact sequence, with F the fundamental group of the fiber andinduced by the monodromy

  • We prove Theorem 1.1 by induction, using the fact that G admits a hierarchy: it may be iteratively split along cyclic groups until vertex groups are Z2

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Summary

Introduction

Given an automorphismof a group F , one may form its mapping torus G D F Ì Z D hF; t j t 1f t D.f /i and obtain an exact sequence 1 ! F ! G ! Z ! 1. Any ' such that no '.ti / is 0 expresses G as the mapping torus of an automorphism of a finitely generated free group. We will see that G cannot be written as the mapping torus of an injective, non-surjective, endomorphism of a finitely generated free group. These groups are exactly the non-solvable GBS1 groups having a non-trivial center [23] (a GBS1 group with trivial center has †.G/ empty). We show that the set of ranks of fibers is an arithmetic progression

The BNS invariant of a graph of groups
Generalized Baumslag–Solitar groups
Hierarchies
Mapping tori of polynomially growing free group automorphisms
UPG automorphisms
Examples
The rank of the fiber
Finite order automorphisms and GBS groups with center

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