## Abstract

An expanding train-track map on a graph of rank n is P-small if its dilatation is bounded above by Pn. We prove that for every P there is a finite list of mapping tori X_{1}, . . ., X_{A}, with A depending only on P and not n, so that the mapping torus associated with every P-small expanding train-track map can be obtained by surgery on some X_{i}. We also show that, given an integer P>0, there is a bound M depending only on P and not n, so that the fundamental group of the mapping torus of any P-small expanding train-track map has a presentation with less than M generators and M relations. We also provide some bounds for the smallest possible dilatation.

Original language | English |
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Pages (from-to) | 44-63 |

Number of pages | 20 |

Journal | Topology and its Applications |

Volume | 180 |

DOIs | |

State | Published - 1 Feb 2015 |

### Bibliographical note

Publisher Copyright:© 2014.

## Keywords

- Geometric group theory
- Mapping tori
- Out(F)

## ASJC Scopus subject areas

- Geometry and Topology