Introduction

Associated to an automorphism φ of any group G one has the algebraic mapping

torus G

φ

Z. In this paper we shall be concerned with the case where G is a finitely

generated free group, denoted F . We seek to understand the complexity of the word

problem in the groups F

φ

Z as measured by their Dehn functions.

The class of groups of the form F

φ

Z has been the subject of intensive

investigation in recent years and a rich structure has begun to emerge in keeping

with the subtlety of the classification of free group automorphisms [4], [6] [7], [23],

[29], [35]. (See [2] and the references therein.) Bestvina–Feighn and Brinkmann

proved that if F

φ

Z doesn’t contain a free abelian subgroup of rank two then it

is hyperbolic [3], [18], i.e. its Dehn function is linear. Epstein and Thurston [22]

proved that if φ is induced by a surface automorphism (in the sense discussed below)

then F

φ

Z is automatic and hence has a quadratic Dehn function. The question

of whether or not all non-hyperbolic groups of the form F

φ

Z have quadratic

Dehn functions has attracted a good deal of attention.

Main Theorem. If F is a finitely generated free group and φ is an automor-

phism of F then F

φ

Z satisfies a quadratic isoperimetric inequality.

Papasoglu [33] proved that if a finitely presented group satisfies a quadratic

isoperimetric inequality, then all of its asymptotic cones are simply connected.

Corollary A. If F is a finitely generated free group and φ is an automorphism

of F then every asymptotic cone of F

φ

Z is simply connected.

Ol’shanskii and Sapir [32, Theorem 2.5] proved that if a multiple HNN ex-

tension of a free group has Dehn function less than

n2

log n (with a somewhat

technical definition of ‘less than’) then it has a solvable conjugacy problem. Our

Main Theorem shows that free-by-cyclic groups fall into this class.

Corollary B. If F is a finitely generated free group and φ is an automorphism

of F , then the conjugacy problem for F

φ

Z is solvable.

Corollary B was first proved in [8] using different methods.

Gromov [26] proved that a finitely presented group is hyperbolic if and only if

its Dehn function is linear. He also proved that if a Dehn function is subquadratic

then it must be linear. Thus if one ranks groups according to the complexity of their

Dehn functions, the groups that have a quadratic Dehn function demand particular

attention. The nature of these groups is far from clear for the moment; in particular

it is unclear what they have in common. It is not known, for example, whether they

all have a solvable conjugacy problem; nor is it known whether the isomorphism

problem is solvable amongst them. Our Main Theorem provides a rich source of

new examples on which to test such questions.

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