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. ix
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