Associated to an automorphism φ of any group G one has the algebraic mapping
torus G
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
Z as measured by their Dehn functions.
The class of groups of the form F
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
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
Z is automatic and hence has a quadratic Dehn function. The question
of whether or not all non-hyperbolic groups of the form F
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
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
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
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
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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