Optimal isoperimetric inequalities for abelian-by-free groups
An isoperimetric inequality for a finitely presented group gives an upper bound for the number of relators needed to show that a word in the generators represents the trivial element of the group, in terms of the length \(n\) of the word. In an equivalent geometric formulation, one considers the combinatorial area of a planar diagram or van Kampen diagram associated to the word. By taking the optimal bound for words of length \(\leq n\) one obtains the Dehn function of the given presentation. Dehn functions of different presentations of the same group are quasi-Lipschitz equivalent. The main theorem of the present paper determines the Dehn functions of groups which are extensions of a finitely generated abelian group \(A\) (the normal subgroup) by a finitely generated free group \(F\): they are quasi-Lipschitz equivalent to either a polynomial or an exponential function. It is polynomial iff there exists a subgroup of finite index of \(F\) whose image in the operation of \(F\) on \(A\)/torsion consists of unipotent elements in \(GL_m (\mathbb{Z})\), and the degree of the polynomial is determined in terms of these unipotent elements. This generalizes the special case of fundamental groups of torus bundles over the circle considered in a paper by the author and \textit{Gersten} [Q. J. Math., Oxf. II. Ser. 47 (1996)]. The main theorem implies upper bounds on Dehn functions for fundamental groups of torus bundles over compact non-positively curved manifolds. To obtain upper bounds on Dehn functions normal forms from the geometric viewpoint of combings are used. Lower bounds are obtained by exploiting the notion of a \(t\)-corridor introduced in the previous paper cited above. The resulting diagrammatic arguments reduce the calculation of lower bounds for the Dehn functions of semidirect products \(\mathbb{Z}^m \rtimes F\) to the task of relating the norm of an element in the image of \(F\) in \(GL_m(\mathbb{Z})\) to the length of the shortest word in its preimage.
scientific article; zbMATH DE number 794439