The quadratic isoperimetric inequality for mapping tori of free group automorphisms.

*(English)*Zbl 1201.20037
Mem. Am. Math. Soc. 955, xii, 152 p. (2010).

In this memoir the authors deal with the group \(F\rtimes_\varphi\mathbb{Z}\), where \(\varphi\) is an automorphism of the free group \(F\). This group is called the algebraic mapping torus.

They prove, in their Main Theorem, that if \(F\) is a finitely generated free group and \(\varphi\) an automorphism of \(F\), then \(F\rtimes_\varphi\mathbb{Z}\) satisfies a quadratic isoperimetric inequality.

One of the corollaries says that the conjugacy problem for \(F\rtimes_\varphi\mathbb{Z}\) is solvable, a result proved before.

This memoir has three parts. In Part 1, they prove the special case where \(\varphi\) is a positive automorphism of \(F\). Part 2 is dedicated to the construction and analysis of a refined topological representative for a suitable iterate of an arbitrary automorphism of a finitely generated free group. In Part 3 they use the techniques developed in Parts 1 and 2 to prove their Main Theorem.

They prove, in their Main Theorem, that if \(F\) is a finitely generated free group and \(\varphi\) an automorphism of \(F\), then \(F\rtimes_\varphi\mathbb{Z}\) satisfies a quadratic isoperimetric inequality.

One of the corollaries says that the conjugacy problem for \(F\rtimes_\varphi\mathbb{Z}\) is solvable, a result proved before.

This memoir has three parts. In Part 1, they prove the special case where \(\varphi\) is a positive automorphism of \(F\). Part 2 is dedicated to the construction and analysis of a refined topological representative for a suitable iterate of an arbitrary automorphism of a finitely generated free group. In Part 3 they use the techniques developed in Parts 1 and 2 to prove their Main Theorem.

Reviewer: Stylianos Andreadakis (Athens)