Quadratic isoperimetric inequality for mapping tori of polynomially growing automorphisms of free groups
Let \(X\) be a finite cell complex and \(\widetilde X\) its universal cover. Assigning to each edge of \(X\) a length of one induces a combinatorial metric on the one-skeleton on \(\widetilde X\) which may be extended to a metric \(d_{\widetilde X}\) on \(\widetilde X\). There is a similar notion of combinatorial area in \(X\). For an edge path \(c\subset X\) or \(c\subset\widetilde X\) denote by \(|c|\) the combinatorial length of \(c\). A function \(f\colon\mathbb{N}\to\mathbb{N}\) is an isoperimetric function for \(X\) if for all \(n\) and all closed edge paths \(c\subset\widetilde X\) with \(|c|\leq n\) there is a disk \(\Delta\colon D^2\to\widetilde X\) with combinatorial area \(\text{Area}(\Delta)\leq f(n)\). We say that \(X\) satisfies the quadratic isoperimetric inequality if it possesses a quadratic isoperimetric function. A group satisfies quadratic isoperimetric inequality if it is the fundamental group of a finite cell complex which satisfies quadratic isoperimetric inequality. The main result of this paper is that the semidirect product \(F_n\triangleleft_\varphi\mathbb{Z}\) (mapping torus) of a free group \(F_n\) of finite rank \(n\) by \(\mathbb{Z}\) via a polynomially growing automorphism \(\varphi\) of \(F_n\), satisfies the quadratic isoperimetric inequality. The method of proof is geometric and is based on work by \textit{M. Bestvina}, \textit{M. Feighn} and \textit{M. Handel} [The Tits alternative for \(\text{Out}(F_n)\). I, Ann. Math. (2) 151, No. 2, 517-623 (2000; Zbl 0984.20025) and II, preprint (1996)]. The author introduces the concept of a Kolchin map which enables her to use induction on the degree of the polynomial growth of the automorphism. So she proves that a linearly growing map \(f\colon G\to G\) satisfies the quadratic isoperimetric inequality. This is the induction step to prove that if \(f\colon G\to G\) is a Kolchin map that grows polynomially with degree \(d\), the mapping torus \(M_f=G\times I|_{(x,0)\sim(f(x),1)}\) satisfies the quadratic isoperimetric inequality. The main result then comes from this and a result contained in the above-mentioned papers of M. Bestvina, M. Feighn and M. Handel.
scientific article; zbMATH DE number 1545125