Isoperimetric functions of finitely generated nilpotent groups
The author makes the following definitions which generalizes the notion of isoperimetric function. Let \(P=\langle{\mathcal F}\mid{\mathcal R}\rangle\) be a finite presentation of a group \(G\), let \(F\) be the free group freely generated by \(\mathcal F\) and let \(H\) be a finitely generated subgroup of \(G\), generated by a subset \({\mathcal E}\subset{\mathcal F}\). Let \(E\) be a subgroup of \(F\) generated by \(\mathcal E\). A rewriting process \(\rho\) from \(G\) to \(H\) relative to \((P,{\mathcal E})\) is a partial map \(\rho\colon F\to E\) defined on all words \(w\in H\) satisfying \(\rho(w)=_G w\) and \(\rho(1)=1\). The distorsion of the rewriting process \(\rho\), \(\delta_\rho\), is defined on the set of natural integers by setting \(\delta_\rho(n)\) to be the maximal length of \(\rho(w)\) for all \(w\in H\) with \(|w|\leq n\). The isoperimetric function \(\Phi_\rho\) of the rewriting process \(\rho\) is defined by \[ \Phi_\rho(n)=\max\{\Delta_P(w^{-1}\rho(w))\mid w\in H\text{ and }|w|\leq n\}. \] In this formula, \(\Delta_P\) is the so-called area function, defined on the normal closure of the set of relators, by \[ \Delta_P(w)=\min\{m\mid w=\prod_{i=1}^m u_i^{-1}r_i^{\varepsilon_i}u_i \text{ for } u_i\in F,\;r_i\in{\mathcal R},\;\varepsilon_i=\pm 1\}. \] The author proves the following Theorem. Let \(G\) be a finitely presented nilpotent group of class \(c\) and \(H\) a subgroup of \(G\). There exists a rewriting process \(\rho\) from \(G\) to \(H\), relative to some finite presentation of \(G\) and some finite set of generators of \(H\), such that \(\delta_\rho(n)\preceq n^c\) and \(\Phi_\rho(n)\preceq n^{2c}.\) (The notation \(f\preceq g\) means that there exists a constant \(K\) such that for all \(n\), we have \(f(n)\leq Kg(Kn)+Kn\).) As a corollary, the author obtains the following Theorem. Let \(G\) be a finitely presented nilpotent group of class \(c\). Then the usual isoperimetric function \(\Phi_G\) of \(G\) satisfies \(\Phi_G(n)\preceq n^{2c}\).
scientific article; zbMATH DE number 1398626