Asymptotic cones and polynomial isoperimetric inequalities
Let \(\Gamma\) be a finitely generated group equipped with a word metric \(d\). \textit{M. Gromov} [Geometric group theory. Vol. 2: Asymptotic invariants of infinite groups, Lond. Math. Soc. Lect. Note Ser. 182 (1993; Zbl 0841.20039)], defines an asymptotic cone \(\text{Cone}_\omega\Gamma\) for \(\Gamma\) by taking a non-principal ultrafilter \(\omega\) on the integers and defining \(\text{Cone}_\omega\Gamma\) to be the limit of the sequence of spaces \(\lim_\omega(\Gamma,(1/n)d)\). In the same paper, Gromov initiated the study of the extent to which properties of \(\Gamma\) are encoded in the geometry and topology of \(\text{Cone}_\omega\Gamma\). He proved for instance that \(\Gamma\) is word hyperbolic if and only if \(\text{Cone}_\omega\Gamma\) is an \(R\)-tree. Gromov proved also that if for every non-principal ultrafilter \(\omega\) the asymptotic cone \(\text{Cone}_\omega\Gamma\) is simply connected, then \(\Gamma\) is finitely presented and satisfies a polynomial isoperimetric inequality. He asked the question of whether the converse result is true, and \textit{P. Papasoglu} solved part of the problem in his paper [J. Differ. Geom. 44, No. 4, 789-806 (1996; Zbl 0893.20029)], where he proved that if \(\Gamma\) satisfies an isoperimetric inequality, then \(\text{Cone}_\omega\Gamma\) is simply connected for every non-principal ultrafilter \(\omega\). In this paper, the author proves the following Theorem. There exist finitely presented groups \(\Gamma\) such that \(\text{Cone}_\omega\Gamma\) is not simply connected for any non-principal ultrafilter \(\omega\), yet \(\Gamma\) does satisfy a polynomial isoperimetric inequality.
scientific article; zbMATH DE number 1307841
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