Density of isoperimetric spectra

TL;DR

This paper demonstrates that for finitely presented groups, the set of k-dimensional isoperimetric exponents is densely distributed in the interval [1, ∞), indicating no gaps similar to Gromov's gap in higher dimensions.

Contribution

It establishes the density of isoperimetric exponents in higher dimensions, showing the absence of gaps like Gromov's in the isoperimetric spectrum.

Findings

The set of k-dimensional isoperimetric exponents is dense in [1, ∞) for k > 1.

No higher-dimensional analogue of Gromov's gap (1,2) exists.

The result applies to finitely presented groups.

Abstract

We show that the set of k-dimensional isoperimetric exponents of finitely presented groups is dense in the interval [1, \infty) for k > 1. Hence there is no higher-dimensional analogue of Gromov's gap (1,2) in the isoperimetric spectrum.