There is only one gap in the isoperimetric spectrum

Abstract

The closure of the set of isoperimetric exponents for finitely presented groups is $ \{1\} \cup [2,\infty $ . For each pair of positive integers $ p \ge q $ , one can construct groups with aspherical presentations for which the Dehn function is $ \simeq n^{2\alpha} $ , where $ \alpha\,=\{\rm log}_{2}(2p/q) $ .