On an isoperimetric inequality for infinite finitely generated groups
Let \(\Gamma\) be an infinite discrete group with a fixed finite symmetric set \(S=S^{-1}\) of generators. Then: \(|A|\leq\sum_{\gamma\in\partial A}\text{dist}(e,\gamma)\) where \(|A|\) denotes the number of elements in \(A\) and \(\partial A=\{\gamma\not\in A;\;s\gamma\in A,\;s\in S\}\); \(\text{dist}(e,\gamma)\) is the minimal number of generators needed to represent \(\gamma\in\Gamma\). The proof rests on the construction of a finitely additive and invariant measure on certain sets of geodesics in \(\Gamma\). Using a result of \textit{J.-C. Sikorav} [Growth of a primitive of a differential form, preprint, Toulouse (1998)] one obtains a result for the universal cover \(\widetilde M^n\) of a compact Riemannian manifold with infinite fundamental group: There is a primitive of the volume form on \(\widetilde M^n\) which has at most linear growth.
scientific article; zbMATH DE number 1472310