Group ring elements with large spectral density

Abstract

Given \({\delta }>0\) we construct a group \(G\) and a group ring element \(S\in \mathbb Z[G]\) such that the spectral measure \(\mu \) of \(S\) fulfils \(\mu ((0,{\varepsilon })) > \frac{C}{|\log ({\varepsilon })|^{1+{\delta }}}\) for small \({\varepsilon }\). In particular the Novikov-Shubin invariant of any such \(S\) is \(0\). The constructed examples show that the best known upper bounds on \(\mu ((0,{\varepsilon }))\) are not far from being optimal.