Group ring elements with large spectral density

Grabowski, Łukasz (2015) Group ring elements with large spectral density. Mathematische Annalen, 363 (1). pp. 637-656. ISSN 0025-5831

Abstract

Given δ>0δ>0 we construct a group GG and a group ring element S∈Z[G]S∈Z[G] such that the spectral measure μμ of SS fulfils μ((0,ε))>C|log(ε)|1+δμ((0,ε))>C|log⁡(ε)|1+δ for small εε. In particular the Novikov-Shubin invariant of any such SS is 00. The constructed examples show that the best known upper bounds on μ((0,ε))μ((0,ε)) are not far from being optimal.