Irrational $l^2$ invariants arising from the lamplighter group

Abstract

We show that the Novikov–Shubin invariant of an element of the integral group ring of the lamplighter group ${\mathbf{Z}}_2\wr \mathbf{Z}$ can be irrational. This disproves a conjecture of Lott and Lück. Furthermore we show that every positive real number is equal to the Novikov–Shubin invariant of some element of the real group ring of ${\mathbf{Z}}_2\wr \mathbf{Z}$. Finally we show that the $l^2$-Betti number of a matrix over the integral group ring of the group ${\mathbf{Z}}_p\wr \mathbf{Z}$, where $p$ is a natural number greater than $1$, can be irrational. As such the groups ${\mathbf{Z}}_p\wr \mathbf{Z}$ become the simplest known examples which give rise to irrational $l^2$-Betti numbers.