Irrational l2-invariants arising from the lamplighter group

TL;DR

This paper demonstrates that certain invariants associated with the lamplighter group can be irrational, disproving previous conjectures and revealing new properties of these mathematical objects.

Contribution

It proves the irrationality of Novikov-Shubin invariants and l2-Betti numbers for lamplighter groups, challenging existing beliefs and expanding understanding of these invariants.

Findings

Novikov-Shubin invariant of lamplighter group elements can be irrational

Every positive real number can be realized as a Novikov-Shubin invariant

l2-Betti number of certain matrices over lamplighter groups can be irrational

Abstract

We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z_2 \wr Z can be irrational. This disproves a conjecture of Lott and Lueck. Furthermore we show that every positive real number is equal to the Novikov-Shubin invariant of some element of the real group ring of Z_2 \wr Z. Finally we show that the l2-Betti number of a matrix over the integral group ring of the group Z_p \wr Z, p>1, can be irrational, and so the groups Z_p \wr Z become the simplest known groups which give rise to irrational l2-Betti numbers.