Irrational l(2)-invariants arising from the lamplighter group

摘要

We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z(2) (sic) Z can be irrational. This disproves a conjecture of Lott and Luck. 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) (sic) Z. Finally we show that the l(2)-Betti number of a matrix over the integral group ring of the group Z(p) (sic) Z, where p is a natural number greater than 1, can be irrational. As such the groups Z(p) (sic) Z become the simplest known examples which give rise to irrational l(2)-Betti numbers.