Groups, Geometry, and Dynamics
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Published online: 2016-06-09
Irrational $l^2$ invariants arising from the lamplighter groupŁukasz Grabowski (1) Lancaster University, UK
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.
Keywords: $l^2$-invariants, Atiyah conjecture, Novikov–Shubin invariants, $l^2$-Betti numbers
Grabowski Łukasz: Irrational $l^2$ invariants arising from the lamplighter group. Groups Geom. Dyn. 10 (2016), 795-817. doi: 10.4171/GGD/366