Irrational invariants arising from the lamplighter group
Łukasz Grabowski
Lancaster University, UK
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Abstract
We show that the Novikov–Shubin invariant of an element of the integral group ring of the lamplighter group 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 . Finally we show that the -Betti number of a matrix over the integral group ring of the group , where is a natural number greater than , can be irrational. As such the groups become the simplest known examples which give rise to irrational -Betti numbers.
Cite this article
Łukasz Grabowski, Irrational invariants arising from the lamplighter group. Groups Geom. Dyn. 10 (2016), no. 2, pp. 795–817
DOI 10.4171/GGD/366