Irrational l2-invariants arising from the lamplighter group

2010/09/01 by Łukasz Grabowski, Grabowski, Łukasz
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #math.GR #math.GT

paper · pdf · doi:10.48550/arxiv.1009.0229

26 pages, 11 figures, v4: changes suggested by a referee (including fixing the proof of Lemma 11); To appear in Groups Geom. Dyn

arxiv created 2015/04/26 · arxiv updated 2015/04/28

Abstract

We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z₂ \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 Z₂ \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.

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