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Irrational l2 invariants arising from the lamplighter group

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Irrational l2 invariants arising from the lamplighter group. / Grabowski, Łukasz.
In: Groups, Geometry, and Dynamics, Vol. 10, No. 2, 2016, p. 795-817.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Grabowski, Ł 2016, 'Irrational l2 invariants arising from the lamplighter group', Groups, Geometry, and Dynamics, vol. 10, no. 2, pp. 795-817. https://doi.org/10.4171/GGD/366

APA

Grabowski, Ł. (2016). Irrational l2 invariants arising from the lamplighter group. Groups, Geometry, and Dynamics, 10(2), 795-817. https://doi.org/10.4171/GGD/366

Vancouver

Grabowski Ł. Irrational l2 invariants arising from the lamplighter group. Groups, Geometry, and Dynamics. 2016;10(2):795-817. doi: 10.4171/GGD/366

Author

Grabowski, Łukasz. / Irrational l2 invariants arising from the lamplighter group. In: Groups, Geometry, and Dynamics. 2016 ; Vol. 10, No. 2. pp. 795-817.

Bibtex

@article{39baf1576824457990970a57b529e1de,
title = "Irrational l2 invariants arising from the lamplighter group",
abstract = "We show that the Novikov–Shubin invariant of an element of the integral group ring of the lamplighter group Z2≀ZZ2≀Z can be irrational. This disproves a conjecture of Lott and L{\"u}ck. 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≀ZZ2≀Z. Finally we show that the l2l2-Betti number of a matrix over the integral group ring of the group Zp≀ZZp≀Z, where pp is a natural number greater than 11, can be irrational. As such the groups Zp≀ZZp≀Z become the simplest known examples which give rise to irrational l2l2-Betti numbers.",
keywords = "l2l2-invariants, Atiyah conjecture, Novikov–Shubin invariants, l2l2-Betti numbers",
author = "{\L}ukasz Grabowski",
year = "2016",
doi = "10.4171/GGD/366",
language = "English",
volume = "10",
pages = "795--817",
journal = "Groups, Geometry, and Dynamics",
issn = "1661-7207",
publisher = "European Mathematical Society Publishing House",
number = "2",

}

RIS

TY - JOUR

T1 - Irrational l2 invariants arising from the lamplighter group

AU - Grabowski, Łukasz

PY - 2016

Y1 - 2016

N2 - We show that the Novikov–Shubin invariant of an element of the integral group ring of the lamplighter group Z2≀ZZ2≀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 Z2≀ZZ2≀Z. Finally we show that the l2l2-Betti number of a matrix over the integral group ring of the group Zp≀ZZp≀Z, where pp is a natural number greater than 11, can be irrational. As such the groups Zp≀ZZp≀Z become the simplest known examples which give rise to irrational l2l2-Betti numbers.

AB - We show that the Novikov–Shubin invariant of an element of the integral group ring of the lamplighter group Z2≀ZZ2≀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 Z2≀ZZ2≀Z. Finally we show that the l2l2-Betti number of a matrix over the integral group ring of the group Zp≀ZZp≀Z, where pp is a natural number greater than 11, can be irrational. As such the groups Zp≀ZZp≀Z become the simplest known examples which give rise to irrational l2l2-Betti numbers.

KW - l2l2-invariants

KW - Atiyah conjecture

KW - Novikov–Shubin invariants

KW - l2l2-Betti numbers

U2 - 10.4171/GGD/366

DO - 10.4171/GGD/366

M3 - Journal article

VL - 10

SP - 795

EP - 817

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

SN - 1661-7207

IS - 2

ER -