L2-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices

Petersen, Henrik Densing; Sauer, Roman; Thom, Andreas

doi:10.1112/topo.12056

Cited by 12 publications

(10 citation statements)

References 39 publications

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“…To even state Theorem 1 in that generality, recent advances in the theory of ℓ 2 -Betti numbers were necessary. ℓ 2 -Betti numbers of discrete groups enjoy a long history but it was not until recently that ℓ 2 -Betti numbers were defined for arbitrary unimodular lcsc groups by Petersen [19], and a systematic theory analogous to the discrete case emerged [13,19,20]. Earlier studies of ℓ 2 -Betti numbers of locally compact groups in specific cases can be found in [4,6,10].…”

Section: Introductionmentioning

confidence: 99%

Vanishing of $\ell ^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

Sauer

Schrödl

2018

Fund. Math.

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We provide a proof that the vanishing of ℓ 2 -Betti numbers of unimodular locally compact second countable groups is an invariant of coarse equivalence. To this end, we define coarse ℓ 2 -cohomology for locally compact groups and show that it is isomorphic to continuous cohomology for unimodular groups and invariant under coarse equivalence.2010 Mathematics Subject Classification. Primary 20F65; Secondary 22D99.

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Section: Introductionmentioning

confidence: 99%

Vanishing of $\ell ^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

Sauer

Schrödl

2018

Fund. Math.

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“…It is based on the new notion of relative soficity. We will not discuss approximation for sequences of lattices in locally compact groups as in [1,19].…”

Section: Introductionmentioning

confidence: 99%

Characters, $$L^2$$L2-Betti numbers and an equivariant approximation theorem

Kionke¹

2018

Math. Ann.

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Let G be a group with a finite subgroup H. We define the L 2multiplicity of an irreducible representation of H in the L 2 -homology of a proper G-CW-complex. These invariants generalize the L 2 -Betti numbers. Our main results are approximation theorems for L 2 -multiplicities which extend the approximation theorems for L 2 -Betti numbers of Lück, Farber and Elek-Szabó respectively. The main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups. We discuss applications to the cohomology of (arithmetic) groups.

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“…In [96] H. D. Petersen, R. Sauer and A. Thom present a general Lück approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. (see [23] for the definition of R[ [G]] and its properties…”

Section: Other Variations Of the Lück Approximationmentioning

confidence: 99%