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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - L2 -Betti numbers arising from the lamplighter group
AU - Ara, P.
AU - Claramunt, J.
PY - 2021/12/31
Y1 - 2021/12/31
N2 - We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute ℓ2-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational ℓ2-Betti numbers arising from the lamplighter group algebra Q[Z2≀ Z]. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational ℓ2-Betti numbers from the algebras Q[Zn≀ Z] , where n≥ 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra O(n¯) , where n¯ is a supernatural number. We compute its ∗ -regular closure, and this allows us to fully characterize the set of O(n¯) -Betti numbers.
AB - We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute ℓ2-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational ℓ2-Betti numbers arising from the lamplighter group algebra Q[Z2≀ Z]. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational ℓ2-Betti numbers from the algebras Q[Zn≀ Z] , where n≥ 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra O(n¯) , where n¯ is a supernatural number. We compute its ∗ -regular closure, and this allows us to fully characterize the set of O(n¯) -Betti numbers.
KW - Atiyah conjecture
KW - Lamplighter
KW - Odometer
KW - Rank function
KW - ℓ2-Betti number
U2 - 10.1007/s10801-021-01044-8
DO - 10.1007/s10801-021-01044-8
M3 - Journal article
VL - 54
SP - 1201
EP - 1245
JO - Journal of Algebraic Combinatorics
JF - Journal of Algebraic Combinatorics
SN - 0925-9899
IS - 4
ER -