PreprintCC BY 4.0arXiv
Counterexamples to torsion-freeness in the K-theory of crystallographic group C*-algebras
Abstract
We give a negative answer to a question of Sánchez Saldaña--Velásquez and Lück asking whether the topological K-theory of the reduced group C*-algebra of a split crystallographic group with finite cyclic holonomy is always torsion-free. For two explicit split crystallographic groups with translation subgroup of rank four and holonomy of order four, we compute the equivariant K-theory of the classifying space for proper actions and find nontrivial two-torsion. The equivariant universal coefficient theorem transfers this torsion to proper equivariant K-homology, and the Baum--Connes conjecture identifies the resulting groups with the topological K-theory of the corresponding reduced group C*-algebras.
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BibTeX
@misc{sarmiento2026counterexamples,
title = {{Counterexamples to torsion-freeness in the $K$-theory of crystallographic group $C^*$-algebras}},
author = {Cristian Sarmiento and Mario Velasquez},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00493},
eprint = {2611.00493},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00493},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counterexamples to torsion-freeness in the K-theory of crystallographic group C^*-algebras
AU - Cristian Sarmiento
AU - Mario Velasquez
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00493
UR - https://arxiv.org/abs/2611.00493
AB - We give a negative answer to a question of Sánchez Saldaña--Velásquez and Lück asking whether the topological K-theory of the reduced group C*-algebra of a split crystallographic group with finite cyclic holonomy is always torsion-free. For two explicit split crystallographic groups with translation subgroup of rank four and holonomy of order four, we compute the equivariant K-theory of the classifying space for proper actions and find nontrivial two-torsion. The equivariant universal coefficient theorem transfers this torsion to proper equivariant K-homology, and the Baum--Connes conjecture identifies the resulting groups with the topological K-theory of the corresponding reduced group C*-algebras.
ER -