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Higher Brin--Thompson groups are Kazhdan
Abstract
We give an algebraic criterion for certain sequences (Gn)n∈ℕ of groups to have Kazhdan's property (T) eventually. As a consequence, we prove that the Brin--Thompson groups nV have Kazhdan's property (T) for large n. The algebraic criterion provides a uniform proof that nV, Aut(Fn) and ELn(R) for a finitely generated associative unital ring R have property (T) for large n. In an appendix, written by Francesco Fournier-Facio, it is shown that the group nV for large n is neither hyperlinear, in particular not sofic, nor MF. Further, it is not Schatten p-approximable for any 1≤ p<∞.
Related
BibTeX
@misc{fournierfacio2026higher,
title = {{Higher Brin--Thompson groups are Kazhdan}},
author = {Francesco Fournier-Facio and Roman Sauer and Eduard Schesler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09524},
eprint = {2610.09524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09524},
}
RIS
TY - UNPB
TI - Higher Brin--Thompson groups are Kazhdan
AU - Francesco Fournier-Facio
AU - Roman Sauer
AU - Eduard Schesler
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.09524
UR - https://arxiv.org/abs/2610.09524
AB - We give an algebraic criterion for certain sequences (G_n)_n∈ℕ of groups to have Kazhdan's property (T) eventually. As a consequence, we prove that the Brin--Thompson groups nV have Kazhdan's property (T) for large n. The algebraic criterion provides a uniform proof that nV, Aut(F_n) and EL_n(R) for a finitely generated associative unital ring R have property (T) for large n. In an appendix, written by Francesco Fournier-Facio, it is shown that the group nV for large n is neither hyperlinear, in particular not sofic, nor MF. Further, it is not Schatten p-approximable for any 1≤ p<∞.
ER -