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Free Burnside groups of large odd exponent have cost 1
Abstract
We prove that for every non-elementary torsion-free hyperbolic group H and sufficiently large odd n>1, the quotient H/Hn has cost $1$. In particular, when H is the free group of rank m≥ 2, we obtain that the free Burnside group B(m,n) has cost $1$ for every odd exponent n≥ 665. It follows that B(m,n) is anti-treeable, and that its first ℓ2-Betti number β1(2)(B(m,n)) vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that β1(2)(B(m,p))=0 for all sufficiently large prime exponents p.
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BibTeX
@misc{donosoechenique2026free,
title = {{Free Burnside groups of large odd exponent have cost 1}},
author = {Miguel Donoso-Echenique and Eduardo Silva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.20472},
eprint = {2608.20472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.20472},
}
RIS
TY - UNPB
TI - Free Burnside groups of large odd exponent have cost 1
AU - Miguel Donoso-Echenique
AU - Eduardo Silva
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.20472
UR - https://arxiv.org/abs/2608.20472
AB - We prove that for every non-elementary torsion-free hyperbolic group H and sufficiently large odd n>1, the quotient H/H^n has cost $1$. In particular, when H is the free group of rank m≥ 2, we obtain that the free Burnside group B(m,n) has cost $1$ for every odd exponent n≥ 665. It follows that B(m,n) is anti-treeable, and that its first ℓ^2-Betti number β_1^(2)(B(m,n)) vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that β_1^(2)(B(m,p))=0 for all sufficiently large prime exponents p.
ER -