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Small undecidable groups and unrecognizable 4-manifolds
Abstract
We construct a $3$-generator $9$-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian--Rabin families of group presentations, one with $4$ generators and $11$ relators, and another with $2$ generators and $10$ relators. As a consequence, #7(S2× S2) is topologically unrecognizable and #9(S2× S2) is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon. The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean~4 formalization of the algebraic results.
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BibTeX
@misc{kegel2026small,
title = {{Small undecidable groups and unrecognizable 4-manifolds}},
author = {Marc Kegel and Shana Yunsheng Li and Qiuyu Ren},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10461},
eprint = {2609.10461},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10461},
}
RIS
TY - UNPB
TI - Small undecidable groups and unrecognizable 4-manifolds
AU - Marc Kegel
AU - Shana Yunsheng Li
AU - Qiuyu Ren
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10461
UR - https://arxiv.org/abs/2609.10461
AB - We construct a $3$-generator $9$-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian--Rabin families of group presentations, one with $4$ generators and $11$ relators, and another with $2$ generators and $10$ relators. As a consequence, #_7(S^2× S^2) is topologically unrecognizable and #_9(S^2× S^2) is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon. The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean~4 formalization of the algebraic results.
ER -