Linearity of non-abelian tensor squares: a counterexample and the braid groups
We answer both parts of Kourovka Problem 19.9. First, for $r\geq4$ the finitely generated linear group $\operatorname{Sp}_{2r}(\mathbb Z)$ has a non-linear non-abelian tensor square, by combining the Brown--Loday description for perfect groups with Deligne's non-residually finite universal central extension. Second, for every $n>3$ the tensor square $B_n\otimes B_n$ of the braid group is linear. The latter follows from the homology map $H_2(B_n;\mathbb Z)\to H_2(S_n;\mathbb Z)$ and the standard splitting of the tensor square when the abelianization has no $2$-torsion.
Lemma 3.1
The induced map \[ \beta_*:H_2(B_n;\Z)\longrightarrow H_2(S_n;\Z) \] is an isomorphism.
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- External referenceR. D. Blyth, F. Fumagalli and M. Morigi, Some structural results on the non-abelian tensor square of groups, J. Group Theory 13 (2010), 83--94.LaTeX source\bibitem{bfm} R. D. Blyth, F. Fumagalli and M. Morigi, Some structural results on the non-abelian tensor square of groups, \emph{J. Group Theory} 13 (2010), 83--94.
- External referenceV. G. Bardakov, A. V. Lavrenov and M. V. Neshchadim, Linearity problem for non-abelian tensor products, Homology Homotopy Appl. 21 (2019), 133--146.LaTeX source\bibitem{bln} V. G. Bardakov, A. V. Lavrenov and M. V. Neshchadim, Linearity problem for non-abelian tensor products, \emph{Homology Homotopy Appl.} 21 (2019), 133--146.
- External referenceS. J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), 471--486.LaTeX source\bibitem{bigelow} S. J. Bigelow, Braid groups are linear, \emph{J. Amer. Math. Soc.} 14 (2001), 471--486.
- External referenceR. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311--335.LaTeX source\bibitem{brownloday} R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, \emph{Topology} 26 (1987), 311--335.
- External referenceP. Deligne, Extensions centrales non residuellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris Ser. A-B 287 (1978), A203--A208.LaTeX source\bibitem{deligne} P. Deligne, Extensions centrales non residuellement finies de groupes arithmetiques, \emph{C. R. Acad. Sci. Paris Ser. A-B} 287 (1978), A203--A208.
- External referenceE. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Novosibirsk, 2026; arXiv:1401.0300.LaTeX source\bibitem{kourovka} E. I. Khukhro and V. D. Mazurov (eds.), \emph{Unsolved Problems in Group Theory: The Kourovka Notebook}, No. 21, Novosibirsk, 2026; arXiv:1401.0300.
- External referenceD. Krammer, Braid groups are linear, Ann. of Math. 155 (2002), 131--156.LaTeX source\bibitem{krammer} D. Krammer, Braid groups are linear, \emph{Ann. of Math.} 155 (2002), 131--156.
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Machine-readable theorem index · 4 statements
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- PF:2026.000006/v1/THM-2.1
There is a finitely generated linear group $G$ such that $G\otimes G$ is not linear.
- PF:2026.000006/v1/LEM-3.1
The induced map \[ \beta_*:H_2(B_n;\Z)\longrightarrow H_2(S_n;\Z) \] is an isomorphism.
- PF:2026.000006/v1/PROP-3.2
The natural map $B_n\wedge B_n\to E_n$ is an isomorphism.
- PF:2026.000006/v1/THM-3.3
For every $n>3$, the non-abelian tensor square $B_n\otimes B_n$ is linear.