Homology, Homotopy and Applications

Volume 21 (2019)

Number 1

Linearity problem for non-abelian tensor products

Pages: 269 – 281

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n1.a12


Valeriy G. Bardakov (Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia; and Novosibirsk State Agrarian University, Novosibirsk, Russia)

Andrei V. Lavrenov (Saint-Petersburg State University, Saint-Petersburg, Russia)

Mikhail V. Neshchadim (Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia)


In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products $G \otimes H$ and tensor squares $G \otimes G$. Using these results we prove that tensor squares of some groups with one relation and some knot groups are linear. We prove that the Peiffer square of a finitely generated linear group is linear. At the end we construct faithful linear representations for the non-abelian tensor square of a free group and free nilpotent group.


non-abelian tensor product, linear group, faithful linear representation

2010 Mathematics Subject Classification

20E05, 20E25, 20G20

Received 11 April 2018

Received revised 16 August 2018

Accepted 29 June 2018

Published 17 October 2018