Linearity problem for non-abelian tensor products

TL;DR

This paper investigates the linearity of non-abelian tensor products, providing examples, conditions for linearity, and constructing faithful representations for specific groups, advancing understanding of tensor product linearity in group theory.

Contribution

It presents the first example of a non-abelian tensor square that is not linear and establishes conditions under which tensor squares are linear, including for certain knot and free groups.

Findings

Example of a non-linear tensor square group.

Conditions ensuring linearity of tensor products.

Faithful linear representations for free and nilpotent groups.

Abstract

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 finitely generated linear groups is linear. At the end we construct faithful linear representations for the non-abelian tensor square of free group and free nilpotent group.