The non abelian tensor product of groups

This is a guide to the bibliography on non abelian tensor products of groups.

The basic idea of the non abelian tensor product is simple.

It was early realised that a direct definition of tensor product of non abelian groups gave nothing new. Let G,H be groups. Define
GH as the group with generators gh, gG, hH and relations

(gg'h) = (gh)(g'h)
(ghh') = (gh)(gh')

for all g,g'G, h,h'H. Then expanding gg'hh' in two ways yields (after some cancellation)

(gh')(g'h) = (g'h)(gh').

From this one finds easily that

 G ⊗H = (Gab)⊗Z (Ha b) .

The start of a new approach was to recognise that if one is interested in non commutative groups then one is certainly interested in the commutator map on a group G

 [ , ] : G ×G → G.

(g,h)→ ghg-1 h-1
This map is not bimultiplicative but instead satisfies
 [gg',h] = [gg',gh][g,h],
 [g,hh'] = [g,h][hg,hh'],
where hg = hgh-1. A map of this type we call a biderivation. It is thus natural to consider the universal object for biderivations. So we now define GG to be the group with generators gh, g,hG and relations
 (gg'⊗h) = (gg'⊗gh)(g ⊗h)
 (g ⊗hh') = (g ⊗h)(hg ⊗hh' )
for all g,g',h,h'G. The natural map G ×GGG, (g,h) → gh is then the universal biderivation and any biderivation
b : G ×GL factors uniquely to give a morphism of groups b' : GGL. In particular the commutator map defines a morphism κ: GGG whose image of course is the commutator subgroup [G,G] of G.

There are other relations satisfied by the commutator, for example [g,g] = 1. It is natural therefore to consider another construction, the exterior product

 G ∧G = (G ⊗G)/{g ⊗g : g ∈ G} .
Again the commutator map yields a morphism of groups
 κ': G ∧G →G.
This map was introduced in essence in 1952 by Clair Miller ('The second homology group of a group; relations among commutators'. Proc. Amer. Math. Soc. 3, (1952). 588-595). who proved that the kernel of κ' is isomorphic to the Schur Multiplicator H2(G).

For a more extended account, see

1. A nonabelian tensor product of groups , slides of a seminar in Warwick, 2001;
2. Applications of a nonabelian tensor product of groups , slides of a seminar at Gottingen, 2011.

Revised 26 January, 2012 