Finiteness conditions for the non-abelian tensor product of groups 1 Raimundo Bastos Universidade de Bras´ ılia - UnB Joint work with Irene Nakaoka (UEM) and Nora´ ı Rocco (UnB) Groups St Andrews 2017 - Birmingham 1 This research was supported by FAPDF-Brazil Raimundo Bastos Finiteness conditions Groups St Andrews 1 / 13
Non-abelian tensor product of groups (Commutator Approach) Non-abelian tensor product of groups Let G and H be groups each of which acts upon the other (on the right), G × H → G , ( g , h ) �→ g h ; H × G → H , ( h , g ) �→ h g and on itself by conjugation, in such a way that for all g , g 1 ∈ G and h , h 1 ∈ H , � h � g 1 �� � g � h 1 . g ( h g 1 ) = g g − 1 �� h h − 1 and h ( g h 1 ) = (1) 1 1 Raimundo Bastos Finiteness conditions Groups St Andrews 2 / 13
Non-abelian tensor product of groups (Commutator Approach) Non-abelian tensor product of groups Let G and H be groups each of which acts upon the other (on the right), G × H → G , ( g , h ) �→ g h ; H × G → H , ( h , g ) �→ h g and on itself by conjugation, in such a way that for all g , g 1 ∈ G and h , h 1 ∈ H , � h � g 1 �� � g � h 1 . g ( h g 1 ) = g g − 1 �� h h − 1 and h ( g h 1 ) = (1) 1 1 In this situation we say that G and H act compatibly on each other. Let H ϕ be an extra copy of H , isomorphic via ϕ : H → H ϕ , h �→ h ϕ , for all h ∈ H . Raimundo Bastos Finiteness conditions Groups St Andrews 2 / 13
Non-abelian tensor product of groups (Commutator Approach) Non-abelian tensor product of groups Let G and H be groups each of which acts upon the other (on the right), G × H → G , ( g , h ) �→ g h ; H × G → H , ( h , g ) �→ h g and on itself by conjugation, in such a way that for all g , g 1 ∈ G and h , h 1 ∈ H , � h � g 1 �� � g � h 1 . g ( h g 1 ) = g g − 1 �� h h − 1 and h ( g h 1 ) = (1) 1 1 In this situation we say that G and H act compatibly on each other. Let H ϕ be an extra copy of H , isomorphic via ϕ : H → H ϕ , h �→ h ϕ , for all h ∈ H . Consider the group η ( G , H ) defined in [Nak00] as η ( G , H ) = � G , H ϕ | [ g , h ϕ ] g 1 = [ g g 1 , ( h g 1 ) ϕ ] , [ g , h ϕ ] h ϕ 1 = [ g h 1 , ( h h 1 ) ϕ ] , ∀ g , g 1 ∈ G , h , h 1 ∈ H � . Raimundo Bastos Finiteness conditions Groups St Andrews 2 / 13
Non-abelian tensor product of groups (Commutator Approach) Non-abelian tensor product of groups It is a well known fact (see [Nak00, Proposition 2.2]) that the subgroup [ G , H ϕ ] of η ( G , H ) is canonically isomorphic with the non-abelian tensor product G ⊗ H , as defined by R. Brown and J.-L. Loday in their seminal paper [BL87], the isomorphism being induced by g ⊗ h �→ [ g , h ϕ ] (see also [EL95]). Raimundo Bastos Finiteness conditions Groups St Andrews 3 / 13
Non-abelian tensor product of groups (Commutator Approach) Non-abelian tensor product of groups It is a well known fact (see [Nak00, Proposition 2.2]) that the subgroup [ G , H ϕ ] of η ( G , H ) is canonically isomorphic with the non-abelian tensor product G ⊗ H , as defined by R. Brown and J.-L. Loday in their seminal paper [BL87], the isomorphism being induced by g ⊗ h �→ [ g , h ϕ ] (see also [EL95]). It is clear that the subgroup [ G , H ϕ ] is normal in η ( G , H ) Raimundo Bastos Finiteness conditions Groups St Andrews 3 / 13
Non-abelian tensor product of groups (Commutator Approach) Non-abelian tensor product of groups It is a well known fact (see [Nak00, Proposition 2.2]) that the subgroup [ G , H ϕ ] of η ( G , H ) is canonically isomorphic with the non-abelian tensor product G ⊗ H , as defined by R. Brown and J.-L. Loday in their seminal paper [BL87], the isomorphism being induced by g ⊗ h �→ [ g , h ϕ ] (see also [EL95]). It is clear that the subgroup [ G , H ϕ ] is normal in η ( G , H ) and one has the decomposition η ( G , H ) = ([ G , H ϕ ] · G ) · H ϕ , (2) where the dots mean (internal) semidirect products. Raimundo Bastos Finiteness conditions Groups St Andrews 3 / 13
Non-abelian tensor product of groups (Commutator Approach) Non-abelian tensor product of groups It is a well known fact (see [Nak00, Proposition 2.2]) that the subgroup [ G , H ϕ ] of η ( G , H ) is canonically isomorphic with the non-abelian tensor product G ⊗ H , as defined by R. Brown and J.-L. Loday in their seminal paper [BL87], the isomorphism being induced by g ⊗ h �→ [ g , h ϕ ] (see also [EL95]). It is clear that the subgroup [ G , H ϕ ] is normal in η ( G , H ) and one has the decomposition η ( G , H ) = ([ G , H ϕ ] · G ) · H ϕ , (2) where the dots mean (internal) semidirect products. We observe that the defining relations of the tensor product can be viewed as abstractions of commutator relations (see also [Kap99]). Raimundo Bastos Finiteness conditions Groups St Andrews 3 / 13
Non-abelian tensor square of groups Non-abelian tensor square of groups We observe that when G = H and all actions are conjugations, η ( G , H ) becomes the group ν ( G ) introduced in [Roc91]. More precisely, ν ( G ) := � G , G ϕ | [ g 1 , g 2 ϕ ] g 3 = [ g 1 g 3 , ( g 2 g 3 ) ϕ ] = [ g 1 , g 2 ϕ ] g 3 ϕ , g i ∈ G � . Raimundo Bastos Finiteness conditions Groups St Andrews 4 / 13
Non-abelian tensor square of groups Non-abelian tensor square of groups We observe that when G = H and all actions are conjugations, η ( G , H ) becomes the group ν ( G ) introduced in [Roc91]. More precisely, ν ( G ) := � G , G ϕ | [ g 1 , g 2 ϕ ] g 3 = [ g 1 g 3 , ( g 2 g 3 ) ϕ ] = [ g 1 , g 2 ϕ ] g 3 ϕ , g i ∈ G � . In particular, ν ( G ) = ([ G , G ϕ ] · G ) · G ϕ , where [ G , G ϕ ] is isomorphic to G ⊗ G , the non-abelian tensor square of G . In the notation of [NR94], we denote by ∆( G ) the diagonal subgroup of the non-abelian tensor square [ G , G ϕ ], ∆( G ) = � [ g , g ϕ ] | g ∈ G � . Raimundo Bastos Finiteness conditions Groups St Andrews 4 / 13
Non-abelian tensor square of groups Non-abelian tensor square of groups We observe that when G = H and all actions are conjugations, η ( G , H ) becomes the group ν ( G ) introduced in [Roc91]. More precisely, ν ( G ) := � G , G ϕ | [ g 1 , g 2 ϕ ] g 3 = [ g 1 g 3 , ( g 2 g 3 ) ϕ ] = [ g 1 , g 2 ϕ ] g 3 ϕ , g i ∈ G � . In particular, ν ( G ) = ([ G , G ϕ ] · G ) · G ϕ , where [ G , G ϕ ] is isomorphic to G ⊗ G , the non-abelian tensor square of G . In the notation of [NR94], we denote by ∆( G ) the diagonal subgroup of the non-abelian tensor square [ G , G ϕ ], ∆( G ) = � [ g , g ϕ ] | g ∈ G � . There is also a connection between ν ( G ) and a group, χ ( G ), introduced by Sidki [Sid80], Raimundo Bastos Finiteness conditions Groups St Andrews 4 / 13
Non-abelian tensor square of groups Non-abelian tensor square of groups We observe that when G = H and all actions are conjugations, η ( G , H ) becomes the group ν ( G ) introduced in [Roc91]. More precisely, ν ( G ) := � G , G ϕ | [ g 1 , g 2 ϕ ] g 3 = [ g 1 g 3 , ( g 2 g 3 ) ϕ ] = [ g 1 , g 2 ϕ ] g 3 ϕ , g i ∈ G � . In particular, ν ( G ) = ([ G , G ϕ ] · G ) · G ϕ , where [ G , G ϕ ] is isomorphic to G ⊗ G , the non-abelian tensor square of G . In the notation of [NR94], we denote by ∆( G ) the diagonal subgroup of the non-abelian tensor square [ G , G ϕ ], ∆( G ) = � [ g , g ϕ ] | g ∈ G � . There is also a connection between ν ( G ) and a group, χ ( G ), introduced by Sidki [Sid80], defined by χ ( G ) := � G , G ϕ | [ g , g ϕ ] = 1 , ∀ g ∈ G � . Raimundo Bastos Finiteness conditions Groups St Andrews 4 / 13
Non-abelian tensor square of groups Some Results Let G and H be groups that act compatibly on each other. (G. Ellis, [Ell87]) If G and H are finite, then the non-abelian tensor product [ G , H ϕ ] is finite; Raimundo Bastos Finiteness conditions Groups St Andrews 5 / 13
Non-abelian tensor square of groups Some Results Let G and H be groups that act compatibly on each other. (G. Ellis, [Ell87]) If G and H are finite, then the non-abelian tensor product [ G , H ϕ ] is finite; (P. Moravec, [Mor08]) If G and H are locally finite, then the non-abelian tensor product [ G , H ϕ ] is locally finite; Raimundo Bastos Finiteness conditions Groups St Andrews 5 / 13
Non-abelian tensor square of groups Some Results Let G and H be groups that act compatibly on each other. (G. Ellis, [Ell87]) If G and H are finite, then the non-abelian tensor product [ G , H ϕ ] is finite; (P. Moravec, [Mor08]) If G and H are locally finite, then the non-abelian tensor product [ G , H ϕ ] is locally finite; Now, consider G = H and all actions are conjugations. (Parvizi and Niroomand, [PN12]) Suppose that G is a finitely generated group. If the non-abelian tensor square [ G , G ϕ ] is finite, then so is G . Raimundo Bastos Finiteness conditions Groups St Andrews 5 / 13
Non-abelian tensor square of groups Question An element α ∈ η ( G , H ) is called a tensor if α = [ a , b ϕ ] for suitable a ∈ G and b ∈ H . If N and K are subgroups of G and H , respectively, let T ⊗ ( N , K ) denote the set of all tensors [ a , b ϕ ] with a ∈ N and b ∈ K . In particular, [ N , K ϕ ] = � T ⊗ ( N , K ) � . Raimundo Bastos Finiteness conditions Groups St Andrews 6 / 13
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