aa r X i v : . [ m a t h . G R ] M a y Discrete Heisenberg group and its automorphismgroup ∗ D.V. Osipov
In this note we give more easy and short proof of a statement previously proved byP. Kahn in [1] that the automorphism group of the discrete Heisenberg group Heis(3 , Z )is isomorphic to the group ( Z ⊕ Z ) ⋊ GL (2 , Z ) . The method which we suggest to constructthis isomorphism gives far more transparent picture of the structure of the automorphismgroup of the group Heis(3 , Z ) .We consider the discrete Heisenberg group G = Heis(3 , Z ) which is the group ofmatrices of the form: a c b ,where a, b, c are from Z . We can consider also the group G as a set of all integer triplesendowed with the group law:( a , b , c )( a , b , c ) = ( a + a , b + b , c + c + a b ). (1)It is clear that [(1 , , , (0 , , , ,
1) and the group G is generated by the elements(1 , ,
0) and (0 , ,
1) .Let Aut( G ) be the group of automorphisms of the group G . The starting point forthis note was the group homomorphism Z → Aut( G ) constructed in [3, §
5] and givenby the formula R d (( a, b, c )) = (cid:18) a + db, b, c + b ( b − d (cid:19) , (2)where R d is the automorphism of the group G induced by an element d ∈ Z . Formula (2)was obtained in [3] after some calculation of automorphisms which are analogs of ”looprotations” in loop groups. But the fact that formula (2) defines a homomorphism from thegroup Z to the group Aut( G ) is also an easy direct consequence of formulas (1) and (2).For the group G we have the following exact sequence of groups1 −→ C −→ G λ −→ H ⊕ P −→
1, (3)where 1 is the trivial group, the group C = { (0 , , c ) | c ∈ Z } ≃ Z is the center of G ,and H = { ( a, , | a ∈ Z } ≃ Z , P = { (0 , b, | b ∈ Z } ≃ Z . ∗ This paper was written with the partial financial support of RFBR (grants no. 13-01-12420 ofi m2,14-01-00178 a) −→ Inn( G ) θ −→ Aut( G ) ϑ −→ GL (2 , Z ), (4)where the group of inner automorphisms Inn( G ) ≃ G/C ≃ Z ⊕ Z , and the homomorphism ϑ is the homomorphism Aut( G ) → Aut( Z ⊕ Z ) , which is induced by the homomorphism λ from exact sequence (3). It is clear that Im( θ ) ⊂ Ker ( ϑ ) .We claim that sequence (4) is exact in the term Aut( G ) . Indeed, it is enough to provethat Ker ( ϑ ) ⊂ Im( θ ) . Consider any ω ∈ Ker ( ϑ ) . We have ω ((1 , , , , c ) and ω ((0 , , , , c ) for some integer c and c . By direct calculations, we obtain( c , − c , , ,
0) = (1 , , c )( c , − c ,
0) and ( c , − c , , ,
0) = (0 , , c )( c , − c ,
0) .Since elements (1 , ,
0) and (0 , ,
0) generate the group G , we obtain that the innerautomorphism defined by the element ( c , − c ,
0) coincides with the automorphism ω .We claim also that the homomorphism ϑ from sequence (4) is surjective. Indeed, byformula (2) we have a homomorphism from the group Z to the group Aut( G ) . It is easyto see that this homomorphism is a section of the homomorphism ϑ over the subgroup (cid:26)(cid:18) d (cid:19)(cid:27) d ∈ Z of the group GL (2 , Z ) . By formula (2), the action of the matrices fromthis subgroup on elements of the group G is given as: (cid:18) d (cid:19) ( a, b, c ) = (cid:18) a + db, b, c + b ( b − d (cid:19) . (5)Symmetrically to the formula (5) we can write the following formula: (cid:18) d (cid:19) ( a, b, c ) = (cid:18) a, da + b, c + a ( a − d (cid:19) . (6)By an easy direct calculation we have that formula (6) defines a correct automorphismof the group G . This automorphism depends on d ∈ Z and defines the homomorphismfrom the subgroup (cid:26)(cid:18) d (cid:19)(cid:27) d ∈ Z of the group GL (2 , Z ) to the group Aut( G ) . Thishomomorphism defines a section of the homomorphism ϑ over this subgroup.Besides, it is easy to see that the homomorphism from the subgroup (cid:26)(cid:18) ± (cid:19)(cid:27) of the group GL (2 , Z ) to the group Aut( G ) given by the formula (cid:18) − (cid:19) ( a, b, c ) = ( − a, b, − c − b ) (7)defines a section of the homomorphism ϑ over this subgroup.By the classical result (see its proof, for example, in [2, Appendix A]), the group SL (2 , Z ) has a presentation: < ρ, τ | ρτ ρ = τ ρτ , ( ρτ ρ ) = 1 > ,2here the element ρ corresponds to the matrix A = (cid:18) (cid:19) and the element τ corresponds to the matrix B = (cid:18) − (cid:19) . The group GL (2 , Z ) is generated by theelements of the group SL (2 , Z ) and the matrix D = (cid:18) − (cid:19) . Therefore the group GL (2 , Z ) has a presentation: < ρ, τ, κ | ρτ ρ = τ ρτ , ( ρτ ρ ) = 1, κτ κ − = τ − , κρκ − = ρ − , κ = 1 > , (8)where the element κ corresponds to the matrix D . Thus we obtained that the homo-morphism ϑ is surjective.We note that the group Aut( G ) contains a distinguished subgroup Aut + ( G ) = ϑ − ( SL (2 , Z )) of index 2 . Since for any ω ∈ Aut( G ) we have ω ((0 , , ω ([(1 , , , (0 , , ω (1 , , , ω (0 , , , , det( ϑ ( ω ))),we obtain that the group Aut + ( G ) consists of elements ω of the group Aut( G ) such that ω ((0 , , , ,
1) . In other words, the group Aut + ( G ) consists of automoprhisms ofthe group G which act identically on the center of the group G . Theorem 1
Partial sections of the homomorphism ϑ given by formulas (5) , (6) and (7) are glued together and define a section of the homomorphism ϑ over the whole group GL (2 , Z ) . Hence, the group Aut( G ) ≃ ( Z ⊕ Z ) ⋊ GL (2 , Z ) , where an action of GL (2 , Z ) on Z ⊕ Z (given by inner automorphisms in the group Aut( G ) ) is the natural matrixaction. Besides, Aut + ( G ) ≃ ( Z ⊕ Z ) ⋊ SL (2 , Z ) . Proof . From the above discussion we see that it is enough to check that automorphismsof the group G which are given by the matrices A = (cid:18) (cid:19) , B = (cid:18) − (cid:19) and D = (cid:18) − (cid:19) (with the help of formulas (5), (6) and (7)) satisfy relations fromformula (8). Since the group G is generated by elements (1 , ,
0) and (0 , ,
0) , it is enoughto check these relations between automorphisms after application to these elements. Wehave
ABA (1 , ,
0) = AB (1 , ,
0) = A (1 , − ,
0) = (0 , − , BAB (1 , ,
0) = BA (1 , − ,
0) = B (0 , − ,
1) = (0 , − , ABA (0 , ,
0) = AB (1 , ,
0) = A (1 , ,
0) = (1 , , BAB (0 , ,
0) = BA (0 , ,
0) = B (1 , ,
0) = (1 , , G given by the composition ABA coincides with the automorphism of G given by the composition BAB . We will check3ow that the composition (
ABA ) defines the identical automorphism of the group G .Indeed, we have checked that ABA (1 , ,
0) = (0 , − ,
1) = (0 , , − . Therefore, usingagain above calculations, we have( ABA ) (1 , ,
0) =
ABA ((0 , , − ) = ( ABA (0 , , − = (1 , , − .Hence we obtain ( ABA ) (1 , ,
0) = (
ABA ) ((1 , , − ) = (1 , , ABA ) (0 , ,
0) =
ABA (1 , ,
0) = (0 , , − , and hence( ABA ) (0 , ,
0) = (
ABA ) ((0 , , − ) = (0 , , DAD − (0 , ,
0) = DA (0 , , −
1) = D (1 , , −
1) = ( − , ,
0) = A − (0 , , DAD − (1 , ,
0) = DA ( − , ,
0) = D ( − , ,
0) = (1 , ,
0) = A − (1 , , DBD − (0 , ,
0) = DB (0 , , −
1) = D (0 , , −
1) = (0 , ,
0) = B − (0 , , DBD − (1 , ,
0) = DB ( − , ,
0) = D ( − , , −
1) = (1 , ,
0) = B − (1 , , G given bythe composition D is the identity automorphism. The theorem is proved. Remark 1
P. Kahn constructed in [1] a section of the homomorphism ϑ over the group GL (2 , Z ) by direct calculations inside the group Aut( G ) (in contrast to our approach,which we started from the explicit homomorphism given by formula (2) and this homo-morphism was obtained in [3, §
5] from an analog of ”loop rotations”). We note, that thesection GL (2 , Z ) → Aut( G ) constructed by P. Kahn does not coincide with the sectionfrom Theorem 1. Let α and α be two sections of the homomorphism ϑ . We define ϕ ( g ) = α ( g ) α ( g ) − ∈ Z ⊕ Z for any g ∈ GL (2 , Z ) . Then ϕ ( g g ) = ϕ ( g ) + g · ϕ ( g ) forany g , g ∈ GL (2 , Z ) , i.e. the map ϕ : GL (2 , Z ) → Z ⊕ Z is a 1 -cocycle. Conversely,for any section α of the homomorphism ϑ and for any 1 -cocycle ϕ : GL (2 , Z ) → Z ⊕ Z we have that the map ( ϕ · α ) : GL (2 , Z ) → Aut( G ) is a homomorphism which is asection of the homomorphism ϑ . Using the Mayer-Vietoris sequence for the presenta-tion SL (2 , Z ) = Z ∗ Z Z as amalgamated free product P. Kahn computed in [1] that H ( SL (2 , Z ) , Z ⊕ Z ) = 0 for the natural matrix action of SL (2 , Z ) on Z ⊕ Z . Hence,from the Lyndon-Hochschild-Serre spectral sequence applied to SL (2 , Z ) ֒ → GL (2 , Z )one immediately obtains that H ( GL (2 , Z ) , Z ⊕ Z ) = 0 . Therefore ϕ is a 1 -coboundary,i.e. there is an element a ∈ Z ⊕ Z such that ϕ ( g ) = ga − a for any g ∈ GL (2 , Z ) . Hencewe obtain that the group of 1 -cocycles of GL (2 , Z ) with values in Z ⊕ Z is isomorphicto the group Z ⊕ Z . Thus, the set of all sections of the homomorphism ϑ is a principalhomogeneous space for the group Z ⊕ Z . 4 eferences [1] Kahn P., Automorphisms of the discrete Heisenberg groups
Braid groups.
With the graphical assistance of O. Dodane.Graduate Texts in Mathematics, 247. Springer, New York, 2008.[3] Osipov D. V., Parshin A. N.,
Representations of discrete Heisenberg group on distri-bution spaces of two-dimensional local fields , Proceedings of the Steklov Institute ofMathematics, vol. 292 (2016), to appear.Steklov Mathematical Institute of RASGubkina str. 8, 119991, Moscow, Russia
E-mail: d − osipov @ mi.ras.rumi.ras.ru