Nonlinearity of some subgroups of the planar Cremona group
aa r X i v : . [ m a t h . AG ] J a n NONLINEARITY OF SOME SUBGROUPS OF THE PLANARCREMONA GROUP
YVES CORNULIER
Abstract.
We give some examples of non-nilpotent locally nilpotent, andhence nonlinear subgroups of the planar Cremona group. Introduction
Let K be a field. The planar Cremona group Cr ( K ) of K is defined as thegroup of birational transformations of the 2-dimensional K -affine space. It canalso be described as the group of K -automorphisms of the field of rational func-tions K ( t , t ). More generally, one defines Cr d ( K ).We provide here two observations about the planar Cremona group. The first isan example of a non-linear finitely generated subgroup of Cr ( C ). The existenceof such a subgroup was known to some experts: for instance it follows froman unpublished construction of S. Cantat (using superrigidity of lattices); ourexample has the additional feature of being 3-solvable. Its non-linearity followsfrom the fact it contains nilpotent subgroups of arbitrary large nilpotency length.We also show there that Cr ( K ) has no nontrivial linear representation over anyfield, extending a result of Cerveau and D´eserti (all representations below areassumed finite-dimensional).We end this short introduction by a few questions.(1) (Cantat) for d ≥
2, and any field K , is Cr d ( K ) locally residually finite(i.e. is every finitely generated subgroup residually finite)?(2) Does there exist a finitely generated subgroup of Aut( C ) with no faithfullinear representation (see Remark 2.4)?(3) Does there exist d and K and an infinite, finitely generated subgroup ofCr d ( K ) such that every linear representation of Γ over any field has afinite image Acknowledgements.
I thank Serge Cantat and Julie Deserti for useful discus-sions. 2.
A nonlinear subgroup of the Cremona group
We provide in this section an example of a finitely generated subgroup ofCr ( C ) that is not linear over any field. It is 3-solvable and actually lies in the Date : February 22, 2013.
Jonqui`eres subgroup, that is, the group of birational transformations preservingthe partition of C by horizontal lines.If f ∈ K ( X ) and g ∈ K ( X ) × , define α f , µ g ∈ Cr ( K ) by α f ( x, y ) = ( x, y + f ( x )); µ g ( x, y ) = ( x, yg ( x )) . We have α f + f ′ = α f α f ′ ; µ gg ′ = µ g µ g ′ ; µ g α f µ − g = α fg . Also for t ∈ K , define s t ∈ Cr ( K ) by ( x, y ) = ( x + t, y ), so that s t α f ( X ) s − t = α f ( X − t ) ; s t µ g ( X ) s − t = µ g ( X − t ) . Consider the subgroup Γ n of Cr ( K ) generated by s and α X n ( n ≥ Lemma 2.1.
The group Γ n is nilpotent of class at most n + 1 ; moreover if K has characteristic zero the nilpotency length of Γ n is exactly n + 1 , and Γ n istorsion-free.Proof. Consider the largest group R n , generated by s and by the abelian sub-group A n consisting of all α P , where P ranges over polynomials of degree at most n . Then A n is normalized by s and [ s , A n ] ⊂ A n − for all n ≥
1, while A = { } .Therefore R n is nilpotent of class at most n +1, and therefore so is Γ n . Conversely,the n -iterated group commutator [ s , [ s , · · · , [ s , α X n ] · · · ]] is equal to α ∆ n X n ,where ∆ is the discrete differential operator ∆ P ( X ) = − P ( X ) + P ( X − K has characteristic zero (or p > n ) then ∆ n X n = 0 and Γ n is not n -nilpotent.In this case it is also clear that R n is torsion-free. (cid:3) Now assume that K has characteristic zero and consider the group G ⊂ Cr ( Q ) ⊂ Cr ( K ) generated by { s , α , µ X } . Proposition 2.2.
The finitely generated group G ⊂ Cr ( Q ) is solvable of lengththree; it is not linear over any field.Proof. From the conjugation relations above it is clear that the subgroup gener-ated by s , all α f and µ g , is solvable of length at most three. If we restrict tothose g of the form Q n ∈ Z ( X − n ) k n (where ( k n ) is finitely supported), we obtaina subgroup containing Γ, that is clearly torsion-free.Since µ nX α µ − nX = α X n , we see that G contains Γ n for all n , which is nilpotentof length exactly n + 1. Therefore it has no linear representation over any field.[Sketch of proof of the latter (well-known) result: in characteristic p >
0, anytorsion-free nilpotent subgroup is abelian, so this discards this case. Otherwisein characteristic zero, since any finite index subgroup of a torsion-free nilpotentgroup of nilpotency length n + 1 still has nilpotency length n + 1, the existence ofa linear representation of G into GL d ( C ) implies the existence of a Lie subalgebraof gl d ( C ) of nilpotency length n + 1 for all n ; this necessarily implies n + 1 ≤ d ,and since n is unbounded this is a contradiction.] ONLINEARITY OF SOME SUBGROUPS OF THE PLANAR CREMONA GROUP 3
The fact that G is not 2-solvable (=metabelian) can be checked by hand, butalso follows from the fact that every torsion-free finitely generated metabeliangroup is linear over a field of characteristic zero [Re]. (cid:3) We easily see Γ n ⊂ Γ n +1 for all n . Denoting Γ ∞ = S Γ n , we see that Γ ∞ islocally nilpotent (that is, all its finitely generated subgroups are nilpotent) andthe above argument works for it. Since Γ ∞ is contained in Aut( C ), we also get: Proposition 2.3. Γ ∞ , and hence Aut ( C ) is not linear over any field. With little further effort, it actually follows from the same argument that Γ ∞ (and hence G ) is not linear over any finite product of fields. Better, it is notlinear over any product of fields (and therefore over any reduced commutativering). This now relies on the fact that nilpotent subgroups of GL d ( K ) for fixed d , have nilpotency length bounded independently of the characteristic of the field K (see [FN]). Remark 2.4.
It is unknown whether there exists a finitely generated subgroupof the group Aut( C ) of polynomial automorphisms of C , that is not linear incharacteristic zero. A construction in the same fashion does not work: indeedlet E be the group of elementary automorphisms, namely of the form ( x, y ) ( αx + P ( y ) , βy + c ) for ( α, β, c, P ) ∈ C ∗ × C ∗ × C × C [ X ]. Then, although E is notlinear (since by the argument above, it contains all Γ n ), every finitely generatedsubgroup of E is linear over C .To see this, write E as a semidirect product ( C ∗ × ( C ∗ ⋉ C )) ⋉ C [ X ], wherethe action on C [ X ] is by ( α, β, c ) · P ( X ) = αP ( βX + c ). In particular, this actionstabilizes the subgroup C n [ X ] of polynomials of degree at most n . Thereforeany finitely generated subgroup of E is contained in the subgroup ( C ∗ × ( C ∗ ⋉ C )) ⋉ C n [ X ] for some n ≥
1. This is a (finite-dimensional) complex Lie groupwhose center is easily shown to be trivial, so its adjoint representation is a faithfulcomplex linear representation.A nice observation by Cerveau and D´eserti [CD, Lemme 5.2] is that the Cre-mona group has no faithful linear representation in characteristic zero. Actually,an easy refinement of the same argument provides a stronger result.
Proposition 2.5. If K is an algebraically closed field, there is no nontrivialfinite-dimensional linear representation of Cr ( K ) over any field. (Note that since the Cremona group is not simple by a recent difficult resultof Cantat and Lamy [CL], the non-existence of a faithful representation does notformally imply the non-existence of a nontrivial representation.) Proof of Proposition 2.5.
In Cr ( K ), there is a natural copy of G = ( K × ) ⋊ Z , where Z acts by the automorphism σ ( x, y ) = ( x, xy ) of ( K × ) . Here, itcorresponds, in affine coordinates, to the group of transformations of the form( x , x ) ( λ x , x n λ x ) for ( λ , λ , n ) ∈ ( K × ) × Z . YVES CORNULIER
Consider an linear representation ρ : G → GL n ( F ), where F is any field (here G isviewed as a discrete group). If p is a prime which is nonzero in K and if ω p ∈ K is aprimitive p -root of unity, set α p ( x , x ) = ( ω p x , ω p x ) and β p ( x , x ) = ( x , ω p x ).Then σα p σ − α − p = β p and commutes with both σ and α p . An argument ofBirkhoff [Bi, Lemma 1] shows that if ρ ( α p ) = 1 then n > p (the short argumentgiven in the proof of [CD, Lemme 5.2] for F of characteristic zero works if it isassumed that p is not the characteristic of F ).Picking p to be greater than n and the characteristics of K and F , this showsthat if we have an arbitrary representation π : Cr ( K ) → GL n ( F ), the restrictionof π to PGL ( K ) is not faithful; since PGL ( K ) is simple, this implies that π is trivial on PGL ( K ); since Cr ( K ) is generated by PGL ( K ) as a normalsubgroup, this yields the conclusion. (cid:3) References [Bi] G. Birkhoff. Lie groups simply isomorphic with no linear group. Bull. Amer. Math.Soc. 42(12) (1936), 883–888.[CD] D. Cerveau, J. D´eserti. Transformations birationnelles de petit degr´e. CoursSp´ecialis´es 19, Soc. Math. France, Paris, 2013.[CL] S. Cantat, S. Lamy. Normal subgroups of the Cremona group. Acta Math. 210 (2013),no. 1, 31–94.[FN] M. Frick, M.F. Newman. Soluble linear groups. Bull. Austral. Math. Soc. 6 (1972),31–44.[Re] V. Remeslennikov. Representation of finitely generated metabelian groups by matri-ces. Algebra i Logika 8 (1969), 72–75 (Russian); English translation in Algebra andLogic 8 (1969), 39–40.
Laboratoire de Math´ematiques, Bˆatiment 425, Universit´e Paris-Sud 11, 91405Orsay, FRANCE
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