Contents Introduction Main Results Automorphisms of Divisible Rigid Groups Denis Ovchinnikov Novosibirsk State University, Russia May, 29, 2013
Contents Introduction Main Results Introduction 1 Main Results 2
Contents Introduction Main Results G ⊲ A , A is abelian. G acts by conjugation: a → a g = g − 1 ag . G / A acts, A is a right Z [ G / A ] -module. u = α 1 g 1 + . . . + α n g n ∈ Z [ G / A ] , a u = ( a g 1 ) α 1 · . . . · ( a g n ) α n . Definition m -rigid group G : there is a normal series G = G 1 > G 2 > . . . > G m > G m + 1 = 1 , G i / G i + 1 are abelian and considering as right Z [ G / G i ] -modules have no torsion. Why rigid? - this series is unique. Given group G is solvable of length exactly m .
Contents Introduction Main Results G ⊲ A , A is abelian. G acts by conjugation: a → a g = g − 1 ag . G / A acts, A is a right Z [ G / A ] -module. u = α 1 g 1 + . . . + α n g n ∈ Z [ G / A ] , a u = ( a g 1 ) α 1 · . . . · ( a g n ) α n . Definition m -rigid group G : there is a normal series G = G 1 > G 2 > . . . > G m > G m + 1 = 1 , G i / G i + 1 are abelian and considering as right Z [ G / G i ] -modules have no torsion. Why rigid? - this series is unique. Given group G is solvable of length exactly m .
Contents Introduction Main Results G ⊲ A , A is abelian. G acts by conjugation: a → a g = g − 1 ag . G / A acts, A is a right Z [ G / A ] -module. u = α 1 g 1 + . . . + α n g n ∈ Z [ G / A ] , a u = ( a g 1 ) α 1 · . . . · ( a g n ) α n . Definition m -rigid group G : there is a normal series G = G 1 > G 2 > . . . > G m > G m + 1 = 1 , G i / G i + 1 are abelian and considering as right Z [ G / G i ] -modules have no torsion. Why rigid? - this series is unique. Given group G is solvable of length exactly m .
Contents Introduction Main Results 1) Free solvable group is rigid, rigid series consists of commutator subgroups. 2) W = A m ≀ ( A m − 1 ≀ ( ... ≀ A 1 ) ... ) , where A i are free abelian groups. Subgroups of rigid groups are rigid too: G � H , H i = H ∩ G i . Corresponding series for H may be shorter. Rigid groups were defined by N.Romanovskiy, he proved that arbitrary rigid group is equationally Noetherian ( Equational Noetherianess of rigid solvable groups , Algebra and Logic, 48(2), 2009, pp. 147-160). This result made possible to develope algebraic gemetry over rigid groups. The dimesion theory for rigid groups was constructed by A.Myasnikov ang N.Romanovskiy ( Krull dimension of solvable groups , J.Algebra, 324 (10), 2010, pp. 2814-2831).
Contents Introduction Main Results 1) Free solvable group is rigid, rigid series consists of commutator subgroups. 2) W = A m ≀ ( A m − 1 ≀ ( ... ≀ A 1 ) ... ) , where A i are free abelian groups. Subgroups of rigid groups are rigid too: G � H , H i = H ∩ G i . Corresponding series for H may be shorter. Rigid groups were defined by N.Romanovskiy, he proved that arbitrary rigid group is equationally Noetherian ( Equational Noetherianess of rigid solvable groups , Algebra and Logic, 48(2), 2009, pp. 147-160). This result made possible to develope algebraic gemetry over rigid groups. The dimesion theory for rigid groups was constructed by A.Myasnikov ang N.Romanovskiy ( Krull dimension of solvable groups , J.Algebra, 324 (10), 2010, pp. 2814-2831).
Contents Introduction Main Results 1) Free solvable group is rigid, rigid series consists of commutator subgroups. 2) W = A m ≀ ( A m − 1 ≀ ( ... ≀ A 1 ) ... ) , where A i are free abelian groups. Subgroups of rigid groups are rigid too: G � H , H i = H ∩ G i . Corresponding series for H may be shorter. Rigid groups were defined by N.Romanovskiy, he proved that arbitrary rigid group is equationally Noetherian ( Equational Noetherianess of rigid solvable groups , Algebra and Logic, 48(2), 2009, pp. 147-160). This result made possible to develope algebraic gemetry over rigid groups. The dimesion theory for rigid groups was constructed by A.Myasnikov ang N.Romanovskiy ( Krull dimension of solvable groups , J.Algebra, 324 (10), 2010, pp. 2814-2831).
Contents Introduction Main Results If G is a solvable torsion free group then the group ring Z G is an Ore domain, so one can consider the Ore skew field of fractions which we denote by Q ( G ) (follows from P.H.Kropholler, P.A.Linnell and J.A.Moody, Applications of a new K -theoretic theorem to soluble group rings, Proc. Amer. Math. Soc., 104 (1988), 675-684). Definition Rigid group G is called divisible if any factor G i / G i + 1 is a divisible module over the ring Z [ G / G i ] , then this factor may be consider as a vector space over skew field of fractions Q ( G / G i ) .
Contents Introduction Main Results If G is a solvable torsion free group then the group ring Z G is an Ore domain, so one can consider the Ore skew field of fractions which we denote by Q ( G ) (follows from P.H.Kropholler, P.A.Linnell and J.A.Moody, Applications of a new K -theoretic theorem to soluble group rings, Proc. Amer. Math. Soc., 104 (1988), 675-684). Definition Rigid group G is called divisible if any factor G i / G i + 1 is a divisible module over the ring Z [ G / G i ] , then this factor may be consider as a vector space over skew field of fractions Q ( G / G i ) .
Contents Introduction Main Results Let α 1 , . . . , α m be nonzero cardinalities. Construct a group M ( α 1 , . . . , α m ) by induction. M ( α 1 ) is a direct sum of α 1 copies of Q . Take A = M ( α 1 , . . . , α m − 1 ) and let T be a vector space with a basis of cardinality α m over the skew field Q ( A ) . Then set (︃ A )︃ 0 M ( α 1 , . . . , α m ) = . 1 T We call such group divisible splittable rigid group because it splits into semidirect product A 1 A 2 . . . A m of abelian groups.
Contents Introduction Main Results Arbitrary m -rigid group can be embedded with preserving linear independence into some divisible splittable m -rigid group (see N.S.Romanovskiy, Divisible rigid groups , Algebra and Logic, 47(6), 2008, pp. 426-434). And any divisible rigid group is splittable, so it is isomorphic to some group M ( α 1 , . . . , α m ) (see A.Myasnikov, N.S.Romanovskiy, Logical aspects of divisible rigid groups , submitted for publication). A.Myasnikov ang N.Romanovskiy also proved that divisible m -rigid groups are exactly algebraic closed or existential closed objects in the class of all ≤ m -rigid groups and that the elementary theory of any such group is decidable.
Contents Introduction Main Results Arbitrary m -rigid group can be embedded with preserving linear independence into some divisible splittable m -rigid group (see N.S.Romanovskiy, Divisible rigid groups , Algebra and Logic, 47(6), 2008, pp. 426-434). And any divisible rigid group is splittable, so it is isomorphic to some group M ( α 1 , . . . , α m ) (see A.Myasnikov, N.S.Romanovskiy, Logical aspects of divisible rigid groups , submitted for publication). A.Myasnikov ang N.Romanovskiy also proved that divisible m -rigid groups are exactly algebraic closed or existential closed objects in the class of all ≤ m -rigid groups and that the elementary theory of any such group is decidable.
Contents Introduction Main Results We study the group of automorphisms of divisible (splittable) m -rigid group G = M ( α 1 , . . . , α m ) . For m = 1 : Aut ( G ) ∼ = GL α 1 ( Q ) . Suppose m ≥ 2 . Fix some splitting A 1 A 2 . . . A m of G into semidirect product of abelian subgroups. Let Q i = Q ( A 1 . . . A i ) and Φ = { ϕ ∈ Aut ( G ) | A i ϕ = A i , i = 1 . . . . , m } . Theorem 1 There is a splitting Φ as a semidirect product Φ 1 Φ 2 . . . Φ m , where Φ i ∼ = GL α i ( Q i ) . We can say that Φ is a general polylinear group.
Contents Introduction Main Results We study the group of automorphisms of divisible (splittable) m -rigid group G = M ( α 1 , . . . , α m ) . For m = 1 : Aut ( G ) ∼ = GL α 1 ( Q ) . Suppose m ≥ 2 . Fix some splitting A 1 A 2 . . . A m of G into semidirect product of abelian subgroups. Let Q i = Q ( A 1 . . . A i ) and Φ = { ϕ ∈ Aut ( G ) | A i ϕ = A i , i = 1 . . . . , m } . Theorem 1 There is a splitting Φ as a semidirect product Φ 1 Φ 2 . . . Φ m , where Φ i ∼ = GL α i ( Q i ) . We can say that Φ is a general polylinear group.
Contents Introduction Main Results We study the group of automorphisms of divisible (splittable) m -rigid group G = M ( α 1 , . . . , α m ) . For m = 1 : Aut ( G ) ∼ = GL α 1 ( Q ) . Suppose m ≥ 2 . Fix some splitting A 1 A 2 . . . A m of G into semidirect product of abelian subgroups. Let Q i = Q ( A 1 . . . A i ) and Φ = { ϕ ∈ Aut ( G ) | A i ϕ = A i , i = 1 . . . . , m } . Theorem 1 There is a splitting Φ as a semidirect product Φ 1 Φ 2 . . . Φ m , where Φ i ∼ = GL α i ( Q i ) . We can say that Φ is a general polylinear group.
Contents Introduction Main Results We study the group of automorphisms of divisible (splittable) m -rigid group G = M ( α 1 , . . . , α m ) . For m = 1 : Aut ( G ) ∼ = GL α 1 ( Q ) . Suppose m ≥ 2 . Fix some splitting A 1 A 2 . . . A m of G into semidirect product of abelian subgroups. Let Q i = Q ( A 1 . . . A i ) and Φ = { ϕ ∈ Aut ( G ) | A i ϕ = A i , i = 1 . . . . , m } . Theorem 1 There is a splitting Φ as a semidirect product Φ 1 Φ 2 . . . Φ m , where Φ i ∼ = GL α i ( Q i ) . We can say that Φ is a general polylinear group.
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