Branch groups and their trees, and ordered sets John Wilson - PowerPoint PPT Presentation

Branch groups and their trees, and ordered sets John Wilson jsw13@cam.ac.uk; John.Wilson@maths.ox.ac.uk; wilson@math.uni-leipzig.de D¨ usseldorf, 26 June 2018 John Wilson June 27, 2018 1 / 1

Definition. A subgroup H of a group G is a precomponent if H commutes with its distinct conjugates. Then � H G � = � H g | g ∈ G } is the central product of the conjugates. Examples: normal subgroups, subgroups of nilpotent groups of class 2, groups H with H / ( H ∩ Z( G )) non-abelian simple. They arise often: • components in finite groups, (precomponents H with H / ( H ∩ Z( G )) simple and H perfect) • the ‘natural’ direct summands of base groups of wreath products • restricted stabilizers for group actions on rooted trees • restricted stabilizers for actions on other sets, e.g. totally ordered sets Aim: unified approach to precomponents via first-order group theory. John Wilson June 27, 2018 2 / 1

Let H � G . H x ∼ H y if ∃ n , ∃ x 0 = x , x 2 , . . . , x n = y with [ H x i − 1 , H x i ] � = 1 for all i . P = � H x | H x ∼ H } is the unique smallest precomp. containing H . Notation: C 2 G ( X ) = C G (C G ( X )). Let P be a precomponent. P ⊳ � P x | x ∈ G � ⊳ G . Also P ⊳ C 2 G ( P ): x ∈ C 2 G ( P ) ⇒ x centralizes C G ( P ) ⇒ x normalises all P g � = P ⇒ x ∈ N G ( P ). When does the obvious graph have (uniformly) bounded diameter? John Wilson June 27, 2018 3 / 1

First-order sentences/formulae ( ∀ x ∀ y ∀ z )([ x , y , z ] = 1) G nilp. of class � 2 Yes! ( ∀ x ∈ G ′ )( ∀ z )([ x , z ] = 1) G nilp. of class � 2 No! ( ∀ x 1 ∀ x 2 ∀ x 3 ∀ x 4 )( ∃ y 1 , y 2 )([ x 1 , x 2 ][ x 3 , x 4 ] = [ y 1 , y 2 ]) every element of G ′ is a commutator ( ∀ x 1 ∀ x 2 ∃ y )( y � = x 1 ∧ y � = x 2 ) | G | � 3 ( ∀ x 1 ∀ x 2 ∀ x 3 ∀ x 4 )( � | G | � 3 1 � i < j � 4 x i = x j ) ( ∀ x )( x 6 = 1 → x = 1) no elements of order 2 , 3 g 4 = 1 ∧ g 2 � = 1 g has order 4 ( ∀ k � = 1)( ∀ g )( ∃ r ∈ N )( ∃ x 1 , . . . , x r )( g = k x 1 k x 2 . . . k x r ) No! John Wilson June 27, 2018 4 / 1

Classes of finite groups defined by a sentence (1) { groups of order � n } , { groups of order � n } , { groups with no elements of order n } , nilpotent groups of class � 2. (2) Felgner’s Theorem (1990). ∃ sentence σ (in the f.-o. language of group theory) such that, for G finite, G | = σ ⇔ G is non-abelian simple. σ = σ 1 ∧ σ 2 with g ∈ G (C G ( x , y )C G ( C G ( x , y ))) g = { 1 } ) , σ 1 : ( ∀ x ∀ y )( x � = 1 ∧ C G ( x , y ) � = { 1 } → � σ 2 : ‘each element is a product of κ 0 commutators’ for a fixed κ 0 ∈ N . (3) Finite soluble groups: They are characterized (among finite groups) by ‘no g � = 1 is a prod. of commutators [ g h , g k ]’; that is, ρ n holds ∀ n ρ n : ( ∀ g ∀ x 1 . . . ∀ x n ∀ y 1 . . . ∀ y n )( g = 1 ∨ g � = [ g x 1 , g y 1 ] . . . [ g x n , g y n ]) . Theorem (JSW 2005). Finite G is soluble iff it satisfies ρ 56 . John Wilson June 27, 2018 5 / 1

Quasisimple groups G is quasisimple if G perfect and G / Z( G ) simple Proposition (JSW 2017). Finite G is quasisimple iff Q satisfies QS 1 ∧ QS 2 ∧ QS 3 : QS 1 : each element is a product of two commutators; QS 2 : ( ∀ x )( ∀ u )[ x , x u ] ∈ Z( G ) → x ∈ Z( G ); QS 3 : G ( x , y )) g = Z( G ). g ∈ G (C G ( x , y )C 2 ∈ Z( G ) ∧ C G ( x , y ) > Z( G )) → � ( ∀ x ∀ y )( x / John Wilson June 27, 2018 6 / 1

Definable sets sets of elements g ∈ G (or in G ( n ) = G × · · · × G ) defined by . . . first-order formulae, possibly with parameters from G . Examples: • Z( G ), defined by ( ∀ y )([ x , y ] = 1) • C G ( h ), defined by [ x , h ] = 1 • Centralizers of definable sets are definable: Say S = { s | ϕ ( s ) } ; then C G ( S ) = { t | ∀ g ( ϕ ( g ) → [ g , t ] = 1) } The ( soluble ) radical R( G ) of a finite group G is the largest soluble normal subgroup of G . Theorem (JSW 2008). There’s a f.-o. formula r ( x ) such that if G is finite and g ∈ G then g ∈ R( G ) iff r ( g ) holds in G . John Wilson June 27, 2018 7 / 1

The sets X h , W h • X h = { [ h − 1 , h g ] | g ∈ G } , W h = � { X h g | g ∈ G , [ X h , X h g ] � = 1 } . ( ∃ y )( x = [ h − 1 , h y ]) ϕ 1 ( h , x ): (defines X h ) ( ∃ t ∃ y 1 ∃ y 2 )( ϕ 1 ( h , y 1 ) ∧ ϕ 1 ( h t , y 2 ) ∧ ϕ 1 ( h t , x ) ∧ [ y 1 , y 2 ] � = 1) ϕ 2 ( h , x ): (defines W h ) ( ∀ y )( ϕ 2 ( h , y ) → [ x , y ] = 1) ϕ 3 ( h , x ): C G ( W h ) C 2 γ ( h , x ): ( ∀ y )( φ 3 ( h , y ) → [ x , y ] = 1) G ( W h ) • ε � ( x , y ): ε � ( h 1 , h 2 ) iff C 2 G ( W h 1 ) � C 2 G ( W h 2 ) { ( h 1 , h 2 ) | ε � ( h 1 , h 2 ) } definable in G × G ; leads to a definable equiv. relation • ∃ β ( x ): β ( h ) iff C 2 G ( W h ) commutes with its distinct conjugates. John Wilson June 27, 2018 8 / 1

G finite: component = quasisimple subgroup Q that commutes with its distinct G -conjugates ( ⇔ Q subnormal). Theorem (JSW 2017). ∃ f.o. formulae π ( h , y ), π ′ ( h ), π ′ c ( h ), π ′ m ( h ) such that for every finite G , the products of components of G are the sets { x | π ( h , x ) } for the h ∈ G satisfying π ′ ( h ) . The components: the sets { x | π ( h , x ) } for which π ′ c ( h ) holds. The non-ab. min. normal subgps.: { x | π ( h , x ) } with π ′ m ( h ). John Wilson June 27, 2018 9 / 1

Define δ r for r � 1 recursively by δ 1 ( x 1 , x 2 ) = [ x 1 , x 2 ] and δ r ( x 1 , . . . , x 2 r ) = [ δ r − 1 ( x 1 , . . . , x 2 r − 1 ) , δ r − 1 ( x 2 r − 1 +1 , . . . , x 2 r )] for r > 1. C 2 γ ( h , x ): ( ∀ y )( φ 3 ( h , y ) → [ x , y ] = 1) G ( W h ) � � 16 α 1 ( h , x ): � ( ∃ y 1 . . . ∃ y 16 )( n =1 γ ( h , y n ) ∧ x = δ 4 ( y 1 , . . . , y 16 )) δ 4 -value in C 2 G ( W h ) ( ∃ y 1 ∃ y 2 )( α 1 ( h , y 1 ) ∧ α 1 ( h , y 1 ) ∧ x = y 1 y 2 ) α ( h , x ): Let G be finite, Q a component. If h ∈ Q \ Z( Q ) then Q = � W h � , so Q � C 2 G ( W h ). Show Q = set of prods. of 2 δ 4 -values in C 2 G ( W h ), so Q = { x | α ( h , x ) } . John Wilson June 27, 2018 10 / 1

Automorphism groups of ordered sets Write Aut O (Ω) for the group of order AMs of a totally ordered set Ω. Theorem (Andrew Glass and JSW, 2017). Suppose that Aut O (Ω) acts transitively on Ω. (a) If Aut O (Ω), Aut O ( R ) satisfy the same first-order sentences then Ω ∼ = R (as ordered set). (b) If Aut O (Ω), Aut O ( Q ) satisfy the same first-order sentences then Ω ∼ = Q or Ω ∼ = R \ Q . Same conclusion by Gurevich and Holland (1981) with the stronger hypothesis that Aut O (Ω) acts transitively on pairs ( α, β ) with α < β . Transitivity is necessary. Let Ω = R × { 0 , 1 } with alphabetic order: ( r 1 , λ 1 ) < ( r 2 , λ 2 ) if r 1 < r 2 or if r 1 = r 2 and λ 1 < λ 2 . Then Aut O ( R × { 0 , 1 } ) ∼ = Aut O ( R ). John Wilson June 27, 2018 11 / 1

Let Ω be totally ordered. For f , g ∈ Aut O (Ω) define f ∨ g , f ∧ g ∈ Aut O (Ω) by α ( f ∨ g ) = max { α f , α g } , α ( f ∧ g ) = min { α f , α g } for all α ∈ Ω . An ℓ - permutation group on Ω is a subgroup G � Aut O (Ω) closed for ∨ , ∧ . Let G be a trans. ℓ -perm. group on Ω. A convex set ∆ ⊆ Ω is an o- block if either ∆ g = ∆ or ∆ g ∩ ∆ = ∅ for each g ∈ G . Stabilizer and rigid stabilizer of o-block ∆ are defined by Stab(∆) := { g ∈ G | ∆ g = ∆ } , rst(∆) := { g ∈ G | supp( g ) ⊆ ∆ } , G is o-primitive if � ∃ o-blocks apart from Ω and singletons. G is o-2 transitive if transitive on all ( α 1 , α 2 ) ∈ Ω × Ω with α 1 < α 2 . o-2-transitivity = ⇒ o-primitivity. ‘McCleary’s Trichotomy’ . Transitive f.d. o-primitive ℓ -permutation groups are o-2 transitive or right regular representations of subgroups of R . John Wilson June 27, 2018 12 / 1

Technicalities Lemma. Let G be o-2 transitive on Ω and g , h ∈ G with supp( h ) ∩ supp( h g ) = ∅ and h � = 1. Then ∃ f , k ∈ G such that [ h − 1 , h f ][ h − g , h gk ] � = [ h − g , h gk ][ h − 1 , h f ] . For g ∈ G and each union Λ of convex g -invariant subsets of Ω, let dep( g , Λ) be the element of Aut O (Ω) that agrees with g on Λ and with the identity elsewhere. Say G fully depressible ( f.d. ) on Ω if dep( g , Λ) ∈ G for all g ∈ G and all such Λ ⊆ Ω. Aut O (Ω) is fully depressible. John Wilson June 27, 2018 13 / 1

Let G be a f.d. transitive ℓ -perm. group on Ω. Write B ( α, β ) for the smallest o-block containing both α, β ∈ Ω. Let T = { B ( α, β ) | α � = β } . Assume Stab(∆) acts on ∆ as a non-abelian group for all ∆ ∈ T . Recall that X h := { [ h − 1 , h g ] | g ∈ G } � and W h = { X h g | g ∈ G , [ X h , X h g ] � = 1 } . For ∆ ∈ T , let Q ∆ = { h ∈ rst(∆) | ( ∃ α ∈ Ω)( B ( α h , α )) = ∆ } . As G transitive and f.d., Q ∆ � = ∅ . Since (rst(∆)) g = rst(∆ g ) commutes with rst(∆) for g / ∈ Stab(∆), we have X h ⊆ rst(∆) and W h ⊆ rst(∆) for all ∆ ∈ T and h ∈ Q ∆ . John Wilson June 27, 2018 14 / 1