Group Rings and Amalgamated Products: The (FA) Property Groups, - PowerPoint PPT Presentation

Group Rings and Amalgamated Products: The (FA) Property Groups, Rings and Associated Structures Doryan Temmerman June 11, 2019 1

Geometric Group Theory 2

Geometric Group Theory EXAMPLE 1 T = Γ = ( Z , +) 3

Geometric Group Theory EXAMPLE 1 T = Γ = ( Z , +) ⇓ T / Γ = 3

Geometric Group Theory HNN EXTENSION B ≤ A groups f : B ֒ → A ⇒ A ∗ f = � A , t | ∀ b ∈ B : b t = f ( b ) � 4

Geometric Group Theory HNN EXTENSION B ≤ A groups f : B ֒ → A ⇒ A ∗ f = � A , t | ∀ b ∈ B : b t = f ( b ) � Γ is a HNN extension ⇒ | Γ ab | = ∞ ⇒ ∃ T on which Γ acts such that T / Γ = 4

Geometric Group Theory HNN EXTENSION B ≤ A groups f : B ֒ → A ⇒ A ∗ f = � A , t | ∀ b ∈ B : b t = f ( b ) � Γ is a HNN extension ⇒ | Γ ab | = ∞ ⇒ ∃ T on which Γ acts such that T / Γ = ⇒ Γ is a HNN extension 4

Geometric Group Theory AMALGAMATED PRODUCT A , B , C groups f : C ֒ → A , g : C ֒ → B ⇒ A ∗ C B = � A , B | ∀ c ∈ C : f ( c ) = g ( c ) � Non-trivial if neither f nor g are surjections. 5

Geometric Group Theory EXAMPLE 2 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: 6

Geometric Group Theory EXAMPLE 2 C 4 ∗ C 3 acts on the tree: Stabilizer of P is C 4 and the stabilizer of Q is C 3 . The stabilizer of y is the trivial group. 6

Geometric Group Theory AMALGAMATED PRODUCT CONT. Theorem (Serre ’68) Q P y A group Γ acts on a tree with as fundamental domain if and only if there exist groups A , B and C such that Γ ∼ = A ∗ C B . Moreover, in this case, A ∼ = Γ P , B ∼ = Γ Q and C ∼ = Γ y , the stabilizers in Γ of P , Q and y respectively. 7

Geometric Group Theory TORSION ELEMENTS AND PROPERTY (FA) Fact Torsion elements of A ∗ f are conjugate to elements of A . Torsion elements of A ∗ C B are conjugate to elements of A or B . 8

Geometric Group Theory TORSION ELEMENTS AND PROPERTY (FA) Fact Torsion elements of A ∗ f are conjugate to elements of A . Torsion elements of A ∗ C B are conjugate to elements of A or B . Definition (Property (FA)) A group Γ is said to have property (FA) if every Γ -action on a tree, without inversion, has a global fix point. 8

Geometric Group Theory TORSION ELEMENTS AND PROPERTY (FA) Fact Torsion elements of A ∗ f are conjugate to elements of A . Torsion elements of A ∗ C B are conjugate to elements of A or B . Definition (Property (FA)) A group Γ is said to have property (FA) if every Γ -action on a tree, without inversion, has a global fix point. Lemma (Serre, ’68) For a finitely generated group Γ holds Γ has property (FA) ⇔ ◮ Γ is not a HNN extension ◮ Γ is not an amalgamated product 8

Group Rings 9

Group Rings PROJECT Question Let G be a finite group. When does U ( Z G ) have (FA) ? 10

Group Rings THE PROBLEM WITH (FA)... Fact Let K be a finite index subgroup of Γ , then K has (FA) ⇒ Γ has (FA) 11

Group Rings THE PROBLEM WITH (FA)... Fact Let K be a finite index subgroup of Γ , then K has (FA) ⇒ Γ has (FA) √ � � �� − 3 1 + SL 2 Z has (FA) 2 f.i. � √ � �� − 3 SL 2 Z does not have (FA) 11

Group Rings THE SOLUTION Definition (Property (HFA)) A group Γ is said to have property (HFA) if every finite index subgroup has property (FA). Fact Let K be a finite index subgroup of Γ , then K has (HFA) ⇔ Γ has (HFA) 12

Group Rings (HFA) INSTEAD OF (FA) Question Let G be a finite group. When does U ( Z G ) have (HFA) ? Idea: reduction to special linear groups SL n ( O ) over orders, i.e. a subring of a Q -algebra which is a free Z -module and contains a Q -basis for the algebra. 13

Group Rings THE REDUCTION ARGUMENT m m Z G ⊆ Q G ∼ � � M n i ( D i ) ⊇ M n i ( O i ) , = i =1 i =1 with O i orders of D i . m � U ( Z G ) GL n i ( O i ) i =1 f.i. f.i. m � U ( Z G ) ∩ GL n i ( O i ) i =1 14

Group Rings THE REDUCTION ARGUMENT (CONT.) Proposition (Bächle-Janssens-Jespers-Kiefer-T.) U ( Z G ) has (HFA) ⇔ G is a cut group and ∀ i : SL n i ( O i ) has (HFA) 15

Group Rings THE REDUCTION ARGUMENT (CONT.) Proposition (Bächle-Janssens-Jespers-Kiefer-T.) U ( Z G ) has (HFA) ⇔ G is a cut group and ∀ i : SL n i ( O i ) has (HFA) Theorem (Margulis, ’91) The groups SL n ( O ) appearing in the context above have property (HFA) when not in one of the following cases: ◮ n = 1 or, √ ◮ n = 2 and D ∼ − d ) or D ∼ � � − a , − b , for some a , b ∈ N > 0 . = Q ( = Q 15

Group Rings THE EXCEPTIONS N = 1 Proposition (Bächle-Janssens-Jespers-Kiefer-T.) For a cut group G , when n i = 1 , D i ∼ � − d i ) for d i ∈ N or = Q ( D i ∼ � � − a i , − b i for some a i , b i ∈ N > 0 . Hence, U ( O i ) and SL n i ( O i ) = Q are finite and have property (HFA) . 16

Group Rings THE EXCEPTIONS N = 2 √ √ √ � � �� − 1]) , SL 2 ( Z [ − 2]) , SL 2 , SL 2 ( O 2 ) , SL 2 ( Z ) , SL 2 ( Z [ Z 1+ − 3 2 SL 2 ( O 3 ) and SL 2 ( O 5 ) . O 2 , O 3 and O 5 are non-commutative maximal orders of � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 respectively , , . Q Q Q 17

Group Rings PROPERTY (HFA) FOR U ( Z G ) Theorem (Bächle-Janssens-Jespers-Kiefer-T.) U ( Z G ) has (HFA) ⇔ G is a cut group and does not have an epimorphic image in a specific list of 10 groups 18

Group Rings PROPERTY (HFA) FOR U ( Z G ) Theorem (Bächle-Janssens-Jespers-Kiefer-T.) U ( Z G ) has (HFA) ⇔ G is a cut group and does not have an epimorphic image in a specific list of 10 groups ⇔ U ( Z G ) has Kazhdan’s property (T) 18

Group Rings PROPERTY (HFA) FOR U ( Z G ) Theorem (Bächle-Janssens-Jespers-Kiefer-T.) U ( Z G ) has (HFA) ⇔ G is a cut group and does not have an epimorphic image in a specific list of 10 groups ⇔ U ( Z G ) has Kazhdan’s property (T) ⇔ All finite index subgroups of U ( Z G ) have finite abelianization 18