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

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Group Rings and Amalgamated Products: The (FA) Property

Groups, Rings and Associated Structures Doryan Temmerman June 11, 2019

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Geometric Group Theory

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Geometric Group Theory

EXAMPLE 1

T = Γ = (Z, +)

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Geometric Group Theory

EXAMPLE 1

T = Γ = (Z, +) ⇓ T/Γ =

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Geometric Group Theory

HNN EXTENSION

B ≤ A groups f : B ֒ → A ⇒ A∗f = A, t | ∀b ∈ B : bt = f(b)

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Geometric Group Theory

HNN EXTENSION

B ≤ A groups f : B ֒ → A ⇒ A∗f = A, t | ∀b ∈ B : bt = f(b) Γ is a HNN extension ⇒ |Γab| = ∞ ⇒ ∃T on which Γ acts such that T/Γ =

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Geometric Group Theory

HNN EXTENSION

B ≤ A groups f : B ֒ → A ⇒ A∗f = A, t | ∀b ∈ B : bt = f(b) Γ is a HNN extension ⇒ |Γab| = ∞ ⇒ ∃T on which Γ acts such that T/Γ = ⇒ Γ is a HNN extension

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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.

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Geometric Group Theory

EXAMPLE 2

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree:

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Geometric Group Theory

EXAMPLE 2

C4 ∗ C3 acts on the tree: Stabilizer of P is C4 and the stabilizer of Q is C3. The stabilizer of y is the trivial group.

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Geometric Group Theory

AMALGAMATED PRODUCT CONT. Theorem (Serre ’68)

A group Γ acts on a tree with as fundamental domain

y P Q

if and only if there exist groups A, B and C such that Γ ∼ = A ∗C B. Moreover, in this case, A ∼ = ΓP, B ∼ = ΓQ and C ∼ = Γy, the stabilizers in Γ of P, Q and y respectively.

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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.

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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.

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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

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Group Rings

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Group Rings

PROJECT Question

Let G be a finite group. When does U(ZG) have (FA)?

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Group Rings

THE PROBLEM WITH (FA)... Fact

Let K be a finite index subgroup of Γ, then K has (FA) ⇒ Γ has (FA)

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Group Rings

THE PROBLEM WITH (FA)... Fact

Let K be a finite index subgroup of Γ, then K has (FA) ⇒ Γ has (FA) SL2

• Z
• 1 +

√ −3 2

• has (FA)

f.i. SL2

• Z

√

−3

• does not have (FA)

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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)

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Group Rings

(HFA) INSTEAD OF (FA) Question

Let G be a finite group. When does U(ZG) have (HFA)? Idea: reduction to special linear groups SLn(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.

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Group Rings

THE REDUCTION ARGUMENT

ZG ⊆ QG ∼ =

m

• i=1

Mni(Di) ⊇

m

• i=1

Mni(Oi), with Oi orders of Di. U(ZG)

m

• i=1

GLni(Oi) U(ZG) ∩

m

• i=1

GLni(Oi)

f.i. f.i.

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Group Rings

THE REDUCTION ARGUMENT (CONT.) Proposition (Bächle-Janssens-Jespers-Kiefer-T.)

U(ZG) has (HFA) ⇔ G is a cut group and ∀i : SLni(Oi) has (HFA)

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Group Rings

THE REDUCTION ARGUMENT (CONT.) Proposition (Bächle-Janssens-Jespers-Kiefer-T.)

U(ZG) has (HFA) ⇔ G is a cut group and ∀i : SLni(Oi) has (HFA)

Theorem (Margulis, ’91)

The groups SLn(O) appearing in the context above have property (HFA) when not in one of the following cases:

◮ n = 1 or, ◮ n = 2 and D ∼

= Q( √ −d) or D ∼ =

• −a,−b

Q

• , for some a, b ∈ N>0.

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Group Rings

THE EXCEPTIONS N = 1 Proposition (Bächle-Janssens-Jespers-Kiefer-T.)

For a cut group G, when ni = 1, Di ∼ = Q(

• −di) for di ∈ N or

Di ∼ =

• −ai,−bi

Q

• for some ai, bi ∈ N>0. Hence, U(Oi) and SLni(Oi)

are finite and have property (HFA).

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Group Rings

THE EXCEPTIONS N = 2

SL2(Z), SL2(Z[ √ −1]), SL2(Z[ √ −2]), SL2

• Z
• 1+

√ −3 2

• , SL2(O2),

SL2(O3) and SL2(O5). O2, O3 and O5 are non-commutative maximal orders of respectively

• −1,−1

Q

• ,
• −1,−3

Q

• ,
• −2,−5

Q

• .

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Group Rings

PROPERTY (HFA) FOR U(ZG) Theorem (Bächle-Janssens-Jespers-Kiefer-T.)

U(ZG) has (HFA) ⇔ G is a cut group and does not have an epimorphic image in a specific list

• f 10 groups

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Group Rings

PROPERTY (HFA) FOR U(ZG) Theorem (Bächle-Janssens-Jespers-Kiefer-T.)

U(ZG) has (HFA) ⇔ G is a cut group and does not have an epimorphic image in a specific list

• f 10 groups

⇔ U(ZG) has Kazhdan’s property (T)

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Group Rings

PROPERTY (HFA) FOR U(ZG) Theorem (Bächle-Janssens-Jespers-Kiefer-T.)

U(ZG) has (HFA) ⇔ G is a cut group and does not have an epimorphic image in a specific list

• f 10 groups

⇔ U(ZG) has Kazhdan’s property (T) ⇔ All finite index subgroups of U(ZG) have finite abelianization