Property (FA) of the unit group of 2 -by- 2 matrices over an order - PowerPoint PPT Presentation

Property (FA) of the unit group of 2 -by- 2 matrices over an order in a quaternion algebra Ann Kiefer joint work with Bächle, Janssens, Jespers and Temmerman 10 June 2019 1

SIMPLE QUESTION � � SL 2 − 1 , − 1 Z 2

SIMPLE QUESTION � � SL 2 − 1 , − 1 acts on Z Hautes Fagnes / High Fens 2

SIMPLE QUESTION � � SL 2 − 1 , − 1 acts on Z Hautes Fagnes / High Fens Does the action have a fixed point? 2

SERRE’S PROPERTY (FA) POINTS FIXES SUR LES ARBRES Definition A group Γ is said to have property (FA) if every action on a tree has a fixed point. 3

SERRE’S PROPERTY (FA) POINTS FIXES SUR LES ARBRES Definition A group Γ is said to have property (FA) if every action on a tree has a fixed point. Theorem (Serre) A finitely generated group Γ has property (FA) if and only if it satisfies the following properties ◮ Γ has finite abelianization, ◮ Γ has no non-trivial decomposition as amalgamated product. 3

THE HEREDITARY VERSION OF (FA) K subgroup of finite index in Γ K has (FA) ⇒ Γ has (FA) Problem: Γ has (FA) ✟ ⇒ K has (FA) ✟ Definition (Property (HFA)) A group Γ is said to have property (HFA) if every finite index subgroup has property (FA). 4

MOTIVATION When does U ( Z G ) has (FA)? When does U ( Z G ) has (HFA)? Cases that have to be handled � � ◮ SL 2 ( O 2 ) , with O 2 maximal order in − 1 , − 1 Q � � ◮ SL 2 ( O 3 ) , with O 3 maximal order in − 1 , − 3 Q � � ◮ SL 2 ( O 5 ) , with O 5 maximal order in − 2 , − 5 Q 5

BACK TO THE BEGINNING � � ◮ SL 2 ( O 2 ) , with O 2 maximal order in − 1 , − 1 Q � � − 1 , − 1 Easier: SL 2 Z 6

BACK TO THE BEGINNING � � ◮ SL 2 ( O 2 ) , with O 2 maximal order in − 1 , − 1 Q � � − 1 , − 1 Easier: SL 2 Z � � − 1 , − 1 SL 2 acts on Z Does the action have a fixed point? 6

STEP BY STEP known: SL 2 ( Z ) = C 4 ∗ C 2 C 6 known: SL 2 ( Z [ i ]) = G 1 ∗ SL 2 ( Z ) G 2 7

STEP BY STEP known: SL 2 ( Z ) = C 4 ∗ C 2 C 6 known: SL 2 ( Z [ i ]) = G 1 ∗ SL 2 ( Z ) G 2 Generalization: Vahlen Group SL + (Γ n ( Z )) n=1 SL 2 ( Z ) n=2 SL 2 ( Z [ i ]) � � n=4 subgroup of finite index in SL 2 − 1 , − 1 Z 7

STEP BY STEP known: SL 2 ( Z ) = C 4 ∗ C 2 C 6 known: SL 2 ( Z [ i ]) = G 1 ∗ SL 2 ( Z ) G 2 Generalization: Vahlen Group SL + (Γ n ( Z )) n=1 SL 2 ( Z ) n=2 SL 2 ( Z [ i ]) � � n=4 subgroup of finite index in SL 2 − 1 , − 1 Z Theorem (Bächle-Janssens-Jespers-K.-Temmerman) SL + (Γ 3 ( Z )) = H 1 ∗ SL 2 ( Z [ i ]) H 2 SL + (Γ 4 ( Z )) = K 1 ∗ SL + (Γ 3 ( Z )) K 2 � � → SL 2 − 1 , − 1 does not have (HFA) Z 7

WHAT ABOUT THE OTHER QUATERNION ALGEBRAS? E 2 group generated by elementary matrices E 2 has finite index in SL 2 ( O ) � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 O maximal order in Q Q Q E 2 ( O ) finite abelianization E 2 ( O ) amalgamated product E 2 ( O ) (FA) E 2 ( O ) (HFA) 8

WHAT ABOUT THE OTHER QUATERNION ALGEBRAS? E 2 group generated by elementary matrices E 2 has finite index in SL 2 ( O ) � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 O maximal order in Q Q Q E 2 ( O ) finite abelianization E 2 ( O ) amalgamated product E 2 ( O ) (FA) E 2 ( O ) (HFA) × 8

WHAT ABOUT THE OTHER QUATERNION ALGEBRAS? E 2 group generated by elementary matrices E 2 has finite index in SL 2 ( O ) � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 O maximal order in Q Q Q E 2 ( O ) finite abelianization � � × E 2 ( O ) amalgamated product E 2 ( O ) (FA) E 2 ( O ) (HFA) × 8

WHAT ABOUT THE OTHER QUATERNION ALGEBRAS? E 2 group generated by elementary matrices E 2 has finite index in SL 2 ( O ) � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 O maximal order in Q Q Q E 2 ( O ) finite abelianization � � × E 2 ( O ) amalgamated product × × ? E 2 ( O ) (FA) E 2 ( O ) (HFA) × 8

WHAT ABOUT THE OTHER QUATERNION ALGEBRAS? E 2 group generated by elementary matrices E 2 has finite index in SL 2 ( O ) � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 O maximal order in Q Q Q E 2 ( O ) finite abelianization � � × E 2 ( O ) amalgamated product × × ? E 2 ( O ) (FA) � � × E 2 ( O ) (HFA) × 8

WHAT ABOUT THE OTHER QUATERNION ALGEBRAS? E 2 group generated by elementary matrices E 2 has finite index in SL 2 ( O ) � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 O maximal order in Q Q Q E 2 ( O ) finite abelianization � � × E 2 ( O ) amalgamated product × × ? E 2 ( O ) (FA) � � × E 2 ( O ) (HFA) × × × 8

WHAT ABOUT THE OTHER QUATERNION ALGEBRAS? E 2 group generated by elementary matrices E 2 has finite index in SL 2 ( O ) � � � � � � − 1 , − 1 − 1 , − 3 − 2 , − 5 O maximal order in Q Q Q E 2 ( O ) finite abelianization � � × E 2 ( O ) amalgamated product × × ? E 2 ( O ) (FA) � � × E 2 ( O ) (HFA) × × × SL 2 ( O ) (FA) � � ? SL 2 ( O ) (HFA) × × × 8

TRAILER When does U ( Z G ) has (FA)? When does U ( Z G ) has (HFA)? Tomorrow 11:00 Doryan’s talk 9

Thank you for your attention 10