The Euclidean Algorithm in Circle/Sphere Packings Arseniy (Senia) - PowerPoint PPT Presentation

The Euclidean Algorithm in Circle/Sphere Packings Arseniy (Senia) Sheydvasser October 25, 2019

Circle/Sphere Inversions 3 3 2 2 1 1 - 3 - 2 - 1 1 2 3 - 3 - 2 - 1 1 2 3 - 1 - 1 - 2 - 2

Circle/Sphere Inversions 3 3 2 2 1 1 - 3 - 2 - 1 1 2 3 - 3 - 2 - 1 1 2 3 - 1 - 1 - 2 - 2 ◮ Choose a circle C with center ( x 0 , y 0 ) and radius R .

Circle/Sphere Inversions 3 3 2 2 1 1 - 3 - 2 - 1 1 2 3 - 3 - 2 - 1 1 2 3 - 1 - 1 - 2 - 2 ◮ Choose a circle C with center ( x 0 , y 0 ) and radius R . ◮ To invert a point ( x , y ) through, measure the distance r between ( x 0 , y 0 ) and ( x , y ), and move ( x , y ) to distance R / r from ( x 0 , y 0 ) (along the same ray).

Circle/Sphere Inversions

Circle/Sphere Inversions

Circle/Sphere Inversions Definition ob( R n ) is the group generated by n -sphere reflections in M¨ R n ∪ {∞} .

Circle/Sphere Inversions Definition ob( R n ) is the group generated by n -sphere reflections in M¨ R n ∪ {∞} . Question ob ( R n ) , and S an n-sphere. What does Let Γ be a subgroup of M¨ the orbit Γ . S look like? Can we compute it effectively?

Motivation Question What analogs of the Apollonian circle packing are there?

Motivation Question What do hyperbolic quotient manifolds H n / Γ look like? 3.0 2.5 2.0 1.5 1.0 0.5 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 1.5 SL (2 , Z ) SL (2 , Z [ i ])

Motivation

Accidental Isomorphisms Question ob ( R n ) ? How do you even represent elements in M¨

Accidental Isomorphisms Question ob ( R n ) ? How do you even represent elements in M¨ ob 0 ( R ) M¨ SL (2 , R ) / {± 1 } � a � b ◮ Let be a matrix in c d ob 0 ( R 2 ) M¨ SL (2 , C ) / {± 1 } SL (2 , R ) or SL (2 , C ). ◮ z �→ ( az + b )( cz + d ) − 1 is ob 0 ( R 3 ) M¨ an orientation-preserving M¨ obius transformation. ◮ z �→ ( az + b )( cz + d ) − 1 is ob 0 ( R 4 ) M¨ SL (2 , H ) / {± 1 } an orientation-reversing M¨ obius transformation. . . .

Accidental Isomorphisms Question ob ( R n ) ? How do you even represent elements in M¨ ob 0 ( R ) M¨ SL (2 , R ) / {± 1 } � a � b ◮ Let be a matrix in c d ob 0 ( R 2 ) M¨ SL (2 , C ) / {± 1 } SL (2 , R ) or SL (2 , C ). ◮ z �→ ( az + b )( cz + d ) − 1 is ob 0 ( R 3 ) M¨ ??? an orientation-preserving M¨ obius transformation. ◮ z �→ ( az + b )( cz + d ) − 1 is ob 0 ( R 4 ) M¨ SL (2 , H ) / {± 1 } an orientation-reversing M¨ obius transformation. . . . ???

Vahlen’s Matrices ◮ Vahlen, 1901: For any n , there is an isomorphism between ob( R n ) and a group of 2 × 2 matrices with entries in a M¨ (subset of a) Clifford algebra, quotiented by {± 1 } .

Vahlen’s Matrices ◮ Vahlen, 1901: For any n , there is an isomorphism between ob( R n ) and a group of 2 × 2 matrices with entries in a M¨ (subset of a) Clifford algebra, quotiented by {± 1 } . ◮ We’ll consider the case n = 3, M¨ ob( R 3 ).

Vahlen’s Matrices ◮ Vahlen, 1901: For any n , there is an isomorphism between ob( R n ) and a group of 2 × 2 matrices with entries in a M¨ (subset of a) Clifford algebra, quotiented by {± 1 } . ◮ We’ll consider the case n = 3, M¨ ob( R 3 ). ◮ Define ( w + xi + yj + zk ) ‡ = w + xi + yj − zk and H + = quaternions fixed by ‡ (i.e. with no k -component).

Vahlen’s Matrices ◮ Vahlen, 1901: For any n , there is an isomorphism between ob( R n ) and a group of 2 × 2 matrices with entries in a M¨ (subset of a) Clifford algebra, quotiented by {± 1 } . ◮ We’ll consider the case n = 3, M¨ ob( R 3 ). ◮ Define ( w + xi + yj + zk ) ‡ = w + xi + yj − zk and H + = quaternions fixed by ‡ (i.e. with no k -component). �� a � � � b � ab ‡ , cd ‡ ∈ H + , ad ‡ − bc ‡ = 1 SL ‡ (2 , H ) = � ∈ Mat(2 , H ) � c d

What is SL ‡ (2 , H ) as a Group? �� a �� � b � ab ‡ , cd ‡ ∈ H + , ad ‡ − bc ‡ = 1 SL ‡ (2 , H ) = � � c d

What is SL ‡ (2 , H ) as a Group? �� a �� � b � ab ‡ , cd ‡ ∈ H + , ad ‡ − bc ‡ = 1 SL ‡ (2 , H ) = � � c d Equivalently, � 0 � 0 � � � �� k k γ T = SL ‡ (2 , H ) = � γ ∈ SL (2 , H ) � γ � − k 0 − k 0

What is SL ‡ (2 , H ) as a Group? �� a �� � b � ab ‡ , cd ‡ ∈ H + , ad ‡ − bc ‡ = 1 SL ‡ (2 , H ) = � � c d Equivalently, � 0 � 0 � � � �� k k γ T = SL ‡ (2 , H ) = � γ ∈ SL (2 , H ) � γ � − k 0 − k 0 Inverses are given as follows: � d ‡ � − 1 � a − b ‡ � b = − c ‡ a ‡ c d

ob( R 3 ) as SL ‡ (2 , H ) M¨ ◮ There is an action on R 3 ∪ {∞} = H + ∪ {∞} defined by � a � b . z = ( az + b )( cz + d ) − 1 c d

ob( R 3 ) as SL ‡ (2 , H ) M¨ ◮ There is an action on R 3 ∪ {∞} = H + ∪ {∞} defined by � a � b . z = ( az + b )( cz + d ) − 1 c d ◮ Every orientation-preserving element of M¨ ob( R 3 ) can be written as z �→ ( az + b )( cz + d ) − 1 . ◮ Every orientation-reversing element of M¨ ob( R 3 ) can be written as z �→ ( az + b )( cz + d ) − 1 .

Arithmetic Groups Definition What sort of subgroups Γ of SL ‡ (2 , H ) should we consider? ◮ We will ask that Γ is discrete. ◮ We will ask that Γ carries some sort of algebraic structure.

Arithmetic Groups Definition What sort of subgroups Γ of SL ‡ (2 , H ) should we consider? ◮ We will ask that Γ is discrete. ◮ We will ask that Γ carries some sort of algebraic structure. ◮ We will ask that Γ is arithmetic .

Arithmetic Groups Definition What sort of subgroups Γ of SL ‡ (2 , H ) should we consider? ◮ We will ask that Γ is discrete. ◮ We will ask that Γ carries some sort of algebraic structure. ◮ We will ask that Γ is arithmetic . ◮ Note that SL ‡ (2 , H ) can be seen as real solutions to a set of polynomial equations. ◮ Roughly, an arithmetic group is the set of integer solutions to that set of polynomial equations.

Arithmetic Groups Definition What sort of subgroups Γ of SL ‡ (2 , H ) should we consider? ◮ We will ask that Γ is discrete. ◮ We will ask that Γ carries some sort of algebraic structure. ◮ We will ask that Γ is arithmetic . ◮ Note that SL ‡ (2 , H ) can be seen as real solutions to a set of polynomial equations. ◮ Roughly, an arithmetic group is the set of integer solutions to that set of polynomial equations. ◮ Not quite true—can only define up to commensurability—but ignore that.

Examples of Arithmetic Groups ◮ Γ = SL (2 , Z ) ◮ Γ( N )

Examples of Arithmetic Groups ◮ Γ = SL (2 , Z ) ◮ Γ( N ) ◮ SL (2 , Z [ i ]) ◮ SL (2 , Z [ √− 2]) 1+ √− 3 � � �� ◮ SL 2 , Z 2

Examples of Arithmetic Groups ◮ Γ = SL (2 , Z ) ◮ Γ( N ) ◮ SL (2 , Z [ i ]) ◮ SL (2 , Z [ √− 2]) 1+ √− 3 � � �� ◮ SL 2 , Z 2 ◮ What about SL ‡ (2 , H )?

Examples of Arithmetic Groups inside SL ‡ (2 , H ) ◮ Classical answer: choose a quadratic form q of signature (4 , 1), and take SO + ( q , Z ) (use the classical isomorphism SO + (4 , 1) ∼ ob( R 3 ) to make sense of this) = M¨

Examples of Arithmetic Groups inside SL ‡ (2 , H ) ◮ Classical answer: choose a quadratic form q of signature (4 , 1), and take SO + ( q , Z ) (use the classical isomorphism SO + (4 , 1) ∼ ob( R 3 ) to make sense of this) = M¨ ◮ Very hard to find any non-trivial elements of this group. ◮ − 4 X 2 1 + 2 X 2 X 1 + X 3 X 1 − 3 X 4 X 1 + 5 X 2 2 + 6 X 2 3 + 7 X 2 4 + 22 X 2 5 − 5 X 2 X 3 + X 2 X 4 + X 3 X 4 − X 2 X 5 + 2 X 3 X 5 + 4 X 4 X 5

Examples of Arithmetic Groups inside SL ‡ (2 , H ) ◮ Classical answer: choose a quadratic form q of signature (4 , 1), and take SO + ( q , Z ) (use the classical isomorphism SO + (4 , 1) ∼ ob( R 3 ) to make sense of this) = M¨ ◮ Very hard to find any non-trivial elements of this group. ◮ − 4 X 2 1 + 2 X 2 X 1 + X 3 X 1 − 3 X 4 X 1 + 5 X 2 2 + 6 X 2 3 + 7 X 2 4 + 22 X 2 5 − 5 X 2 X 3 + X 2 X 4 + X 3 X 4 − X 2 X 5 + 2 X 3 X 5 + 4 X 4 X 5 ◮ There is a better way!

Examples of Arithmetic Groups inside SL ‡ (2 , H ) ◮ Let O be an order of H that is closed under ‡ (i.e. O = O ‡ ). ◮ Then SL ‡ (2 , O ) = SL ‡ (2 , H ) ∩ Mat(2 , O ) is an arithmetic group.

Examples of Arithmetic Groups inside SL ‡ (2 , H ) ◮ Let O be an order of H that is closed under ‡ (i.e. O = O ‡ ). ◮ Then SL ‡ (2 , O ) = SL ‡ (2 , H ) ∩ Mat(2 , O ) is an arithmetic group. ◮ Here, an order means a sub-ring that is also a lattice. √ √ √ √ √ 2 i ⊕ Z 1 + 2 i + 5 j 2 i + 10 k ◮ Example: O = Z ⊕ Z ⊕ Z 2 2

Maximal ‡ -Orders ◮ Why ask that O = O ‡ ? � d ‡ � − 1 − b ‡ � a b � ◮ Recall that = − c ‡ a ‡ c d

Maximal ‡ -Orders ◮ Why ask that O = O ‡ ? � d ‡ � − 1 − b ‡ � a b � ◮ Recall that = − c ‡ a ‡ c d Definition If O is an order of H closed under ‡ , we say that O is a ‡ - order . If O is not contained inside any larger ‡ -order, we say that it is a maximal ‡ - order .