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Subgroup structure of branch groups Alejandra Garrido University of Oxford Groups St Andrews, St Andrews 311 August 2013 Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 1 / 17 What is a branch

  1. Subgroup structure of branch groups Alejandra Garrido University of Oxford Groups St Andrews, St Andrews 3–11 August 2013 Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 1 / 17

  2. What is a branch group? Outline What is a branch group? 1 Why do we study them? 2 Subgroup structure 3 Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 2 / 17

  3. What is a branch group? What is a branch group? 2 definitions: Algebraic Groups whose lattice of subnormal subgroups is similar to the structure of a regular rooted tree. Geometric Groups acting level-transitively on a spherically homogeneous rooted tree T and having a subnormal subgroup structure similar to that of Aut ( T ) . Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 3 / 17

  4. What is a branch group? Definition ( m n ) n ≥ 0 sequence of integers ≥ 2. T is a spherically homogeneous rooted tree of type ( m n ) n if T is a tree with root v 0 of degree m 0 s.t. every vertex at distance n ≥ 1 from v 0 has degree m n + 1. L n = vertices at distance n from root v 0 L 0 m 0 . . . L 1 v T v m 1 . . . . . . . . . . . . L 2 . . . . . . . . . . . . . . . . . . T v is subtree rooted at v Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 4 / 17

  5. What is a branch group? Rigid Stabilisers Definition T - spherically homogeneous tree of type ( m n ) n . G acts faithfully on T . rist G ( v ) := { g ∈ G : g fixes all vertices outside T v } is the rigid stabiliser of v ∈ T . rist G ( n ) := � v ∈L n rist G ( v ) is the rigid stabiliser of level n . v T v Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 5 / 17

  6. What is a branch group? Branch Group Definition and Example Definition G acts as a branch group on T iff for every n : 1 G acts transitively on L n (‘acts level-transitively on T ’) 2 | G : rist G ( n ) | < ∞ Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 6 / 17

  7. What is a branch group? Branch Group Definition and Example Definition G acts as a branch group on T iff for every n : 1 G acts transitively on L n (‘acts level-transitively on T ’) 2 | G : rist G ( n ) | < ∞ Definition G is branch if it acts faithfully as a branch group on some T . Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 6 / 17

  8. What is a branch group? Branch Group Definition and Example Definition G acts as a branch group on T iff for every n : 1 G acts transitively on L n (‘acts level-transitively on T ’) 2 | G : rist G ( n ) | < ∞ Definition G is branch if it acts faithfully as a branch group on some T . Example For all n , A = Aut ( T ) acts transitively on L n with kernel rist A ( n ) . Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 6 / 17

  9. What is a branch group? Example: Gupta–Sidki p -groups T = T ( p ) , p - odd prime a := ( 1 2 . . . p ) on L 1 G := � a , b � b := ( a , a − 1 , 1 , . . . , 1 , b ) . Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 7 / 17

  10. What is a branch group? Example: Gupta–Sidki p -groups T = T ( p ) , p - odd prime a := ( 1 2 . . . p ) on L 1 G := � a , b � b := ( a , a − 1 , 1 , . . . , 1 , b ) . a . . . . . . . . . Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 7 / 17

  11. What is a branch group? Example: Gupta–Sidki p -groups T = T ( p ) , p - odd prime a := ( 1 2 . . . p ) on L 1 G := � a , b � b := ( a , a − 1 , 1 , . . . , 1 , b ) . b a a − 1 b a a − 1 b . . . . . . . . . . . . . . . Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 7 / 17

  12. What is a branch group? Example: (First) Grigorchuk Group T = T ( 2 ) - binary tree ( m n = 2) a := ( 12 ) on L 1 d := ( 1 , b ) b := ( a , c ) c := ( a , d ) Γ := � a , b , c , d � Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 8 / 17

  13. What is a branch group? Example: (First) Grigorchuk Group T = T ( 2 ) - binary tree ( m n = 2) a := ( 12 ) on L 1 d := ( 1 , b ) b := ( a , c ) c := ( a , d ) Γ := � a , b , c , d � a . . . . . . . . . . . . Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 8 / 17

  14. What is a branch group? Example: (First) Grigorchuk Group T = T ( 2 ) - binary tree ( m n = 2) a := ( 12 ) on L 1 d := ( 1 , b ) b := ( a , c ) c := ( a , d ) Γ := � a , b , c , d � b a a 1 Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 8 / 17

  15. What is a branch group? Example: (First) Grigorchuk Group T = T ( 2 ) - binary tree ( m n = 2) a := ( 12 ) on L 1 d := ( 1 , b ) b := ( a , c ) c := ( a , d ) Γ := � a , b , c , d � c a 1 a Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 8 / 17

  16. What is a branch group? Example: (First) Grigorchuk Group T = T ( 2 ) - binary tree ( m n = 2) a := ( 12 ) on L 1 d := ( 1 , b ) b := ( a , c ) c := ( a , d ) Γ := � a , b , c , d � d 1 a a Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 8 / 17

  17. Why do we study them? Outline What is a branch group? 1 Why do we study them? 2 Subgroup structure 3 Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 9 / 17

  18. Why do we study them? Solve Open Problems General Burnside Problem Is every finitely generated torsion group finite? Gupta–Sidki, ’83 Gupta–Sidki p-groups are just infinite p-groups. Every finite p-group is contained in the Gupta–Sidki p-group. Grigorchuk, ’80 Grigorchuk group is a just infinite 2-group. Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 10 / 17

  19. Why do we study them? Solve Open Problems General Burnside Problem Is every finitely generated torsion group finite? Gupta–Sidki, ’83 Gupta–Sidki p-groups are just infinite p-groups. Every finite p-group is contained in the Gupta–Sidki p-group. Grigorchuk, ’80 Grigorchuk group is a just infinite 2-group. Other problems Grigorchuk group is first group shown to have intermediate word growth. Also first group shown to be amenable but not elementary amenable. Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 10 / 17

  20. Why do we study them? Just infinite groups Definition G is just infinite if it is infinite and all its proper quotients are finite. Lemma Every infinite finitely generated (f.g.) group has a just infinite quotient. Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 11 / 17

  21. Why do we study them? Just infinite groups Definition G is just infinite if it is infinite and all its proper quotients are finite. Lemma Every infinite finitely generated (f.g.) group has a just infinite quotient. Theorem (Wilson, ’70) The class of f.g. just infinite groups splits into 3 classes: Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 11 / 17

  22. Why do we study them? Just infinite groups Definition G is just infinite if it is infinite and all its proper quotients are finite. Lemma Every infinite finitely generated (f.g.) group has a just infinite quotient. Theorem (Wilson, ’70) The class of f.g. just infinite groups splits into 3 classes: (just infinite) branch groups Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 11 / 17

  23. Why do we study them? Just infinite groups Definition G is just infinite if it is infinite and all its proper quotients are finite. Lemma Every infinite finitely generated (f.g.) group has a just infinite quotient. Theorem (Wilson, ’70) The class of f.g. just infinite groups splits into 3 classes: (just infinite) branch groups groups with a finite index subgroup H s.t. H = � k L for some k and L is hereditarily just infinite (all subgroups of finite index are just infinite) simple Proved by looking at lattice of subnormal subgroups. Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 11 / 17

  24. Subgroup structure Outline What is a branch group? 1 Why do we study them? 2 Subgroup structure 3 Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 12 / 17

  25. Subgroup structure Commensurability Definition G and H are (abstractly) commensurable if they have isomorphic finite index subgroups. Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 13 / 17

  26. Subgroup structure Commensurability Definition G and H are (abstractly) commensurable if they have isomorphic finite index subgroups. How many commensurability classes of (f.g.) subgroups can a group have? Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 13 / 17

  27. Subgroup structure Commensurability Definition G and H are (abstractly) commensurable if they have isomorphic finite index subgroups. How many commensurability classes of (f.g.) subgroups can a group have? 1 class finite groups Alejandra Garrido (Oxford) Subgroup structure of branch groups St Andrews, August 2013 13 / 17

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