Finitely presented groups 1 Max Neunhöffer LMS Short Course on Computational Group Theory 29 July – 2 August 2013 Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 1 / 12 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. A; B j ABA−1B−1 Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 Finitely presented groups Introduction and definition C j Ck = 1 is the cyclic group of order k. − − A; B j ABA 1B 1 is the integral lattice Z2 with addition. “Definition” G := hX j R1; R2;:::; Rk i is the “largest” group that has X as generating system, such that the relations R1;:::; Rk hold. Every group with a generating system X, for which the relations R1;:::; Rk hold, is a quotient of G. P; Q j Pn; Q2; PQPQ is the dihedral group of order 2n. S; T j S3; T 2 is the modular group PSL(2; Z). D; E; F j DEFED; EFD; D2EF is the trivial group f1g. Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 X^ ∗: set of finite words in X^ , concatenation as product R ⊆ X^ ∗: finite set of relations Define ∼ as finest equivalence relation on X^ ∗ with uv ∼ uxx−1v for all u; v 2 X^ ∗ and all x 2 X^ , and uv ∼ urv for all u; v 2 X^ ∗ and all r 2 R. Definition/Proposition (Finitely presented group) The set G := X^ ∗= ∼ of equivalence classes is a group with concatenation as product. It is called the finitely presented group with generators X and relations R and is denoted by G := hX j Ri. : X^ ∗ ! G; w 7! w is the natural map from w to its equivalence class. e.g. in G = S; T j S6; T 2; STST : ; ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T Finitely presented groups Introduction and definition : X: finite set of generators, X^ := X [ X −1 with involution x 7! x−1 Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 R ⊆ X^ ∗: finite set of relations Define ∼ as finest equivalence relation on X^ ∗ with uv ∼ uxx−1v for all u; v 2 X^ ∗ and all x 2 X^ , and uv ∼ urv for all u; v 2 X^ ∗ and all r 2 R. Definition/Proposition (Finitely presented group) The set G := X^ ∗= ∼ of equivalence classes is a group with concatenation as product. It is called the finitely presented group with generators X and relations R and is denoted by G := hX j Ri. : X^ ∗ ! G; w 7! w is the natural map from w to its equivalence class. e.g. in G = S; T j S6; T 2; STST : ; ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T Finitely presented groups Introduction and definition : X: finite set of generators, X^ := X [ X −1 with involution x 7! x−1 X^ ∗: set of finite words in X^ , concatenation as product Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 Define ∼ as finest equivalence relation on X^ ∗ with uv ∼ uxx−1v for all u; v 2 X^ ∗ and all x 2 X^ , and uv ∼ urv for all u; v 2 X^ ∗ and all r 2 R.