Finitely Presented Groups 1

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 | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

G := hX | 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 | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 − − A, B | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

P, Q | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | 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”

P, Q | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1 is the cyclic group of order k.

A, B | ABA−1B−1

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 “Deﬁnition”

P, Q | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1 is the cyclic group of order k.

− − A, B | 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 | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1 is the cyclic group of order k.

− − A, B | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

G := hX | 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 | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1 is the cyclic group of order k.

− − A, B | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1 is the cyclic group of order k.

− − A, B | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

P, Q | Pn, Q2, PQPQ

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1 is the cyclic group of order k.

− − A, B | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

P, Q | 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 | DEFED, EFD, D2EF is the trivial group {1}.

Finitely presented groups Introduction and deﬁnition

C | Ck = 1 is the cyclic group of order k.

− − A, B | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

P, Q | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | 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 | Ck = 1 is the cyclic group of order k.

− − A, B | ABA 1B 1 is the integral lattice Z2 with addition. “Deﬁnition”

P, Q | Pn, Q2, PQPQ is the dihedral group of order 2n.

S, T | S3, T 2 is the modular group PSL(2, Z).

D, E, F | DEFED, EFD, D2EF is the trivial group {1}.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 2 / 12 Xˆ ∗: set of finite words in Xˆ , concatenation as product R ⊆ Xˆ ∗: finite set of relations Define ∼ as finest equivalence relation on Xˆ ∗ with uv ∼ uxx−1v for all u, v ∈ Xˆ ∗ and all x ∈ Xˆ , and uv ∼ urv for all u, v ∈ Xˆ ∗ and all r ∈ R. Definition/Proposition (Finitely presented group) The set G := Xˆ ∗/ ∼ 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 | Ri. : Xˆ ∗ → G, w 7→ w is the natural map from w to its equivalence class.

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

Finitely presented groups Introduction and deﬁnition . X: ﬁnite set of generators, Xˆ := 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 ⊆ Xˆ ∗: finite set of relations Define ∼ as finest equivalence relation on Xˆ ∗ with uv ∼ uxx−1v for all u, v ∈ Xˆ ∗ and all x ∈ Xˆ , and uv ∼ urv for all u, v ∈ Xˆ ∗ and all r ∈ R. Definition/Proposition (Finitely presented group) The set G := Xˆ ∗/ ∼ 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 | Ri. : Xˆ ∗ → G, w 7→ w is the natural map from w to its equivalence class.

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

Finitely presented groups Introduction and definition . X: finite set of generators, Xˆ := X ∪ X −1 with involution x 7→ x−1 Xˆ ∗: set of finite words in Xˆ , concatenation as product

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 Define ∼ as finest equivalence relation on Xˆ ∗ with uv ∼ uxx−1v for all u, v ∈ Xˆ ∗ and all x ∈ Xˆ , and uv ∼ urv for all u, v ∈ Xˆ ∗ and all r ∈ R. Definition/Proposition (Finitely presented group) The set G := Xˆ ∗/ ∼ 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 | Ri. : Xˆ ∗ → G, w 7→ w is the natural map from w to its equivalence class.

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 Deﬁnition/Proposition (Finitely presented group) The set G := Xˆ ∗/ ∼ 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 | Ri. : Xˆ ∗ → G, w 7→ w is the natural map from w to its equivalence class.

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 : Xˆ ∗ → G, w 7→ w is the natural map from w to its equivalence class.

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

Finitely presented groups Introduction and definition . X: finite set of generators, Xˆ := X ∪ X −1 with involution x 7→ x−1 Xˆ ∗: set of finite words in Xˆ , concatenation as product R ⊆ Xˆ ∗: finite set of relations Define ∼ as finest equivalence relation on Xˆ ∗ with uv ∼ uxx−1v for all u, v ∈ Xˆ ∗ and all x ∈ Xˆ , and uv ∼ urv for all u, v ∈ Xˆ ∗ and all r ∈ R. Definition/Proposition (Finitely presented group) The set G := Xˆ ∗/ ∼ 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 | Ri.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 = SSTSTTST ∼ SSTSST = S2TS2T

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 ∼ SSTSST = S2TS2T

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 Finitely presented groups Introduction and definition . X: finite set of generators, Xˆ := X ∪ X −1 with involution x 7→ x−1 Xˆ ∗: set of finite words in Xˆ , concatenation as product R ⊆ Xˆ ∗: finite set of relations Define ∼ as finest equivalence relation on Xˆ ∗ with uv ∼ uxx−1v for all u, v ∈ Xˆ ∗ and all x ∈ Xˆ , and uv ∼ urv for all u, v ∈ Xˆ ∗ and all r ∈ R. Definition/Proposition (Finitely presented group) The set G := Xˆ ∗/ ∼ 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 | Ri. : Xˆ ∗ → G, w 7→ w is the natural map from w to its equivalence class.

e.g. in G = S, T | S6, T 2, STST :

∅ ∼ STST ∼ SSTST TST = SSTSTTST ∼ SSTSST = S2TS2T

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 3 / 12 In other words: any map f : X → H with (∗) can be extended to a group homomorphism ˜f : G → H in a unique way.

This means: G has all groups as a quotient that have a generating system X which fulﬁlls the relations R. hA, B |i is the free group on two generators, it has any 2-generated group as a quotient. C | Ck has all cyclic groups of order o dividing k as quotients.

Finitely presented groups Universal property

Theorem (Universal property (Dyck, 1882)) Let G := hX | Ri be a ﬁnitely presented group. Let H be any group, f : X → H any map and set f (x−1) := f (x)−1 for x ∈ X. If f (x1) · f (x2) ····· f (xk ) = 1 ∈ H (∗)

for all r = x1x2 ··· xk ∈ R, then there is a unique group homomorphism ˜f : G → H with ˜f (x) = f (x) for all x ∈ X.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 4 / 12 This means: G has all groups as a quotient that have a generating system X which fulﬁlls the relations R. hA, B |i is the free group on two generators, it has any 2-generated group as a quotient. C | Ck has all cyclic groups of order o dividing k as quotients.

Finitely presented groups Universal property

for all r = x1x2 ··· xk ∈ R, then there is a unique group homomorphism ˜f : G → H with ˜f (x) = f (x) for all x ∈ X. In other words: any map f : X → H with (∗) can be extended to a group homomorphism ˜f : G → H in a unique way.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 4 / 12 hA, B |i is the free group on two generators, it has any 2-generated group as a quotient. C | Ck has all cyclic groups of order o dividing k as quotients.

Finitely presented groups Universal property

This means: G has all groups as a quotient that have a generating system X which fulﬁlls the relations R.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 4 / 12 C | Ck has all cyclic groups of order o dividing k as quotients.

Finitely presented groups Universal property

This means: G has all groups as a quotient that have a generating system X which fulﬁlls the relations R. hA, B |i is the free group on two generators, it has any 2-generated group as a quotient.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 4 / 12 Finitely presented groups Universal property

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 4 / 12 Problem (Conjugacy problem (Dehn 1911)) For G := hX | Ri and two words v, w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not there is a word u ∈ Xˆ ∗ such that w = uvu−1 ∈ G.

Problem (Isomorphism problem (Dehn 1911)) For G := hX | Ri and H := hY | Si, decide in ﬁnitely many steps whether or not G =∼ H, or even: for a map f : X → Yˆ ∗, decide in ﬁnitely many steps whether or not f induces an isomorphism ˜f : G → H.

We would like to have algorithms for these decision problems ...

Finitely presented groups Fundamental problems

In a seminal paper, Max Dehn formulated three fundamental problems: Problem (Word problem (Dehn 1911)) For G := hX | Ri and a word w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not w = 1 ∈ G.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 5 / 12 Problem (Isomorphism problem (Dehn 1911)) For G := hX | Ri and H := hY | Si, decide in ﬁnitely many steps whether or not G =∼ H, or even: for a map f : X → Yˆ ∗, decide in ﬁnitely many steps whether or not f induces an isomorphism ˜f : G → H.

We would like to have algorithms for these decision problems ...

Finitely presented groups Fundamental problems

Problem (Conjugacy problem (Dehn 1911)) For G := hX | Ri and two words v, w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not there is a word u ∈ Xˆ ∗ such that w = uvu−1 ∈ G.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 5 / 12 or even: for a map f : X → Yˆ ∗, decide in ﬁnitely many steps whether or not f induces an isomorphism ˜f : G → H.

We would like to have algorithms for these decision problems ...

Finitely presented groups Fundamental problems

Problem (Conjugacy problem (Dehn 1911)) For G := hX | Ri and two words v, w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not there is a word u ∈ Xˆ ∗ such that w = uvu−1 ∈ G.

Problem (Isomorphism problem (Dehn 1911)) For G := hX | Ri and H := hY | Si, decide in ﬁnitely many steps whether or not G =∼ H,

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 5 / 12 We would like to have algorithms for these decision problems ...

Finitely presented groups Fundamental problems

Problem (Conjugacy problem (Dehn 1911)) For G := hX | Ri and two words v, w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not there is a word u ∈ Xˆ ∗ such that w = uvu−1 ∈ G.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 5 / 12 Finitely presented groups Fundamental problems

Problem (Conjugacy problem (Dehn 1911)) For G := hX | Ri and two words v, w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not there is a word u ∈ Xˆ ∗ such that w = uvu−1 ∈ G.

We would like to have algorithms for these decision problems ...

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 5 / 12 Lemma Let G = hX | Ri, then w = 1 for a w ∈ Xˆ ∗ if and only if k Y −1 w ∼ wi ri wi in F = hX |i i=1 ˆ ∗ −1 for a k ∈ N and some wi ∈ X and ri ∈ R ∪ R . For w with w = 1 let A(w) be the minimal such k.

Deﬁnition (Dehn function)

δ(`) := max{A(w) | w ∈ Xˆ ∗ of length ` and w = 1}

Finitely presented groups Fundamental problems

Problem (Word problem (Dehn 1911)) For G := hX | Ri and a word w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not w = 1 ∈ G.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 6 / 12 For w with w = 1 let A(w) be the minimal such k.

Deﬁnition (Dehn function)

δ(`) := max{A(w) | w ∈ Xˆ ∗ of length ` and w = 1}

Finitely presented groups Fundamental problems

Problem (Word problem (Dehn 1911)) For G := hX | Ri and a word w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not w = 1 ∈ G.

Lemma Let G = hX | Ri, then w = 1 for a w ∈ Xˆ ∗ if and only if k Y −1 w ∼ wi ri wi in F = hX |i i=1 ˆ ∗ −1 for a k ∈ N and some wi ∈ X and ri ∈ R ∪ R .

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 6 / 12 Deﬁnition (Dehn function)

δ(`) := max{A(w) | w ∈ Xˆ ∗ of length ` and w = 1}

Finitely presented groups Fundamental problems

Problem (Word problem (Dehn 1911)) For G := hX | Ri and a word w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not w = 1 ∈ G.

Lemma Let G = hX | Ri, then w = 1 for a w ∈ Xˆ ∗ if and only if k Y −1 w ∼ wi ri wi in F = hX |i i=1 ˆ ∗ −1 for a k ∈ N and some wi ∈ X and ri ∈ R ∪ R . For w with w = 1 let A(w) be the minimal such k.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 6 / 12 Finitely presented groups Fundamental problems

Problem (Word problem (Dehn 1911)) For G := hX | Ri and a word w ∈ Xˆ ∗, decide in ﬁnitely many steps whether or not w = 1 ∈ G.

Deﬁnition (Dehn function)

δ(`) := max{A(w) | w ∈ Xˆ ∗ of length ` and w = 1}

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 6 / 12 Theorem (Shapiro) The word problem for G = hX | Ri is algorithmically solvable if and only if the Dehn function δ is recursively computable.

The growth rate of δ is a measure for the difﬁculty of the word problem.

The conjugacy problem and the isomorphism problem are unsolvable in the same way. Even the question if hX | Ri = {1} is in general unsolvable.

Nevertheless, in GAP you can work with ﬁnitely presented groups: http://tinyurl.com/MNGAPsess/GAP_FP_1.g

Finitely presented groups The word problem is insolvable Theorem (Novikov 1955, Boone 1958) There is a ﬁnite presentation G = hX | Ri, for which it is proven that there is no algorithm to solve the word problem.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 7 / 12 The growth rate of δ is a measure for the difﬁculty of the word problem.

The conjugacy problem and the isomorphism problem are unsolvable in the same way. Even the question if hX | Ri = {1} is in general unsolvable.

Nevertheless, in GAP you can work with ﬁnitely presented groups: http://tinyurl.com/MNGAPsess/GAP_FP_1.g

Theorem (Shapiro) The word problem for G = hX | Ri is algorithmically solvable if and only if the Dehn function δ is recursively computable.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 7 / 12 The conjugacy problem and the isomorphism problem are unsolvable in the same way. Even the question if hX | Ri = {1} is in general unsolvable.

Nevertheless, in GAP you can work with ﬁnitely presented groups: http://tinyurl.com/MNGAPsess/GAP_FP_1.g

Theorem (Shapiro) The word problem for G = hX | Ri is algorithmically solvable if and only if the Dehn function δ is recursively computable.

The growth rate of δ is a measure for the difﬁculty of the word problem.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 7 / 12 Even the question if hX | Ri = {1} is in general unsolvable.

Nevertheless, in GAP you can work with ﬁnitely presented groups: http://tinyurl.com/MNGAPsess/GAP_FP_1.g

Theorem (Shapiro) The word problem for G = hX | Ri is algorithmically solvable if and only if the Dehn function δ is recursively computable.

The growth rate of δ is a measure for the difﬁculty of the word problem.

The conjugacy problem and the isomorphism problem are unsolvable in the same way.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 7 / 12 Nevertheless, in GAP you can work with ﬁnitely presented groups: http://tinyurl.com/MNGAPsess/GAP_FP_1.g

Theorem (Shapiro) The word problem for G = hX | Ri is algorithmically solvable if and only if the Dehn function δ is recursively computable.

The growth rate of δ is a measure for the difﬁculty of the word problem.

The conjugacy problem and the isomorphism problem are unsolvable in the same way. Even the question if hX | Ri = {1} is in general unsolvable.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 7 / 12 Finitely presented groups The word problem is insolvable Theorem (Novikov 1955, Boone 1958) There is a ﬁnite presentation G = hX | Ri, for which it is proven that there is no algorithm to solve the word problem.

Theorem (Shapiro) The word problem for G = hX | Ri is algorithmically solvable if and only if the Dehn function δ is recursively computable.

The growth rate of δ is a measure for the difﬁculty of the word problem.

The conjugacy problem and the isomorphism problem are unsolvable in the same way. Even the question if hX | Ri = {1} is in general unsolvable.

Nevertheless, in GAP you can work with ﬁnitely presented groups: http://tinyurl.com/MNGAPsess/GAP_FP_1.g

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 7 / 12 Rotating or inverting a relator does not change the group. If vw ∈ R with v, w ∈ Xˆ ∗, then v = w −1 ∈ G. In other words: vu ∼ vww −1u ∼ w −1u, so we can replace v by w −1 in any other relator. In particular, if v ∈ Xˆ and w does not contain the letter v, we can replace v everywhere by w −1. We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Changing the presentation Tietze transformations Let G := hX | Ri be a ﬁnite presentation.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 If vw ∈ R with v, w ∈ Xˆ ∗, then v = w −1 ∈ G. In other words: vu ∼ vww −1u ∼ w −1u, so we can replace v by w −1 in any other relator. In particular, if v ∈ Xˆ and w does not contain the letter v, we can replace v everywhere by w −1. We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Changing the presentation Tietze transformations Let G := hX | Ri be a ﬁnite presentation. Rotating or inverting a relator does not change the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 In other words: vu ∼ vww −1u ∼ w −1u, so we can replace v by w −1 in any other relator. In particular, if v ∈ Xˆ and w does not contain the letter v, we can replace v everywhere by w −1. We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 so we can replace v by w −1 in any other relator. In particular, if v ∈ Xˆ and w does not contain the letter v, we can replace v everywhere by w −1. We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 In particular, if v ∈ Xˆ and w does not contain the letter v, we can replace v everywhere by w −1. We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Changing the presentation Tietze transformations Let G := hX | Ri be a ﬁnite presentation. Rotating or inverting a relator does not change the group. If vw ∈ R with v, w ∈ Xˆ ∗, then v = w −1 ∈ G. In other words: vu ∼ vww −1u ∼ w −1u, so we can replace v by w −1 in any other relator. In particular, if v ∈ Xˆ and w does not contain the letter v, we can replace v everywhere by w −1. We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 Changing the presentation Tietze transformations Let G := hX | Ri be a ﬁnite presentation. Rotating or inverting a relator does not change the group. If vw ∈ R with v, w ∈ Xˆ ∗, then v = w −1 ∈ G. In other words: vu ∼ vww −1u ∼ w −1u, so we can replace v by w −1 in any other relator. In particular, if v ∈ Xˆ and w does not contain the letter v, we can replace v everywhere by w −1. We can then remove the relator vw and the generators v, v −1 from Xˆ . Conversely, if a word w shows up frequently in the relators, we can add generators v, v −1 to Xˆ and add the relator v −1w. We can then replace w everywhere by v. Furthermore, we can always add relators that follow from others and remove redundand relators. All these steps are called “Tietze transformations”, they change the presentation, but not the isomorphism type of the group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 8 / 12 fully automatically, or semi-automatically. The aim is always to simplify the presentation, which usually means fewer generators and/or shorter relators.

GAP distinguishes between objects representing a ﬁnitely presented group, and objects representing a ﬁnite presentation of a group.

http://tinyurl.com/MNGAPsess/GAP_FP_2.g

Changing the presentation Tietze transformations

GAP allows you to work with presentations and do Tietze transformations manually,

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 9 / 12 semi-automatically. The aim is always to simplify the presentation, which usually means fewer generators and/or shorter relators.

GAP distinguishes between objects representing a ﬁnitely presented group, and objects representing a ﬁnite presentation of a group.

http://tinyurl.com/MNGAPsess/GAP_FP_2.g

Changing the presentation Tietze transformations

GAP allows you to work with presentations and do Tietze transformations manually, fully automatically, or

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 9 / 12 The aim is always to simplify the presentation, which usually means fewer generators and/or shorter relators.

GAP distinguishes between objects representing a ﬁnitely presented group, and objects representing a ﬁnite presentation of a group.

http://tinyurl.com/MNGAPsess/GAP_FP_2.g

Changing the presentation Tietze transformations

GAP allows you to work with presentations and do Tietze transformations manually, fully automatically, or semi-automatically.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 9 / 12 GAP distinguishes between objects representing a ﬁnitely presented group, and objects representing a ﬁnite presentation of a group.

http://tinyurl.com/MNGAPsess/GAP_FP_2.g

Changing the presentation Tietze transformations

GAP allows you to work with presentations and do Tietze transformations manually, fully automatically, or semi-automatically. The aim is always to simplify the presentation, which usually means fewer generators and/or shorter relators.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 9 / 12 http://tinyurl.com/MNGAPsess/GAP_FP_2.g

Changing the presentation Tietze transformations

GAP distinguishes between objects representing a ﬁnitely presented group, and objects representing a ﬁnite presentation of a group.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 9 / 12 Changing the presentation Tietze transformations

GAP distinguishes between objects representing a ﬁnitely presented group, and objects representing a ﬁnite presentation of a group.

http://tinyurl.com/MNGAPsess/GAP_FP_2.g

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 9 / 12 G/G0 is the largest abelian quotient of G, we have: Proposition If ϕ : G → A is any group homomorphism to an abelian group A, then there is a unique group homomorphism ϕ˜ : G/G0 → A with ϕ = πϕ˜:

π G / G/G0

ϕ ϕ˜ A }

where π : G → G/G0 is the natural map. (Note G0 ⊆ ker ϕ)

We immediately get: D E G/G0 =∼ H := X | R, {xyx−1y −1 | x, y ∈ X}

Abelian quotients Theory The rest of this talk is about studying G/G0 where G = hX | Ri and G0 is the derived subgroup.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 10 / 12 (Note G0 ⊆ ker ϕ)

We immediately get: D E G/G0 =∼ H := X | R, {xyx−1y −1 | x, y ∈ X}

Abelian quotients Theory The rest of this talk is about studying G/G0 where G = hX | Ri and G0 is the derived subgroup. G/G0 is the largest abelian quotient of G, we have: Proposition If ϕ : G → A is any group homomorphism to an abelian group A, then there is a unique group homomorphism ϕ˜ : G/G0 → A with ϕ = πϕ˜:

π G / G/G0

ϕ ϕ˜ A }

where π : G → G/G0 is the natural map.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 10 / 12 We immediately get: D E G/G0 =∼ H := X | R, {xyx−1y −1 | x, y ∈ X}

π G / G/G0

ϕ ϕ˜ A }

where π : G → G/G0 is the natural map. (Note G0 ⊆ ker ϕ)

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 10 / 12 Abelian quotients Theory The rest of this talk is about studying G/G0 where G = hX | Ri and G0 is the derived subgroup. G/G0 is the largest abelian quotient of G, we have: Proposition If ϕ : G → A is any group homomorphism to an abelian group A, then there is a unique group homomorphism ϕ˜ : G/G0 → A with ϕ = πϕ˜:

π G / G/G0

ϕ ϕ˜ A }

where π : G → G/G0 is the natural map. (Note G0 ⊆ ker ϕ)

We immediately get: D E G/G0 =∼ H := X | R, {xyx−1y −1 | x, y ∈ X}

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 10 / 12 We write elements of Zn as integer vectors. Since Zn is abelian, the normal closure of R ⊆ Zn is hRi. That is, H as quotient of Zn is described by a matrix   r1  r2  M :=   ∈ k×n  .  Z  .  rk

∼ n where H = Z /(row space of M). If N := S · M · T is the Smith normal form of M (with S, T invertible), ∼ n then H = Z /(row space of N) and we can read off the structure.

Abelian quotients Theory However, H := X | R, {xyx−1y −1 | x, y ∈ X} is naturally a quotient of the free abelian group

D −1 −1 E ∼ n X | {xyx y | x, y ∈ X} = Z (with addition),

where n = |X|.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 11 / 12 Since Zn is abelian, the normal closure of R ⊆ Zn is hRi. That is, H as quotient of Zn is described by a matrix   r1  r2  M :=   ∈ k×n  .  Z  .  rk

∼ n where H = Z /(row space of M). If N := S · M · T is the Smith normal form of M (with S, T invertible), ∼ n then H = Z /(row space of N) and we can read off the structure.

Abelian quotients Theory However, H := X | R, {xyx−1y −1 | x, y ∈ X} is naturally a quotient of the free abelian group

D −1 −1 E ∼ n X | {xyx y | x, y ∈ X} = Z (with addition),

where n = |X|. We write elements of Zn as integer vectors.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 11 / 12 That is, H as quotient of Zn is described by a matrix   r1  r2  M :=   ∈ k×n  .  Z  .  rk

∼ n where H = Z /(row space of M). If N := S · M · T is the Smith normal form of M (with S, T invertible), ∼ n then H = Z /(row space of N) and we can read off the structure.

Abelian quotients Theory However, H := X | R, {xyx−1y −1 | x, y ∈ X} is naturally a quotient of the free abelian group

D −1 −1 E ∼ n X | {xyx y | x, y ∈ X} = Z (with addition),

where n = |X|. We write elements of Zn as integer vectors. Since Zn is abelian, the normal closure of R ⊆ Zn is hRi.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 11 / 12 If N := S · M · T is the Smith normal form of M (with S, T invertible), ∼ n then H = Z /(row space of N) and we can read off the structure.

Abelian quotients Theory However, H := X | R, {xyx−1y −1 | x, y ∈ X} is naturally a quotient of the free abelian group

D −1 −1 E ∼ n X | {xyx y | x, y ∈ X} = Z (with addition),

where n = |X|. We write elements of Zn as integer vectors. Since Zn is abelian, the normal closure of R ⊆ Zn is hRi. That is, H as quotient of Zn is described by a matrix   r1  r2  M :=   ∈ k×n  .  Z  .  rk

∼ n where H = Z /(row space of M).

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 11 / 12 ∼ n then H = Z /(row space of N) and we can read off the structure.

Abelian quotients Theory However, H := X | R, {xyx−1y −1 | x, y ∈ X} is naturally a quotient of the free abelian group

D −1 −1 E ∼ n X | {xyx y | x, y ∈ X} = Z (with addition),

∼ n where H = Z /(row space of M). If N := S · M · T is the Smith normal form of M (with S, T invertible),

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 11 / 12 Abelian quotients Theory However, H := X | R, {xyx−1y −1 | x, y ∈ X} is naturally a quotient of the free abelian group

D −1 −1 E ∼ n X | {xyx y | x, y ∈ X} = Z (with addition),

∼ n where H = Z /(row space of M). If N := S · M · T is the Smith normal form of M (with S, T invertible), ∼ n then H = Z /(row space of N) and we can read off the structure. Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 11 / 12 The matrix S recombines the rows of M (changing the generators of the row space) and the matrix T changes the generating set of Zn. Both manipulations can be traced explicitly, yielding an explicit, computable isomorphism 0 ∼ n G/G = Z /(row space of N).

The diagonal entries d1, d2,... dk fulﬁll d1 | d2 | · · · | dr and dr+1 = 0 = dr+2 = ··· = dk for some r and are called the invariant factors or abelian invariants of G.

GAP can do all this: http://tinyurl.com/MNGAPsess/GAP_FP_3.g

Abelian quotients Practice More concretely, if for example  1 ····   · 2 ···  N =   = S · M · T ,  ·· 6 ··  ··· 0 · ∼ then H = Z/(2Z) ⊕ Z/(6Z) ⊕ Z ⊕ Z.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 12 / 12 The diagonal entries d1, d2,... dk fulﬁll d1 | d2 | · · · | dr and dr+1 = 0 = dr+2 = ··· = dk for some r and are called the invariant factors or abelian invariants of G.

GAP can do all this: http://tinyurl.com/MNGAPsess/GAP_FP_3.g

Abelian quotients Practice More concretely, if for example  1 ····   · 2 ···  N =   = S · M · T ,  ·· 6 ··  ··· 0 · ∼ then H = Z/(2Z) ⊕ Z/(6Z) ⊕ Z ⊕ Z. The matrix S recombines the rows of M (changing the generators of the row space) and the matrix T changes the generating set of Zn. Both manipulations can be traced explicitly, yielding an explicit, computable isomorphism 0 ∼ n G/G = Z /(row space of N).

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 12 / 12 GAP can do all this: http://tinyurl.com/MNGAPsess/GAP_FP_3.g

The diagonal entries d1, d2,... dk fulﬁll d1 | d2 | · · · | dr and dr+1 = 0 = dr+2 = ··· = dk for some r and are called the invariant factors or abelian invariants of G.

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 12 / 12 Abelian quotients Practice More concretely, if for example  1 ····   · 2 ···  N =   = S · M · T ,  ·· 6 ··  ··· 0 · ∼ then H = Z/(2Z) ⊕ Z/(6Z) ⊕ Z ⊕ Z. The matrix S recombines the rows of M (changing the generators of the row space) and the matrix T changes the generating set of Zn. Both manipulations can be traced explicitly, yielding an explicit, computable isomorphism 0 ∼ n G/G = Z /(row space of N).

The diagonal entries d1, d2,... dk fulﬁll d1 | d2 | · · · | dr and dr+1 = 0 = dr+2 = ··· = dk for some r and are called the invariant factors or abelian invariants of G.

GAP can do all this: http://tinyurl.com/MNGAPsess/GAP_FP_3.g

Max Neunhöffer (University of St Andrews) Finitely presented groups 1 29 July 2013 12 / 12