Rewriting in Artin groups Sarah Rees University of Newcastle Paul - - PowerPoint PPT Presentation

SLIDE 1

Rewriting in Artin groups

Sarah Rees

University of Newcastle

Paul Schupp Fest, Stevens Institute of Technology, June 2017

SLIDE 2

Introduction

This is about the word problem in Artin groups, focusing on recent attempts to find a uniform approach. I’ll start at work of Artin, then Garside, Deligne, Brieskorn&Saito, move on via Appel&Schupp, to very recent work, by myself and Derek Holt, by Eddy Godelle and Patrick Dehornoy, also by Blasco, Huang&Osajda.

SLIDE 3

Introduction

This is about the word problem in Artin groups, focusing on recent attempts to find a uniform approach. I’ll start at work of Artin, then Garside, Deligne, Brieskorn&Saito, move on via Appel&Schupp, to very recent work, by myself and Derek Holt, by Eddy Godelle and Patrick Dehornoy, also by Blasco, Huang&Osajda. A f.g. Artin group G = G(Γ) is defined by a presentation x1, x2, · · · , xn |

mij

• xixjxi · · · =

mij

• xjxixj · · ·,

i = j ∈ {1, 2, . . . , n} mij ∈ N ∪ {∞}, mij ≥ 2, with relations called braid relations.

SLIDE 4

Introduction

This is about the word problem in Artin groups, focusing on recent attempts to find a uniform approach. I’ll start at work of Artin, then Garside, Deligne, Brieskorn&Saito, move on via Appel&Schupp, to very recent work, by myself and Derek Holt, by Eddy Godelle and Patrick Dehornoy, also by Blasco, Huang&Osajda. A f.g. Artin group G = G(Γ) is defined by a presentation x1, x2, · · · , xn |

mij

• xixjxi · · · =

mij

• xjxixj · · ·,

i = j ∈ {1, 2, . . . , n} mij ∈ N ∪ {∞}, mij ≥ 2, with relations called braid relations. Adding relations x2

i = 1, ∀i gives a Coxeter group W = W (Γ).

SLIDE 5

Introduction

This is about the word problem in Artin groups, focusing on recent attempts to find a uniform approach. I’ll start at work of Artin, then Garside, Deligne, Brieskorn&Saito, move on via Appel&Schupp, to very recent work, by myself and Derek Holt, by Eddy Godelle and Patrick Dehornoy, also by Blasco, Huang&Osajda. A f.g. Artin group G = G(Γ) is defined by a presentation x1, x2, · · · , xn |

mij

• xixjxi · · · =

mij

• xjxixj · · ·,

i = j ∈ {1, 2, . . . , n} mij ∈ N ∪ {∞}, mij ≥ 2, with relations called braid relations. Adding relations x2

i = 1, ∀i gives a Coxeter group W = W (Γ).

The Coxeter matrix (mij) defines an edge-labelling of the complete graph Γ

• n X = {xi : i = 1, 2, . . . , n}, where {xi, xj} is labelled mij. That’s often

drawn with 2-edges omitted, 3-edges unlabelled, 4-edges represented

• doubled. And sometimes the subgraph Γ \ {∞-edges} is relevant.

SLIDE 6

The study of Artin groups started with braid groups

Emil Artin introduced the type An groups as braid groups (1926, 1947). x1, x2, · · · , xn | xixi+1xi = xi+1xixi+1, xixj = xjxi, ∀i + 1 < j

r

1

r

2

r

• 3. . . . . .

r

n−1

r

n

SLIDE 7

The study of Artin groups started with braid groups

Emil Artin introduced the type An groups as braid groups (1926, 1947). x1, x2, · · · , xn | xixi+1xi = xi+1xixi+1, xixj = xjxi, ∀i + 1 < j

r

1

r

2

r

• 3. . . . . .

r

n−1

r

n

Artin represented G(An) faithfully as the group of braids on n + 1 strings, providing a natural solution to the word problem. Chow (1948) found the centre, ∆2 when n > 1.

q q q q q q q q

Generator x1 1 2 3 4

q q q q q q q q

1 2 3 4 Half twist, ∆, x1x2x3x1x2x1 = (x2x1x3)2

✑ ✑ ✑

SLIDE 8

The study of Artin groups started with braid groups

Emil Artin introduced the type An groups as braid groups (1926, 1947). x1, x2, · · · , xn | xixi+1xi = xi+1xixi+1, xixj = xjxi, ∀i + 1 < j

r

1

r

2

r

• 3. . . . . .

r

n−1

r

n

Artin represented G(An) faithfully as the group of braids on n + 1 strings, providing a natural solution to the word problem. Chow (1948) found the centre, ∆2 when n > 1.

q q q q q q q q

Generator x1 1 2 3 4

q q q q q q q q

1 2 3 4 Half twist, ∆, x1x2x3x1x2x1 = (x2x1x3)2

✑ ✑ ✑

The term ‘Artin groups’ was introduced when the spherical type groups were studied by Garside (1969), Deligne, Brieskorn&Saito (1972).

SLIDE 9

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,

SLIDE 10

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,
• acts freely, properly discontinuously on a 2n-dim. conn.

manifoldYW = (Ω × V ) \

i(Hi × Hi), where Ω is a convex open

subset of V known as the Tits cone,

SLIDE 11

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,
• acts freely, properly discontinuously on a 2n-dim. conn.

manifoldYW = (Ω × V ) \

i(Hi × Hi), where Ω is a convex open

subset of V known as the Tits cone,

• has word problem that is soluble in time O(n2) (better?),

SLIDE 12

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,
• acts freely, properly discontinuously on a 2n-dim. conn.

manifoldYW = (Ω × V ) \

i(Hi × Hi), where Ω is a convex open

subset of V known as the Tits cone,

• has word problem that is soluble in time O(n2) (better?),

that is, given w = xi1 · · · xin, can decide w =W (Γ) 1? ,

SLIDE 13

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,
• acts freely, properly discontinuously on a 2n-dim. conn.

manifoldYW = (Ω × V ) \

i(Hi × Hi), where Ω is a convex open

subset of V known as the Tits cone,

• has word problem that is soluble in time O(n2) (better?),

that is, given w = xi1 · · · xin, can decide w =W (Γ) 1? ,

• has soluble conjugacy problem,

SLIDE 14

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,
• acts freely, properly discontinuously on a 2n-dim. conn.

manifoldYW = (Ω × V ) \

i(Hi × Hi), where Ω is a convex open

subset of V known as the Tits cone,

• has word problem that is soluble in time O(n2) (better?),

that is, given w = xi1 · · · xin, can decide w =W (Γ) 1? ,

• has soluble conjugacy problem, that is, given w, v, can decide

∃ u ? wu =W (Γ) uv,

SLIDE 15

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,
• acts freely, properly discontinuously on a 2n-dim. conn.

manifoldYW = (Ω × V ) \

i(Hi × Hi), where Ω is a convex open

subset of V known as the Tits cone,

• has word problem that is soluble in time O(n2) (better?),

that is, given w = xi1 · · · xin, can decide w =W (Γ) 1? ,

• has soluble conjugacy problem, that is, given w, v, can decide

∃ u ? wu =W (Γ) uv,

• is shortlex automatic, with regular set of geodesics,

SLIDE 16

Coxeter groups are well studied, with uniform properties.

The Coxeter group W (Γ)

• is linear, generated by reflections in hyperplanes in V := Rn,
• acts freely, properly discontinuously on a 2n-dim. conn.

manifoldYW = (Ω × V ) \

i(Hi × Hi), where Ω is a convex open

subset of V known as the Tits cone,

• has word problem that is soluble in time O(n2) (better?),

that is, given w = xi1 · · · xin, can decide w =W (Γ) 1? ,

• has soluble conjugacy problem, that is, given w, v, can decide

∃ u ? wu =W (Γ) uv,

• is shortlex automatic, with regular set of geodesics,

that is, both a sensible normal form for W (Γ) and the set of all geodesics can be recognised by finite state automata.

SLIDE 17

Artin groups are difficult

We don’t know whether G(Γ) has soluble WP, whether it has torsion.

SLIDE 18

Artin groups are difficult

We don’t know whether G(Γ) has soluble WP, whether it has torsion. We do know that

• (van der Lek, 1983) For XW = YW /W (Γ), we have π1(XW ) = G(Γ).

The related K(π, 1) conjecture (Arnold, Brieskorn, Pham, Thom), that all higher homotopy groups are trivial, proved for ‘spherical type’ (Deligne), ‘FC type’ (Charney&Davis), remains open in general.

SLIDE 19

Artin groups are difficult

We don’t know whether G(Γ) has soluble WP, whether it has torsion. We do know that

• (van der Lek, 1983) For XW = YW /W (Γ), we have π1(XW ) = G(Γ).

The related K(π, 1) conjecture (Arnold, Brieskorn, Pham, Thom), that all higher homotopy groups are trivial, proved for ‘spherical type’ (Deligne), ‘FC type’ (Charney&Davis), remains open in general.

• (Crisp&Paris, 2000) x2

i , ∀i is free, modulo obvious commutation

relations (Tits’ conjecture, emerged from Appel&Schupp’s work),

SLIDE 20

Artin groups are difficult

We don’t know whether G(Γ) has soluble WP, whether it has torsion. We do know that

• (van der Lek, 1983) For XW = YW /W (Γ), we have π1(XW ) = G(Γ).

The related K(π, 1) conjecture (Arnold, Brieskorn, Pham, Thom), that all higher homotopy groups are trivial, proved for ‘spherical type’ (Deligne), ‘FC type’ (Charney&Davis), remains open in general.

• (Crisp&Paris, 2000) x2

i , ∀i is free, modulo obvious commutation

relations (Tits’ conjecture, emerged from Appel&Schupp’s work),

• (Paris, 2001) the Artin monoid M(Γ) (of positive words) embeds in

the group G(Γ),

SLIDE 21

Artin groups are difficult

We don’t know whether G(Γ) has soluble WP, whether it has torsion. We do know that

• (van der Lek, 1983) For XW = YW /W (Γ), we have π1(XW ) = G(Γ).

The related K(π, 1) conjecture (Arnold, Brieskorn, Pham, Thom), that all higher homotopy groups are trivial, proved for ‘spherical type’ (Deligne), ‘FC type’ (Charney&Davis), remains open in general.

• (Crisp&Paris, 2000) x2

i , ∀i is free, modulo obvious commutation

relations (Tits’ conjecture, emerged from Appel&Schupp’s work),

• (Paris, 2001) the Artin monoid M(Γ) (of positive words) embeds in

the group G(Γ),

• (Charney&Paris, 2014) parabolic subgroups (generated by subsets of

{x1, . . . , xn}) are also Artin groups, and are convex.

SLIDE 22

Artin groups are difficult

We don’t know whether G(Γ) has soluble WP, whether it has torsion. We do know that

• (van der Lek, 1983) For XW = YW /W (Γ), we have π1(XW ) = G(Γ).

The related K(π, 1) conjecture (Arnold, Brieskorn, Pham, Thom), that all higher homotopy groups are trivial, proved for ‘spherical type’ (Deligne), ‘FC type’ (Charney&Davis), remains open in general.

• (Crisp&Paris, 2000) x2

i , ∀i is free, modulo obvious commutation

relations (Tits’ conjecture, emerged from Appel&Schupp’s work),

• (Paris, 2001) the Artin monoid M(Γ) (of positive words) embeds in

the group G(Γ),

• (Charney&Paris, 2014) parabolic subgroups (generated by subsets of

{x1, . . . , xn}) are also Artin groups, and are convex. Proofs of these use various actions of G(Γ) or a related Artin group.

SLIDE 23

Artin groups are difficult

We don’t know whether G(Γ) has soluble WP, whether it has torsion. We do know that

• (van der Lek, 1983) For XW = YW /W (Γ), we have π1(XW ) = G(Γ).

The related K(π, 1) conjecture (Arnold, Brieskorn, Pham, Thom), that all higher homotopy groups are trivial, proved for ‘spherical type’ (Deligne), ‘FC type’ (Charney&Davis), remains open in general.

• (Crisp&Paris, 2000) x2

i , ∀i is free, modulo obvious commutation

relations (Tits’ conjecture, emerged from Appel&Schupp’s work),

• (Paris, 2001) the Artin monoid M(Γ) (of positive words) embeds in

the group G(Γ),

• (Charney&Paris, 2014) parabolic subgroups (generated by subsets of

{x1, . . . , xn}) are also Artin groups, and are convex. Proofs of these use various actions of G(Γ) or a related Artin group. But we can solve many problems only for certain classes of Artin groups, with different methods known for the different classes.

SLIDE 24

Artin groups of spherical type

G(Γ) is of spherical type (aka finite type) when |W (Γ)| < ∞. Irreducible diagrams are in the following list:-

SLIDE 25

Artin groups of spherical type

G(Γ) is of spherical type (aka finite type) when |W (Γ)| < ∞. Irreducible diagrams are in the following list:- An

r

1

r

2

r

• 3. . . . . .

r

n−1

r

n

Cn

r

1

r

2

r

• 3. . . . . .

r

n−1

r

n

Dn

r

1

r

2

r

• 3. . . . . .

r

n−2

r n r

n−1

En (n = 6, 7, 8)

r

1

r

. . . . . . . . .

2

r

n−3

r

n−2

r n r

n−1

F4

r

1

r

2

r

3

r

4

G2(m)

r

1

m r

2

H3

r

1

r

2

5 r

3

H4

r

1

r

2

r

3

5 r

4

SLIDE 26

Artin groups of spherical type

Garside (1969) found normal forms for braid groups; Brieskorn&Saito and Deligne (1972) extended to spherical type.

SLIDE 27

Artin groups of spherical type

Garside (1969) found normal forms for braid groups; Brieskorn&Saito and Deligne (1972) extended to spherical type. Each Artin group of spherical type is a Garside group, the group of fractions of M(Γ). The monoid is a Garside monoid, admitting left and right division, left and right gcds and lcms, and more.

SLIDE 28

Artin groups of spherical type

Garside (1969) found normal forms for braid groups; Brieskorn&Saito and Deligne (1972) extended to spherical type. Each Artin group of spherical type is a Garside group, the group of fractions of M(Γ). The monoid is a Garside monoid, admitting left and right division, left and right gcds and lcms, and more. Any word in G(Γ) rewrites to the normal form uv−1, u, v ∈ M(Γ), gcdr(u, v) = 1,

SLIDE 29

Artin groups of spherical type

Garside (1969) found normal forms for braid groups; Brieskorn&Saito and Deligne (1972) extended to spherical type. Each Artin group of spherical type is a Garside group, the group of fractions of M(Γ). The monoid is a Garside monoid, admitting left and right division, left and right gcds and lcms, and more. Any word in G(Γ) rewrites to the normal form uv−1, u, v ∈ M(Γ), gcdr(u, v) = 1, since for a, b ∈ M(Γ), when lcmr(a, b) = aα = bβ, lcml(a, b) = γa = δb, then a−1b =G(Γ) αβ−1 and ab−1 =G(Γ) γ−1δ

SLIDE 30

Artin groups of spherical type

Garside (1969) found normal forms for braid groups; Brieskorn&Saito and Deligne (1972) extended to spherical type. Each Artin group of spherical type is a Garside group, the group of fractions of M(Γ). The monoid is a Garside monoid, admitting left and right division, left and right gcds and lcms, and more. Any word in G(Γ) rewrites to the normal form uv−1, u, v ∈ M(Γ), gcdr(u, v) = 1, since for a, b ∈ M(Γ), when lcmr(a, b) = aα = bβ, lcml(a, b) = γa = δb, then a−1b =G(Γ) αβ−1 and ab−1 =G(Γ) γ−1δ So the word problem is soluble: ∃ a uniform solution for all Artin groups of spherical type.

SLIDE 31

Just beyond spherical

Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ A3

r

1

r

2

r

3

r

4

˜ B3

r

1

r

2

r

3

r 4

SLIDE 32

Just beyond spherical

Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ A3

r

1

r

2

r

3

r

4

˜ B3

r

1

r

2

r

3

r 4

They are subgroups of Garside groups (McCammond& Sulway, 2013), and hence have soluble word problem (and more).

SLIDE 33

Just beyond spherical

Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ A3

r

1

r

2

r

3

r

4

˜ B3

r

1

r

2

r

3

r 4

They are subgroups of Garside groups (McCammond& Sulway, 2013), and hence have soluble word problem (and more). Artin groups of FC type are defined by Coxeter diagrams Γ for which the complete subgraphs of Γ \ {∞-edges} must be of spherical type, such as

r

1

∞ r

2

r

3

r

4

r

1

∞ r

2

∞ r

3

r4

• ❅

❅ ❅ ❅ r

1

∞ r

2

r

3

r4

SLIDE 34

Just beyond spherical

Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ A3

r

1

r

2

r

3

r

4

˜ B3

r

1

r

2

r

3

r 4

They are subgroups of Garside groups (McCammond& Sulway, 2013), and hence have soluble word problem (and more). Artin groups of FC type are defined by Coxeter diagrams Γ for which the complete subgraphs of Γ \ {∞-edges} must be of spherical type, such as

r

1

∞ r

2

r

3

r

4

r

1

∞ r

2

∞ r

3

r4

• ❅

❅ ❅ ❅ r

1

∞ r

2

r

3

r4

FC includes right-angled groups (RAAGs), (Baudisch 1977, Green 1990).

SLIDE 35

Just beyond spherical

Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ A3

r

1

r

2

r

3

r

4

˜ B3

r

1

r

2

r

3

r 4

They are subgroups of Garside groups (McCammond& Sulway, 2013), and hence have soluble word problem (and more). Artin groups of FC type are defined by Coxeter diagrams Γ for which the complete subgraphs of Γ \ {∞-edges} must be of spherical type, such as

r

1

∞ r

2

r

3

r

4

r

1

∞ r

2

∞ r

3

r4

• ❅

❅ ❅ ❅ r

1

∞ r

2

r

3

r4

FC includes right-angled groups (RAAGs), (Baudisch 1977, Green 1990). FC groups are amalgamated products of spherical type groups over parabolic subgroups.

SLIDE 36

Just beyond spherical

Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ A3

r

1

r

2

r

3

r

4

˜ B3

r

1

r

2

r

3

r 4

They are subgroups of Garside groups (McCammond& Sulway, 2013), and hence have soluble word problem (and more). Artin groups of FC type are defined by Coxeter diagrams Γ for which the complete subgraphs of Γ \ {∞-edges} must be of spherical type, such as

r

1

∞ r

2

r

3

r

4

r

1

∞ r

2

∞ r

3

r4

• ❅

❅ ❅ ❅ r

1

∞ r

2

r

3

r4

FC includes right-angled groups (RAAGs), (Baudisch 1977, Green 1990). FC groups are amalgamated products of spherical type groups over parabolic subgroups. So they have soluble word problem and more (Altobelli&Charney, 1996, 2000). The even groups (mij ∈ 2Z) are poly-free (Hermiller&˘ Suni´ c,2007, Blasco,Martinez-Perez&Paris 2017).

SLIDE 37

Rewriting in groups of FC type

It follows from the decomposition of groups of FC type as amalgamated products of spherical type groups that, if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, a, b, α, β, γ, δ ∈ M(Γ), between subwords over generators of parabolic subgroups of spherical type.

SLIDE 38

Rewriting in groups of FC type

It follows from the decomposition of groups of FC type as amalgamated products of spherical type groups that, if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, a, b, α, β, γ, δ ∈ M(Γ), between subwords over generators of parabolic subgroups of spherical type. In fact, we need only use rewrites on 2-generator subwords derived directly from braid relations, and where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse.

SLIDE 39

Rewriting in groups of FC type

It follows from the decomposition of groups of FC type as amalgamated products of spherical type groups that, if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, a, b, α, β, γ, δ ∈ M(Γ), between subwords over generators of parabolic subgroups of spherical type. In fact, we need only use rewrites on 2-generator subwords derived directly from braid relations, and where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse. e.g.if mij = 3, aba → bab, aba−1 → b−1ab are OK, but not ab → baba−1.

SLIDE 40

Rewriting in groups of FC type

It follows from the decomposition of groups of FC type as amalgamated products of spherical type groups that, if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, a, b, α, β, γ, δ ∈ M(Γ), between subwords over generators of parabolic subgroups of spherical type. In fact, we need only use rewrites on 2-generator subwords derived directly from braid relations, and where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse. e.g.if mij = 3, aba → bab, aba−1 → b−1ab are OK, but not ab → baba−1. We have observed that:

Theorem (Dehornoy&Godelle, 2013)

Artin groups of FC type satisfy property H.

SLIDE 41

Rewriting in groups of FC type

It follows from the decomposition of groups of FC type as amalgamated products of spherical type groups that, if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, a, b, α, β, γ, δ ∈ M(Γ), between subwords over generators of parabolic subgroups of spherical type. In fact, we need only use rewrites on 2-generator subwords derived directly from braid relations, and where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse. e.g.if mij = 3, aba → bab, aba−1 → b−1ab are OK, but not ab → baba−1. We have observed that:

Theorem (Dehornoy&Godelle, 2013)

Artin groups of FC type satisfy property H. Property H seems to suggest a kind of negative curvature (hyperbolicity).

SLIDE 42

Larger type groups seem to need a different approach

An Artin group G(Γ) is of large type if mij ≥ 3. These groups were introduced by Appel&Schupp (1983).

SLIDE 43

Larger type groups seem to need a different approach

An Artin group G(Γ) is of large type if mij ≥ 3. These groups were introduced by Appel&Schupp (1983).

Theorem (Appel&Schupp, 1983, Appel, 1984)

Large type Artin groups have soluble word and conjugacy problems, are torsion-free, have x2

i , ∀i free, and much more.

SLIDE 44

Larger type groups seem to need a different approach

An Artin group G(Γ) is of large type if mij ≥ 3. These groups were introduced by Appel&Schupp (1983).

Theorem (Appel&Schupp, 1983, Appel, 1984)

Large type Artin groups have soluble word and conjugacy problems, are torsion-free, have x2

i , ∀i free, and much more.

Standard presentations for two generator subgroups Gij satisfy small cancellation conditions, and so does a presentation for G with infinitely many 2-generator relations. So a Dehn algorithm solves the word problem.

SLIDE 45

Larger type groups seem to need a different approach

An Artin group G(Γ) is of large type if mij ≥ 3. These groups were introduced by Appel&Schupp (1983).

Theorem (Appel&Schupp, 1983, Appel, 1984)

Large type Artin groups have soluble word and conjugacy problems, are torsion-free, have x2

i , ∀i free, and much more.

Standard presentations for two generator subgroups Gij satisfy small cancellation conditions, and so does a presentation for G with infinitely many 2-generator relations. So a Dehn algorithm solves the word problem. If G is extra-large (mij ≥ 4) we have C(8); it’s elementary that Dehn’s algorithm applies. Using some rather technical arguments, Appel (1984) extended the results of Appel&Schupp from extra-large to large type.

SLIDE 46

Larger type groups seem to need a different approach

An Artin group G(Γ) is of large type if mij ≥ 3. These groups were introduced by Appel&Schupp (1983).

Theorem (Appel&Schupp, 1983, Appel, 1984)

Large type Artin groups have soluble word and conjugacy problems, are torsion-free, have x2

i , ∀i free, and much more.

Standard presentations for two generator subgroups Gij satisfy small cancellation conditions, and so does a presentation for G with infinitely many 2-generator relations. So a Dehn algorithm solves the word problem. If G is extra-large (mij ≥ 4) we have C(8); it’s elementary that Dehn’s algorithm applies. Using some rather technical arguments, Appel (1984) extended the results of Appel&Schupp from extra-large to large type. Kapovich&Schupp (2003): if mij ≥ 7, G is relatively hyperbolic in the sense of Farb, relative to its 2-generator parabolic subgroups.

SLIDE 47

Beyond large type

Pride (1986) used similar geometric methods to deal with triangle free Artin groups, for which Γ \ {∞-edges} contains no triangles. He proved word and conjugacy problems soluble, verified Tits’ conjecture, etc.

SLIDE 48

Beyond large type

Pride (1986) used similar geometric methods to deal with triangle free Artin groups, for which Γ \ {∞-edges} contains no triangles. He proved word and conjugacy problems soluble, verified Tits’ conjecture, etc. Peifer (1996) used Appel&Schupp’s small cancellation techniques to prove that extra-large type groups are biautomatic (⇒ automatic, soluble word problem, soluble conjugacy problem).

SLIDE 49

Beyond large type

Pride (1986) used similar geometric methods to deal with triangle free Artin groups, for which Γ \ {∞-edges} contains no triangles. He proved word and conjugacy problems soluble, verified Tits’ conjecture, etc. Peifer (1996) used Appel&Schupp’s small cancellation techniques to prove that extra-large type groups are biautomatic (⇒ automatic, soluble word problem, soluble conjugacy problem). And Chermak (1998) studied locally non-spherical Artin groups, for which no rank 3 subdiagram of Γ is spherical. An exponential agorithm solves the word problem, rewriting 2-generator subwords.

SLIDE 50

Geometry of large type groups

Appel&Schupp had revealed non-positively curved geometry within large type groups, and there was more to find, using results of Gersten&Short.

SLIDE 51

Geometry of large type groups

Appel&Schupp had revealed non-positively curved geometry within large type groups, and there was more to find, using results of Gersten&Short.

Theorem (Gersten&Short, 1991,1992)

The fundamental groups of piecewise Euclidean 2-complexes of types A1 × A1, A2, B2 and G2 (corresponding to tesselations of the Euclidean plane by squares, equilateral triangles, and triangles with angles (π/2, π/4, π/4), (π/2, π/3, π/6)) are automatic, and biautomatic.

SLIDE 52

Geometry of large type groups

Appel&Schupp had revealed non-positively curved geometry within large type groups, and there was more to find, using results of Gersten&Short.

Theorem (Gersten&Short, 1991,1992)

The fundamental groups of piecewise Euclidean 2-complexes of types A1 × A1, A2, B2 and G2 (corresponding to tesselations of the Euclidean plane by squares, equilateral triangles, and triangles with angles (π/2, π/4, π/4), (π/2, π/3, π/6)) are automatic, and biautomatic.

Theorem (Brady&McCammond, 2000)

Various Artin groups of large type, including all that are 3-generated, act appropriately on piecewise Euclidean non-positively curved 2-complexes of types A2 or B2, and hence are biautomatic.

SLIDE 53

Geometry of large type groups

Appel&Schupp had revealed non-positively curved geometry within large type groups, and there was more to find, using results of Gersten&Short.

Theorem (Gersten&Short, 1991,1992)

The fundamental groups of piecewise Euclidean 2-complexes of types A1 × A1, A2, B2 and G2 (corresponding to tesselations of the Euclidean plane by squares, equilateral triangles, and triangles with angles (π/2, π/4, π/4), (π/2, π/3, π/6)) are automatic, and biautomatic.

Theorem (Brady&McCammond, 2000)

Various Artin groups of large type, including all that are 3-generated, act appropriately on piecewise Euclidean non-positively curved 2-complexes of types A2 or B2, and hence are biautomatic. B-McC complexes are made by attaching angles and lengths to presentation complexes for non-standard presentations, all relators of length 3.

SLIDE 54

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic.

SLIDE 55

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic. The clique complex of a graph is k-systolic if connected, simply connected and locally k-large (no minimal ℓ-cycle in nbd of vtx with 3 < ℓ < k).

SLIDE 56

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic. The clique complex of a graph is k-systolic if connected, simply connected and locally k-large (no minimal ℓ-cycle in nbd of vtx with 3 < ℓ < k).

Theorem (Huang&Osajda, arXiv 2017)

Artin groups of almost large type (no 2 in any triangle in Γ \ {∞-edges}, at most two 2s in any square) are systolic.

SLIDE 57

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic. The clique complex of a graph is k-systolic if connected, simply connected and locally k-large (no minimal ℓ-cycle in nbd of vtx with 3 < ℓ < k).

Theorem (Huang&Osajda, arXiv 2017)

Artin groups of almost large type (no 2 in any triangle in Γ \ {∞-edges}, at most two 2s in any square) are systolic. Consequences: all Artin groups of almost large type

• are biautomatic,

SLIDE 58

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic. The clique complex of a graph is k-systolic if connected, simply connected and locally k-large (no minimal ℓ-cycle in nbd of vtx with 3 < ℓ < k).

Theorem (Huang&Osajda, arXiv 2017)

Artin groups of almost large type (no 2 in any triangle in Γ \ {∞-edges}, at most two 2s in any square) are systolic. Consequences: all Artin groups of almost large type

• are biautomatic,
• have all virtually soluble subgroups being virtually (cyclic or Z2),

SLIDE 59

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic. The clique complex of a graph is k-systolic if connected, simply connected and locally k-large (no minimal ℓ-cycle in nbd of vtx with 3 < ℓ < k).

Theorem (Huang&Osajda, arXiv 2017)

Artin groups of almost large type (no 2 in any triangle in Γ \ {∞-edges}, at most two 2s in any square) are systolic. Consequences: all Artin groups of almost large type

• are biautomatic,
• have all virtually soluble subgroups being virtually (cyclic or Z2),
• have CG(t), for |t| infinite, commensurate with Fn × Z or Z,

SLIDE 60

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic. The clique complex of a graph is k-systolic if connected, simply connected and locally k-large (no minimal ℓ-cycle in nbd of vtx with 3 < ℓ < k).

Theorem (Huang&Osajda, arXiv 2017)

Artin groups of almost large type (no 2 in any triangle in Γ \ {∞-edges}, at most two 2s in any square) are systolic. Consequences: all Artin groups of almost large type

• are biautomatic,
• have all virtually soluble subgroups being virtually (cyclic or Z2),
• have CG(t), for |t| infinite, commensurate with Fn × Z or Z,
• satisfy Novikov, Burghelea and Bass conjectures, and more . . .

SLIDE 61

Systolic complexes and groups

A group is k-systolic if it acts simplicially, properly discontinuously and cocompactly on a k-systolic simplical complex, called systolic if 6-systolic. The clique complex of a graph is k-systolic if connected, simply connected and locally k-large (no minimal ℓ-cycle in nbd of vtx with 3 < ℓ < k).

Theorem (Huang&Osajda, arXiv 2017)

Artin groups of almost large type (no 2 in any triangle in Γ \ {∞-edges}, at most two 2s in any square) are systolic. Consequences: all Artin groups of almost large type

• are biautomatic,
• have all virtually soluble subgroups being virtually (cyclic or Z2),
• have CG(t), for |t| infinite, commensurate with Fn × Z or Z,
• satisfy Novikov, Burghelea and Bass conjectures, and more . . .

NB: Brady-McCammond complexes are also systolic.

SLIDE 62

Back to rewriting: sufficiently large groups

An Artin group is of sufficiently large type if 2-edges can only occur within triangles with three 2s or at least one ∞.

SLIDE 63

Back to rewriting: sufficiently large groups

An Artin group is of sufficiently large type if 2-edges can only occur within triangles with three 2s or at least one ∞. e.g. all groups of large or almost large type, all RAAGs, and most (but not all) of the groups studied by Pride and Chermak.

SLIDE 64

Back to rewriting: sufficiently large groups

An Artin group is of sufficiently large type if 2-edges can only occur within triangles with three 2s or at least one ∞. e.g. all groups of large or almost large type, all RAAGs, and most (but not all) of the groups studied by Pride and Chermak.

Theorem (Holt&Rees, 2012,2013)

When G(Γ) is sufficiently large, any word can be reduced to a geodesic using a particular combination of τ-moves on 2-generator subwords, commuting moves, and free reduction.

SLIDE 65

Back to rewriting: sufficiently large groups

An Artin group is of sufficiently large type if 2-edges can only occur within triangles with three 2s or at least one ∞. e.g. all groups of large or almost large type, all RAAGs, and most (but not all) of the groups studied by Pride and Chermak.

Theorem (Holt&Rees, 2012,2013)

When G(Γ) is sufficiently large, any word can be reduced to a geodesic using a particular combination of τ-moves on 2-generator subwords, commuting moves, and free reduction. τ-moves relate in pairs certain two-generator critical words w, for which integers p(w) and n(w) (recording lengths of maximal positive and negative alternating subwords, relative to mij) satisfy p(w) + n(w) = mij.

SLIDE 66

Back to rewriting: sufficiently large groups

An Artin group is of sufficiently large type if 2-edges can only occur within triangles with three 2s or at least one ∞. e.g. all groups of large or almost large type, all RAAGs, and most (but not all) of the groups studied by Pride and Chermak.

Theorem (Holt&Rees, 2012,2013)

When G(Γ) is sufficiently large, any word can be reduced to a geodesic using a particular combination of τ-moves on 2-generator subwords, commuting moves, and free reduction. τ-moves relate in pairs certain two-generator critical words w, for which integers p(w) and n(w) (recording lengths of maximal positive and negative alternating subwords, relative to mij) satisfy p(w) + n(w) = mij. That equation ensures that critical words are just geodesic.

SLIDE 67

Back to rewriting: sufficiently large groups

An Artin group is of sufficiently large type if 2-edges can only occur within triangles with three 2s or at least one ∞. e.g. all groups of large or almost large type, all RAAGs, and most (but not all) of the groups studied by Pride and Chermak.

Theorem (Holt&Rees, 2012,2013)

When G(Γ) is sufficiently large, any word can be reduced to a geodesic using a particular combination of τ-moves on 2-generator subwords, commuting moves, and free reduction. τ-moves relate in pairs certain two-generator critical words w, for which integers p(w) and n(w) (recording lengths of maximal positive and negative alternating subwords, relative to mij) satisfy p(w) + n(w) = mij. That equation ensures that critical words are just geodesic. A non-geodesic word in a suff. large group must contain a critical subword.

SLIDE 68

Critical words and τ-moves: a little more detail

We call a word on two generators a := xi and b := xj critical if either w

• r its reverse has the form

(

p

aba · · ·)(y1 · · · yk)(

n

• · · · c−1d−1)

where {a, b} = {c, d}, p + n = mij;

SLIDE 69

Critical words and τ-moves: a little more detail

We call a word on two generators a := xi and b := xj critical if either w

• r its reverse has the form

(

p

aba · · ·)(y1 · · · yk)(

n

• · · · c−1d−1)

where {a, b} = {c, d}, p + n = mij; a τ-move swaps the above with its critical friend (

n

• b−1a−1 · · ·)δ(y1 · · · yk)(

p

· · · cdc),

SLIDE 70

Critical words and τ-moves: a little more detail

We call a word on two generators a := xi and b := xj critical if either w

• r its reverse has the form

(

p

aba · · ·)(y1 · · · yk)(

n

• · · · c−1d−1)

where {a, b} = {c, d}, p + n = mij; a τ-move swaps the above with its critical friend (

n

• b−1a−1 · · ·)δ(y1 · · · yk)(

p

· · · cdc), where the permutation δ() (of order 1 or 2) of both generators and words is induced by conjugation by the element ∆ij :=

mij

aba · · ·.

SLIDE 71

a, b, c, d | aba = bab, aca = cac, bcbc = cbcb, bd = db; reducing w = a−1baddc−1bcaba.

SLIDE 72

a, b, c, d | aba = bab, aca = cac, bcbc = cbcb, bd = db; reducing w = a−1baddc−1bcaba.

a−1ba is critical, and bab−1 its critical friend.

SLIDE 73

a, b, c, d | aba = bab, aca = cac, bcbc = cbcb, bd = db; reducing w = a−1baddc−1bcaba.

a−1ba is critical, and bab−1 its critical friend. We apply a sequence of 4 τ-moves. Each of the first three produces a new critical subword, and so provokes the next move. The last τ-move provokes a free reduction.

SLIDE 74

a, b, c, d | aba = bab, aca = cac, bcbc = cbcb, bd = db; reducing w = a−1baddc−1bcaba.

a−1ba is critical, and bab−1 its critical friend. We apply a sequence of 4 τ-moves. Each of the first three produces a new critical subword, and so provokes the next move. The last τ-move provokes a free reduction.

a−1

q

b a

q

b a b−1

q

d d

q

d d b−1

q

c−1 b c b−1

q

c b c−1

q

a b

q

a b a−1

q

a

SLIDE 75

a, b, c, d | aba = bab, aca = cac, bcbc = cbcb, bd = db; reducing w = a−1baddc−1bcaba.

a−1ba is critical, and bab−1 its critical friend. We apply a sequence of 4 τ-moves. Each of the first three produces a new critical subword, and so provokes the next move. The last τ-move provokes a free reduction.

a−1

q

b a

q

b a b−1

q

d d

q

d d b−1

q

c−1 b c b−1

q

c b c−1

q

a b

q

a b a−1

q

a

Using combinations of rightward and leftward sequences we have rapid reduction of any input word to a shortlex geodesic.

SLIDE 76

Consequences of rewrite system for sufficiently large groups

• groups are proved shortlex automatic over standard generating set,

any order,

SLIDE 77

Consequences of rewrite system for sufficiently large groups

• groups are proved shortlex automatic over standard generating set,

any order,

• fast O(n2) solution to word problem, and O(n3) solution to conjugacy

problem for extra-large type (using Appel-Schupp approach),

SLIDE 78

Consequences of rewrite system for sufficiently large groups

• groups are proved shortlex automatic over standard generating set,

any order,

• fast O(n2) solution to word problem, and O(n3) solution to conjugacy

problem for extra-large type (using Appel-Schupp approach),

• proof of rapid decay property and hence (using B-McC complex)

Baum-Connes conjecture for many large type groups (NB: Baum-Connes is stronger than Novikov, but implies it),

SLIDE 79

Consequences of rewrite system for sufficiently large groups

• groups are proved shortlex automatic over standard generating set,

any order,

• fast O(n2) solution to word problem, and O(n3) solution to conjugacy

problem for extra-large type (using Appel-Schupp approach),

• proof of rapid decay property and hence (using B-McC complex)

Baum-Connes conjecture for many large type groups (NB: Baum-Connes is stronger than Novikov, but implies it),

• (Blasco) proof of poly-freeness for even Artin groups of large type.

SLIDE 80

Consequences of rewrite system for sufficiently large groups

• groups are proved shortlex automatic over standard generating set,

any order,

• fast O(n2) solution to word problem, and O(n3) solution to conjugacy

problem for extra-large type (using Appel-Schupp approach),

• proof of rapid decay property and hence (using B-McC complex)

Baum-Connes conjecture for many large type groups (NB: Baum-Connes is stronger than Novikov, but implies it),

• (Blasco) proof of poly-freeness for even Artin groups of large type.
• can we deduce torsion-free?

SLIDE 81

Consequences of rewrite system for sufficiently large groups

• groups are proved shortlex automatic over standard generating set,

any order,

• fast O(n2) solution to word problem, and O(n3) solution to conjugacy

problem for extra-large type (using Appel-Schupp approach),

• proof of rapid decay property and hence (using B-McC complex)

Baum-Connes conjecture for many large type groups (NB: Baum-Connes is stronger than Novikov, but implies it),

• (Blasco) proof of poly-freeness for even Artin groups of large type.
• can we deduce torsion-free?

NB: New work of Blasco may generalise critical sequence method a little beyond sufficiently large.

SLIDE 82

Can we find a uniform rewrite system for all Artin groups?

And if so, can we solve the word problem?

SLIDE 83

Can we find a uniform rewrite system for all Artin groups?

And if so, can we solve the word problem? We have

Conjecture (Dehornoy, 2011)

Every Artin group satisfies Property H.

SLIDE 84

Can we find a uniform rewrite system for all Artin groups?

And if so, can we solve the word problem? We have

Conjecture (Dehornoy, 2011)

Every Artin group satisfies Property H. i.e. if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, for braid relations aα = bβ, γa = βb, where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse.

SLIDE 85

Can we find a uniform rewrite system for all Artin groups?

And if so, can we solve the word problem? We have

Conjecture (Dehornoy, 2011)

Every Artin group satisfies Property H. i.e. if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, for braid relations aα = bβ, γa = βb, where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse. In fact the property holds for almost all groups for which the word problem is currently known to be soluble (Euclidean diagrams are excluded):

SLIDE 86

Can we find a uniform rewrite system for all Artin groups?

And if so, can we solve the word problem? We have

Conjecture (Dehornoy, 2011)

Every Artin group satisfies Property H. i.e. if w =G 1, then w → ǫ using free reduction plus rewrites of the form a−1b → αβ−1, ab−1 → γ−1δ, for braid relations aα = bβ, γa = βb, where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse. In fact the property holds for almost all groups for which the word problem is currently known to be soluble (Euclidean diagrams are excluded):

Theorem (Dehornoy&Godelle, 2013, Godelle&Rees, 2016)

Property H holds for all Artin groups G(Γ) for which the complete subgraphs of Γ \ {∞-edges} are either spherical or large.

SLIDE 87

Dehornoy’s new idea: multifraction reduction

Recent work of Dehornoy et al. explores multifractions.

SLIDE 88

Dehornoy’s new idea: multifraction reduction

Recent work of Dehornoy et al. explores multifractions. Recall that, if M(Γ) is Garside, any word in G(Γ) rewrites to the normal form ab−1, a, b ∈ M(Γ), gcdr(a, b) = 1.

SLIDE 89

Dehornoy’s new idea: multifraction reduction

Recent work of Dehornoy et al. explores multifractions. Recall that, if M(Γ) is Garside, any word in G(Γ) rewrites to the normal form ab−1, a, b ∈ M(Γ), gcdr(a, b) = 1. In a general Artin monoid M(Γ), ∃ the (right) lcms needed to reduce to a single fraction;

SLIDE 90

Dehornoy’s new idea: multifraction reduction

Recent work of Dehornoy et al. explores multifractions. Recall that, if M(Γ) is Garside, any word in G(Γ) rewrites to the normal form ab−1, a, b ∈ M(Γ), gcdr(a, b) = 1. In a general Artin monoid M(Γ), ∃ the (right) lcms needed to reduce to a single fraction; Dehornoy considers the monoid F(M) of multifractions. Where F(M) := {a1/a2/ · · · /ak := a1a−1

2

· · · a(−1)k−1

k

, aj ∈ M(Γ)},

SLIDE 91

Dehornoy’s new idea: multifraction reduction

Recent work of Dehornoy et al. explores multifractions. Recall that, if M(Γ) is Garside, any word in G(Γ) rewrites to the normal form ab−1, a, b ∈ M(Γ), gcdr(a, b) = 1. In a general Artin monoid M(Γ), ∃ the (right) lcms needed to reduce to a single fraction; Dehornoy considers the monoid F(M) of multifractions. Where F(M) := {a1/a2/ · · · /ak := a1a−1

2

· · · a(−1)k−1

k

, aj ∈ M(Γ)}, then G(Γ) ∼ = U(M) ∼ = F(M)/ ≈ . U(M) is the enveloping group of M, and 1 ≈ a/a ≈ 1/a/a ≈ ∅.

SLIDE 92

Dehornoy’s new idea: multifraction reduction

Recent work of Dehornoy et al. explores multifractions. Recall that, if M(Γ) is Garside, any word in G(Γ) rewrites to the normal form ab−1, a, b ∈ M(Γ), gcdr(a, b) = 1. In a general Artin monoid M(Γ), ∃ the (right) lcms needed to reduce to a single fraction; Dehornoy considers the monoid F(M) of multifractions. Where F(M) := {a1/a2/ · · · /ak := a1a−1

2

· · · a(−1)k−1

k

, aj ∈ M(Γ)}, then G(Γ) ∼ = U(M) ∼ = F(M)/ ≈ . U(M) is the enveloping group of M, and 1 ≈ a/a ≈ 1/a/a ≈ ∅. Can we find a normal form for the elements of F(M), and an effective mechanism to rewrite to it?

SLIDE 93

Dehornoy’s new idea: multifraction reduction

Recent work of Dehornoy et al. explores multifractions. Recall that, if M(Γ) is Garside, any word in G(Γ) rewrites to the normal form ab−1, a, b ∈ M(Γ), gcdr(a, b) = 1. In a general Artin monoid M(Γ), ∃ the (right) lcms needed to reduce to a single fraction; Dehornoy considers the monoid F(M) of multifractions. Where F(M) := {a1/a2/ · · · /ak := a1a−1

2

· · · a(−1)k−1

k

, aj ∈ M(Γ)}, then G(Γ) ∼ = U(M) ∼ = F(M)/ ≈ . U(M) is the enveloping group of M, and 1 ≈ a/a ≈ 1/a/a ≈ ∅. Can we find a normal form for the elements of F(M), and an effective mechanism to rewrite to it? If Γ is spherical, we can do this with k = 2.

SLIDE 94

When M(Γ) isn’t Garside . . .

For i = 2j, if ai and ai+1 don’t have a common right multiple,

SLIDE 95

When M(Γ) isn’t Garside . . .

For i = 2j, if ai and ai+1 don’t have a common right multiple, maybe some left divisor x of ai+1 has a common right multiple with ai, so that ai+1 = xbi+1, lcmr(x, ai) = xbi = aix′, bi−1 = ai−1x′ ?

SLIDE 96

When M(Γ) isn’t Garside . . .

For i = 2j, if ai and ai+1 don’t have a common right multiple, maybe some left divisor x of ai+1 has a common right multiple with ai, so that ai+1 = xbi+1, lcmr(x, ai) = xbi = aix′, bi−1 = ai−1x′ ? Then a1/ · · · /ai−1/ai/ai+1/ · · · /ak and a1/ · · · /bi−1/bi/bi+1/ · · · /ak represent the same element.

SLIDE 97

When M(Γ) isn’t Garside . . .

For i = 2j, if ai and ai+1 don’t have a common right multiple, maybe some left divisor x of ai+1 has a common right multiple with ai, so that ai+1 = xbi+1, lcmr(x, ai) = xbi = aix′, bi−1 = ai−1x′ ? Then a1/ · · · /ai−1/ai/ai+1/ · · · /ak and a1/ · · · /bi−1/bi/bi+1/ · · · /ak represent the same element. Similary, for i = 2j + 1, if some right divisor x of ai+1 has a common left multiple with ai, so that ai+1 = bi+1x, lcmr(ai, x) = bix = x′ai, bi−1 = x′ai−1,

SLIDE 98

When M(Γ) isn’t Garside . . .

For i = 2j, if ai and ai+1 don’t have a common right multiple, maybe some left divisor x of ai+1 has a common right multiple with ai, so that ai+1 = xbi+1, lcmr(x, ai) = xbi = aix′, bi−1 = ai−1x′ ? Then a1/ · · · /ai−1/ai/ai+1/ · · · /ak and a1/ · · · /bi−1/bi/bi+1/ · · · /ak represent the same element. Similary, for i = 2j + 1, if some right divisor x of ai+1 has a common left multiple with ai, so that ai+1 = bi+1x, lcmr(ai, x) = bix = x′ai, bi−1 = x′ai−1, then a1/ . . . ai−1/ai/ai+1/ · · · /ak and a1/ . . . bi−1/bi/bi+1/ · · · /ak represent the same element.

SLIDE 99

A system of rewrite rules for F(M)

We define a rewrite system R to perform rewrites of the type a1/a2/ · · · /ai−1/ai/ai+1/ · · · /ak → a1/a2/ · · · /bi−1/bi/bi+1/ · · · /ak, given the situations we have just described:

s

ai−1

✟✟✟ ✟ ✯

bi−1

❍❍❍ ❍ ❥ s

x′

❄ s ☞ s

ai

✛

x

❄

ai+1

❍❍❍ ❍ ❥ s

bi

✛

bi+1

✟✟✟ ✟ ✯ s s

ai−1 ✟

✟ ✟ ✟ ✙

bi−1 ❍

❍ ❍ ❍ ❨ s

x′✻

s ☞ s

ai

✲

x

✻

ai+1

❍ ❍ ❍ ❍ ❨ s

bi

✲

bi+1

✟ ✟ ✟ ✟ ✙ s

SLIDE 100

A system of rewrite rules for F(M)

We define a rewrite system R to perform rewrites of the type a1/a2/ · · · /ai−1/ai/ai+1/ · · · /ak → a1/a2/ · · · /bi−1/bi/bi+1/ · · · /ak, given the situations we have just described:

s

ai−1

✟✟✟ ✟ ✯

bi−1

❍❍❍ ❍ ❥ s

x′

❄ s ☞ s

ai

✛

x

❄

ai+1

❍❍❍ ❍ ❥ s

bi

✛

bi+1

✟✟✟ ✟ ✯ s s

ai−1 ✟

✟ ✟ ✟ ✙

bi−1 ❍

❍ ❍ ❍ ❨ s

x′✻

s ☞ s

ai

✲

x

✻

ai+1

❍ ❍ ❍ ❍ ❨ s

bi

✲

bi+1

✟ ✟ ✟ ✟ ✙ s

We also use rules that delete trailing 1s at the right-hand end of a multifraction.

SLIDE 101

A system of rewrite rules for F(M)

We define a rewrite system R to perform rewrites of the type a1/a2/ · · · /ai−1/ai/ai+1/ · · · /ak → a1/a2/ · · · /bi−1/bi/bi+1/ · · · /ak, given the situations we have just described:

s

ai−1

✟✟✟ ✟ ✯

bi−1

❍❍❍ ❍ ❥ s

x′

❄ s ☞ s

ai

✛

x

❄

ai+1

❍❍❍ ❍ ❥ s

bi

✛

bi+1

✟✟✟ ✟ ✯ s s

ai−1 ✟

✟ ✟ ✟ ✙

bi−1 ❍

❍ ❍ ❍ ❨ s

x′✻

s ☞ s

ai

✲

x

✻

ai+1

❍ ❍ ❍ ❍ ❨ s

bi

✲

bi+1

✟ ✟ ✟ ✟ ✙ s

We also use rules that delete trailing 1s at the right-hand end of a multifraction. Note that when x′ = 1, then x is a common left (right) divisior of ai, ai+1, and so we are performing free reduction between two successive sections.

SLIDE 102

When is R confluent?

SLIDE 103

When is R confluent?

The 3-Ore condition on the Artin monoid M makes R confluent.

SLIDE 104

When is R confluent?

The 3-Ore condition on the Artin monoid M makes R confluent.

Theorem A (Dehornoy, 2016)

Let M be a Noetherian gcd monoid satisfying the ‘3-Ore condition’ (that 3 elements admitting pairwise common right (left) multiples must admit a single common right (left) multiple). Then M ֒ → U(M) and, for w ∈ U(M), w ↔ irreducible multifraction a1/a2/ · · · /ak, k ≥ 0, ai = 1,

SLIDE 105

When is R confluent?

The 3-Ore condition on the Artin monoid M makes R confluent.

Theorem A (Dehornoy, 2016)

Let M be a Noetherian gcd monoid satisfying the ‘3-Ore condition’ (that 3 elements admitting pairwise common right (left) multiples must admit a single common right (left) multiple). Then M ֒ → U(M) and, for w ∈ U(M), w ↔ irreducible multifraction a1/a2/ · · · /ak, k ≥ 0, ai = 1, where irreducible means: gcdr(a1, a2) = 1, and for i even (odd), if x left (right) divides ai+1, then x, ai have no common right (left) multiple.

SLIDE 106

When is R confluent?

The 3-Ore condition on the Artin monoid M makes R confluent.

Theorem A (Dehornoy, 2016)

Let M be a Noetherian gcd monoid satisfying the ‘3-Ore condition’ (that 3 elements admitting pairwise common right (left) multiples must admit a single common right (left) multiple). Then M ֒ → U(M) and, for w ∈ U(M), w ↔ irreducible multifraction a1/a2/ · · · /ak, k ≥ 0, ai = 1, where irreducible means: gcdr(a1, a2) = 1, and for i even (odd), if x left (right) divides ai+1, then x, ai have no common right (left) multiple. Every Artin monoid is a Noetherian gcd monoid; those of FC type satisfy the 3-Ore condition. The stronger 2-Ore condition (that any two elements admit left and right lcms) is satisfied by the monoids of spherical type.

SLIDE 107

But what happens beyond FC type?

We know we can solve the word problem for some Artin groups beyond FC type.

SLIDE 108

But what happens beyond FC type?

We know we can solve the word problem for some Artin groups beyond FC

• type. In particular we have a well described solution for all sufficiently

large groups.

SLIDE 109

But what happens beyond FC type?

We know we can solve the word problem for some Artin groups beyond FC

• type. In particular we have a well described solution for all sufficiently

large groups. But the rewrite system R is not convergent beyond FC-type.

SLIDE 110

But what happens beyond FC type?

We know we can solve the word problem for some Artin groups beyond FC

• type. In particular we have a well described solution for all sufficiently

large groups. But the rewrite system R is not convergent beyond FC-type.

Example

Where G = G( A2) = a, b, c | aba = bab, bcb = cbc, cac = aca, 1/c/aba → ac/ca/ba, and 1/c/aba =F(M) 1/c/bab → bc/cb/ab But neither ac/ca/ba nor bc/cb/ab reduces further using R.

SLIDE 111

But what happens beyond FC type?

We know we can solve the word problem for some Artin groups beyond FC

• type. In particular we have a well described solution for all sufficiently

large groups. But the rewrite system R is not convergent beyond FC-type.

Example

Where G = G( A2) = a, b, c | aba = bab, bcb = cbc, cac = aca, 1/c/aba → ac/ca/ba, and 1/c/aba =F(M) 1/c/bab → bc/cb/ab But neither ac/ca/ba nor bc/cb/ab reduces further using R. But note that c(ba)−1 = 1 in G( A2). To solve the word problem, R needs

• nly to reduce to geodesic form those words that represent the identity,

that is, to be semi-convergent.

SLIDE 112

But what happens beyond FC type?

We know we can solve the word problem for some Artin groups beyond FC

• type. In particular we have a well described solution for all sufficiently

large groups. But the rewrite system R is not convergent beyond FC-type.

Example

Where G = G( A2) = a, b, c | aba = bab, bcb = cbc, cac = aca, 1/c/aba → ac/ca/ba, and 1/c/aba =F(M) 1/c/bab → bc/cb/ab But neither ac/ca/ba nor bc/cb/ab reduces further using R. But note that c(ba)−1 = 1 in G( A2). To solve the word problem, R needs

• nly to reduce to geodesic form those words that represent the identity,

that is, to be semi-convergent. We look for ways to extend R so that its extension is at least semi-convergent on a wider class of groups.

SLIDE 113

Extending R to allow padding

We can modify R to allow padding, that is, the addition of an even number of 1s (representing the identity) at the beginning of a

• multifraction. Whenever a = a1/ · · · /ak represents a word w, so does

12l/a :=

2l

• 1/1/ · · · /1 /a1/ · · · /ak.

SLIDE 114

Extending R to allow padding

We can modify R to allow padding, that is, the addition of an even number of 1s (representing the identity) at the beginning of a

• multifraction. Whenever a = a1/ · · · /ak represents a word w, so does

12l/a :=

2l

• 1/1/ · · · /1 /a1/ · · · /ak.

If we apply the rules of R to 12l/a rather than to a, we are able to split elements ai = uv of a into the two parts u and v, rewriting as follows:

r

t

✟✟✟ ✟ ✯

tu

❍❍❍ ❍ ❥ r

u

❄ r ✟ r

1

✛

u

❄

uv

❍❍❍ ❍ ❥ r

1

✛

v

✟✟✟ ✟ ✯ r r

t

✟ ✟ ✟ ✟ ✙

vt

❍ ❍ ❍ ❍ ❨ r

v ✻

r ✟ r

1✲ v

✻

uv

❍ ❍ ❍ ❍ ❨ r

1

✲

u

✟ ✟ ✟ ✟ ✙ r

SLIDE 115

Extending R to allow padding

We can modify R to allow padding, that is, the addition of an even number of 1s (representing the identity) at the beginning of a

• multifraction. Whenever a = a1/ · · · /ak represents a word w, so does

12l/a :=

2l

• 1/1/ · · · /1 /a1/ · · · /ak.

If we apply the rules of R to 12l/a rather than to a, we are able to split elements ai = uv of a into the two parts u and v, rewriting as follows:

r

t

✟✟✟ ✟ ✯

tu

❍❍❍ ❍ ❥ r

u

❄ r ✟ r

1

✛

u

❄

uv

❍❍❍ ❍ ❥ r

1

✛

v

✟✟✟ ✟ ✯ r r

t

✟ ✟ ✟ ✟ ✙

vt

❍ ❍ ❍ ❍ ❨ r

v ✻

r ✟ r

1✲ v

✻

uv

❍ ❍ ❍ ❍ ❨ r

1

✲

u

✟ ✟ ✟ ✟ ✙ r

We need to be able to split off some 2-generator subwords in order to simulate τ-moves by rules within our multifraction rewrite system R.

SLIDE 116

With padding, R can deal with sufficiently large type

Theorem (Dehornoy,Holt&Rees, 2017)

For any sufficiently large Artin group G, whenever a is a multifraction representing a non-geodesic word w, the rewrite system R reduces the multifraction 16|w|/a to a shorter multifraction.

SLIDE 117

With padding, R can deal with sufficiently large type

Theorem (Dehornoy,Holt&Rees, 2017)

For any sufficiently large Artin group G, whenever a is a multifraction representing a non-geodesic word w, the rewrite system R reduces the multifraction 16|w|/a to a shorter multifraction.

Corollary (Dehornoy,Holt&Rees, 2017)

Once R is extended to allow padding, it solves the word problem in all Artin groups of sufficiently large type.

SLIDE 118

With padding, R can deal with sufficiently large type

Theorem (Dehornoy,Holt&Rees, 2017)

For any sufficiently large Artin group G, whenever a is a multifraction representing a non-geodesic word w, the rewrite system R reduces the multifraction 16|w|/a to a shorter multifraction.

Corollary (Dehornoy,Holt&Rees, 2017)

Once R is extended to allow padding, it solves the word problem in all Artin groups of sufficiently large type. In fact, we can also prove

Proposition (Dehornoy,Holt&Rees, 2017)

The padded extension of R is semi-convergent for an Artin group G(Γ) iff G satisfies property H.

SLIDE 119

Is this the route to a uniform solution to the word problem for Artin groups?

Conjecture (Dehornoy, 2016)

Some modification of R is semi-convergent for all Artin groups.