Garside groups and some of their properties Definition of a Garside - PowerPoint PPT Presentation

Right reversing method Garside groups and some of their properties lcm of x 2 1 and x 2 Fabienne 4 Chouraqui The lcm is: x 1 x 1 Definition of x 2 1 x 2 2 = x 4 a Garside 1 = In M monoid x 2 4 x 2 3 = x 4 (group) 4 x 1 x 3 = x 4 x 2 x 4 x 3 x 2 Questions about the x 2 x 1 = x 4 x 3 Garside gps x 2 1 \ x 2 4 = x 2 x 2 x 4 x 1 x 2 = x 3 x 4 2 A class of Garside x 1 x 3 = x 4 x 2 groups x 2 4 \ x 2 1 = x 2 the QYBE x 4 x 1 x 2 groups 3 Coxeter-like x 3 x 3 quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside element ∆ Garside groups and some of their properties ∆ in M is a Garside element if Fabienne Chouraqui ∆ is balanced, Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside element ∆ Garside groups and some of their properties ∆ in M is a Garside element if Fabienne Chouraqui ∆ is balanced, i.e. the set of left divisors of ∆ = the set of its right divisors = Div(∆) Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside element ∆ Garside groups and some of their properties ∆ in M is a Garside element if Fabienne Chouraqui ∆ is balanced, i.e. the set of left divisors of ∆ = the set of its right divisors = Div(∆) Definition of a Garside monoid Div(∆) is finite. (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside element ∆ Garside groups and some of their properties ∆ in M is a Garside element if Fabienne Chouraqui ∆ is balanced, i.e. the set of left divisors of ∆ = the set of its right divisors = Div(∆) Definition of a Garside monoid Div(∆) is finite. (group) Div(∆) is a generating set of M . Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside element ∆ Garside groups and some of their properties ∆ in M is a Garside element if Fabienne Chouraqui ∆ is balanced, i.e. the set of left divisors of ∆ = the set of its right divisors = Div(∆) Definition of a Garside monoid Div(∆) is finite. (group) Div(∆) is a generating set of M . Questions about the Garside gps A class of Example Garside groups X 4 the QYBE 1 is a Garside element. Why? groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside element ∆ Garside groups and some of their properties ∆ in M is a Garside element if Fabienne Chouraqui ∆ is balanced, i.e. the set of left divisors of ∆ = the set of its right divisors = Div(∆) Definition of a Garside monoid Div(∆) is finite. (group) Div(∆) is a generating set of M . Questions about the Garside gps A class of Example Garside groups X 4 the QYBE 1 is a Garside element. Why? groups Since in M , X 4 1 = X 4 2 = X 4 3 = X 4 4 = ... Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside monoid Garside groups and some of their properties A monoid M is Garside if Fabienne Chouraqui 1 is the unique invertible element. Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside monoid Garside groups and some of their properties A monoid M is Garside if Fabienne Chouraqui 1 is the unique invertible element. Definition of M is left and right cancellative. a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside monoid Garside groups and some of their properties A monoid M is Garside if Fabienne Chouraqui 1 is the unique invertible element. Definition of M is left and right cancellative. a Garside monoid (group) Any 2 elements in M have a right and left lcm. Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside monoid Garside groups and some of their properties A monoid M is Garside if Fabienne Chouraqui 1 is the unique invertible element. Definition of M is left and right cancellative. a Garside monoid (group) Any 2 elements in M have a right and left lcm. Questions Any 2 elements in M have a right and left gcd. about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside monoid Garside groups and some of their properties A monoid M is Garside if Fabienne Chouraqui 1 is the unique invertible element. Definition of M is left and right cancellative. a Garside monoid (group) Any 2 elements in M have a right and left lcm. Questions Any 2 elements in M have a right and left gcd. about the Garside gps M has a Garside element. A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Definition of a Garside monoid Garside groups and some of their properties A monoid M is Garside if Fabienne Chouraqui 1 is the unique invertible element. Definition of M is left and right cancellative. a Garside monoid (group) Any 2 elements in M have a right and left lcm. Questions Any 2 elements in M have a right and left gcd. about the Garside gps M has a Garside element. A class of Garside groups the QYBE groups A Garside group is the group of fractions of a Garside monoid. Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

A criteria for recognizing Garside monoids Garside groups and some of their properties Theorem (P.Dehornoy) Fabienne Chouraqui A monoid M is Garside if and only if Definition of 1 is the unique invertible element. a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

A criteria for recognizing Garside monoids Garside groups and some of their properties Theorem (P.Dehornoy) Fabienne Chouraqui A monoid M is Garside if and only if Definition of 1 is the unique invertible element. a Garside monoid (group) M is left and right cancellative. Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

A criteria for recognizing Garside monoids Garside groups and some of their properties Theorem (P.Dehornoy) Fabienne Chouraqui A monoid M is Garside if and only if Definition of 1 is the unique invertible element. a Garside monoid (group) M is left and right cancellative. Questions Any two elements in M with a right common multiple about the Garside gps admit a right lcm. A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

A criteria for recognizing Garside monoids Garside groups and some of their properties Theorem (P.Dehornoy) Fabienne Chouraqui A monoid M is Garside if and only if Definition of 1 is the unique invertible element. a Garside monoid (group) M is left and right cancellative. Questions Any two elements in M with a right common multiple about the Garside gps admit a right lcm. A class of Garside M has a finite generating set S closed under complement, groups the QYBE that is if X , Y ∈ S then the complement X \ Y is in S. groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

What are the advantages of being a Garside group? Garside groups and some of their properties Fabienne Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

What are the advantages of being a Garside group? Garside If the group G is Garside, then groups and some of their properties G is torsion-free [P.Dehornoy 1998] Fabienne Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

What are the advantages of being a Garside group? Garside If the group G is Garside, then groups and some of their properties G is torsion-free [P.Dehornoy 1998] Fabienne Chouraqui G is bi-automatic [P.Dehornoy 2002] Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

What are the advantages of being a Garside group? Garside If the group G is Garside, then groups and some of their properties G is torsion-free [P.Dehornoy 1998] Fabienne Chouraqui G is bi-automatic [P.Dehornoy 2002] G has word and conjugacy problem solvable Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

What are the advantages of being a Garside group? Garside If the group G is Garside, then groups and some of their properties G is torsion-free [P.Dehornoy 1998] Fabienne Chouraqui G is bi-automatic [P.Dehornoy 2002] G has word and conjugacy problem solvable Definition of a Garside monoid G has finite homological dimension [P.Dehornoy and (group) Y.Lafont 2003][R.Charney, J. Meier and K. Whittlesey Questions about the 2004] Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

What are the advantages of being a Garside group? Garside If the group G is Garside, then groups and some of their properties G is torsion-free [P.Dehornoy 1998] Fabienne Chouraqui G is bi-automatic [P.Dehornoy 2002] G has word and conjugacy problem solvable Definition of a Garside monoid G has finite homological dimension [P.Dehornoy and (group) Y.Lafont 2003][R.Charney, J. Meier and K. Whittlesey Questions about the 2004] Garside gps A class of Garside Examples of Garside groups groups the QYBE groups Coxeter-like Braid groups [Garside] quotient groups Artin groups of finite type [Deligne, Brieskorn-Saito] Orderability Torus link groups [Picantin] of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some questions about the Garside groups Garside groups and some of their properties Do Garside groups admit a finite quotient that plays the same Fabienne role S n plays for B n or the Coxeter groups for finite-type Artin Chouraqui groups? Definition of a Garside question raised by D.Bessis. monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some questions about the Garside groups Garside groups and some of their properties Do Garside groups admit a finite quotient that plays the same Fabienne role S n plays for B n or the Coxeter groups for finite-type Artin Chouraqui groups? Definition of a Garside question raised by D.Bessis. monoid (group) Questions Are all the Garside groups left-orderable? about the Garside gps question raised by P.Dehornoy, I.Dynnikov, D.Rolfsen, B.Wiest. A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some questions about the Garside groups Garside groups and some of their properties Do Garside groups admit a finite quotient that plays the same Fabienne role S n plays for B n or the Coxeter groups for finite-type Artin Chouraqui groups? Definition of a Garside question raised by D.Bessis. monoid (group) Questions Are all the Garside groups left-orderable? about the Garside gps question raised by P.Dehornoy, I.Dynnikov, D.Rolfsen, B.Wiest. A class of Garside groups Are all the Garside groups linear groups? the QYBE groups Coxeter-like question raised by M.Elder. quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The quantum Yang-Baxter equation - QYBE Garside groups and some of their Let R : V ⊗ V → V ⊗ V be a linear operator, where V is a properties vector space. Fabienne Chouraqui The QYBE is the equality R 12 R 13 R 23 = R 23 R 13 R 12 of linear transformations on V ⊗ V ⊗ V , where R ij means R acting on Definition of a Garside monoid the i − th and j − th components. (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The quantum Yang-Baxter equation - QYBE Garside groups and some of their Let R : V ⊗ V → V ⊗ V be a linear operator, where V is a properties vector space. Fabienne Chouraqui The QYBE is the equality R 12 R 13 R 23 = R 23 R 13 R 12 of linear transformations on V ⊗ V ⊗ V , where R ij means R acting on Definition of a Garside monoid the i − th and j − th components. (group) Questions A set-theoretical solution ( X , S ) of this equation [Drinfeld] about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The quantum Yang-Baxter equation - QYBE Garside groups and some of their Let R : V ⊗ V → V ⊗ V be a linear operator, where V is a properties vector space. Fabienne Chouraqui The QYBE is the equality R 12 R 13 R 23 = R 23 R 13 R 12 of linear transformations on V ⊗ V ⊗ V , where R ij means R acting on Definition of a Garside monoid the i − th and j − th components. (group) Questions A set-theoretical solution ( X , S ) of this equation [Drinfeld] about the Garside gps A class of V is a vector space spanned by a set X . Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The quantum Yang-Baxter equation - QYBE Garside groups and some of their Let R : V ⊗ V → V ⊗ V be a linear operator, where V is a properties vector space. Fabienne Chouraqui The QYBE is the equality R 12 R 13 R 23 = R 23 R 13 R 12 of linear transformations on V ⊗ V ⊗ V , where R ij means R acting on Definition of a Garside monoid the i − th and j − th components. (group) Questions A set-theoretical solution ( X , S ) of this equation [Drinfeld] about the Garside gps A class of V is a vector space spanned by a set X . Garside groups R is the linear operator induced by a mapping the QYBE groups S : X × X → X × X . Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Properties of a solution ( X , S ) Garside groups and Let X = { x 1 , ..., x n } and let S be defined in the following way: some of their properties S ( i , j ) = ( g i ( j ) , f j ( i )), where f i , g i : X → X . Fabienne Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Properties of a solution ( X , S ) Garside groups and Let X = { x 1 , ..., x n } and let S be defined in the following way: some of their properties S ( i , j ) = ( g i ( j ) , f j ( i )), where f i , g i : X → X . Fabienne Chouraqui Proposition [Etingof, Schedler, Soloviev - 1999] Definition of a Garside monoid (group) ( X , S ) is non-degenerate ⇔ f i and g i are bijective, Questions 1 ≤ i ≤ n . about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Properties of a solution ( X , S ) Garside groups and Let X = { x 1 , ..., x n } and let S be defined in the following way: some of their properties S ( i , j ) = ( g i ( j ) , f j ( i )), where f i , g i : X → X . Fabienne Chouraqui Proposition [P.Etingof, T.Schedler, A.Soloviev - 1999] Definition of a Garside monoid (group) ( X , S ) is non-degenerate ⇔ f i and g i are bijective, Questions 1 ≤ i ≤ n . about the Garside gps ( X , S ) is involutive ⇔ S 2 = Id X × X . A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Properties of a solution ( X , S ) Garside groups and Let X = { x 1 , ..., x n } and let S be defined in the following way: some of their properties S ( i , j ) = ( g i ( j ) , f j ( i )), where f i , g i : X → X . Fabienne Chouraqui Proposition [P.Etingof, T.Schedler, A.Soloviev - 1999] Definition of a Garside monoid (group) ( X , S ) is non-degenerate ⇔ f i and g i are bijective, Questions 1 ≤ i ≤ n . about the Garside gps ( X , S ) is involutive ⇔ S 2 = Id X × X . A class of ( X , S ) is braided ⇔ S 12 S 23 S 12 = S 23 S 12 S 23 Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Properties of a solution ( X , S ) Garside groups and Let X = { x 1 , ..., x n } and let S be defined in the following way: some of their properties S ( i , j ) = ( g i ( j ) , f j ( i )), where f i , g i : X → X . Fabienne Chouraqui Proposition [P.Etingof, T.Schedler, A.Soloviev - 1999] Definition of a Garside monoid (group) ( X , S ) is non-degenerate ⇔ f i and g i are bijective, Questions 1 ≤ i ≤ n . about the Garside gps ( X , S ) is involutive ⇔ g g i ( j ) f j ( i ) = i and f f j ( i ) g i ( j ) = j , A class of Garside 1 ≤ i , j ≤ n . groups the QYBE ( X , S ) is braided ⇔ g i g j = g g i ( j ) g f j ( i ) and f j f i = f f j ( i ) f g i ( j ) groups and f g fj ( i ) ( k ) g i ( j ) = g f gj ( k ) ( i ) f k ( j ), 1 ≤ i , j , k ≤ n . Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The QYBE group: the structure group of ( X , S ) Garside groups and some of their properties Assumption: ( X , S ) is a non-degenerate, involutive and braided Fabienne solution. Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The QYBE group: the structure group of ( X , S ) Garside groups and some of their properties Assumption: ( X , S ) is a non-degenerate, involutive and braided Fabienne solution. Chouraqui The structure group G of ( X , S ) [Etingof, Schedler, Soloviev] Definition of a Garside monoid The generators: X = { x 1 , x 2 , .., x n } . (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The QYBE group: the structure group of ( X , S ) Garside groups and some of their properties Assumption: ( X , S ) is a non-degenerate, involutive and braided Fabienne solution. Chouraqui The structure group G of ( X , S ) [Etingof, Schedler, Soloviev] Definition of a Garside monoid The generators: X = { x 1 , x 2 , .., x n } . (group) Questions The defining relations: x i x j = x k x l whenever about the Garside gps S ( i , j ) = ( k , l ) A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The QYBE group: the structure group of ( X , S ) Garside groups and some of their properties Assumption: ( X , S ) is a non-degenerate, involutive and braided Fabienne solution. Chouraqui The structure group G of ( X , S ) [Etingof, Schedler, Soloviev] Definition of a Garside monoid The generators: X = { x 1 , x 2 , .., x n } . (group) Questions The defining relations: x i x j = x k x l whenever about the Garside gps S ( i , j ) = ( k , l ) A class of Garside groups the QYBE There are exactly n ( n − 1) defining relations. groups 2 Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The example Garside groups and some of their Let X = { x 1 , x 2 , x 3 , x 4 } . properties Fabienne The functions that define S Chouraqui f 1 = g 1 = f 3 = g 3 = (1 , 2 , 3 , 4) Definition of a Garside f 2 = g 2 = f 4 = g 4 = (1 , 4 , 3 , 2) monoid (group) ( X , S ) is a non-degenerate, involutive and braided solution. Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The example Garside groups and some of their Let X = { x 1 , x 2 , x 3 , x 4 } . properties Fabienne The functions that define S Chouraqui f 1 = g 1 = f 3 = g 3 = (1 , 2 , 3 , 4) Definition of a Garside f 2 = g 2 = f 4 = g 4 = (1 , 4 , 3 , 2) monoid (group) ( X , S ) is a non-degenerate, involutive and braided solution. Questions about the Garside gps The defining relations in G and in M A class of Garside x 2 1 = x 2 x 2 3 = x 2 groups 2 4 x 1 x 2 = x 3 x 4 x 1 x 3 = x 4 x 2 the QYBE groups x 2 x 4 = x 3 x 1 x 2 x 1 = x 4 x 3 Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The correspondence between QYBE groups and Garside groups Garside groups and Theorem (F.C. 2009) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided Fabienne Chouraqui set-theoretical solution of the quantum Yang-Baxter equation with structure group G. Then G is Garside. Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The correspondence between QYBE groups and Garside groups Garside groups and Theorem (F.C. 2009) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided Fabienne Chouraqui set-theoretical solution of the quantum Yang-Baxter equation with structure group G. Then G is Garside. Definition of a Garside monoid (group) Questions Assume that Mon � X | R � is a Garside monoid such that: about the Garside gps - the cardinality of R is n ( n − 1) / 2 A class of - each side of a relation in R has length 2. Garside groups - if the word x i x j appears in R , then it appears only once. the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

The correspondence between QYBE groups and Garside groups Garside groups and Theorem (F.C. 2009) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided Fabienne Chouraqui set-theoretical solution of the quantum Yang-Baxter equation with structure group G. Then G is Garside. Definition of a Garside monoid (group) Questions Assume that Mon � X | R � is a Garside monoid such that: about the Garside gps - the cardinality of R is n ( n − 1) / 2 A class of - each side of a relation in R has length 2. Garside groups - if the word x i x j appears in R , then it appears only once. the QYBE groups Then G = Gp � X | R � is the structure group of a Coxeter-like quotient non-degenerate, involutive and braided solution ( X , S ), with groups | X | = n . Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (1) Garside groups and some of their properties The original Coxeter group Fabienne Chouraqui There exits a short exact sequence: 1 → P n → B n → S n → 1 Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (1) Garside groups and some of their properties The original Coxeter group Fabienne Chouraqui There exits a short exact sequence: 1 → P n → B n → S n → 1 Definition of More generally, finite-type Artin groups have a finite quotient a Garside monoid group: the finite Coxeter group. (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (1) Garside groups and some of their properties The original Coxeter group Fabienne Chouraqui There exits a short exact sequence: 1 → P n → B n → S n → 1 Definition of More generally, finite-type Artin groups have a finite quotient a Garside monoid group: the finite Coxeter group. (group) Questions about the What is so special with this finite quotient group? Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (1) Garside groups and some of their properties The original Coxeter group Fabienne Chouraqui There exits a short exact sequence: 1 → P n → B n → S n → 1 Definition of More generally, finite-type Artin groups have a finite quotient a Garside monoid group: the finite Coxeter group. (group) Questions about the What is so special with this finite quotient group? Garside gps A class of There exits a bijection between the elements in the finite Garside groups quotient group ( S n or finite Coxeter) and the set Div(∆) in B n the QYBE groups or finite-type Artin group. Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (2) Garside groups and some of their properties The question raised by D.Bessis Fabienne Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (2) Garside groups and some of their properties The question raised by D.Bessis Fabienne Chouraqui Do Garside groups admit a finite quotient that plays the same Definition of role S n plays for B n or the Coxeter groups for finite-type Artin a Garside monoid groups? (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (2) Garside groups and some of their properties The question raised by D.Bessis Fabienne Chouraqui Do Garside groups admit a finite quotient that plays the same Definition of role S n plays for B n or the Coxeter groups for finite-type Artin a Garside monoid groups? (group) Questions about the Our answer: yes for QYBE groups with additional condition ( C ) Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Do Coxeter-like quotient groups exist for Garside groups? (2) Garside groups and some of their properties The question raised by D.Bessis Fabienne Chouraqui Do Garside groups admit a finite quotient that plays the same Definition of role S n plays for B n or the Coxeter groups for finite-type Artin a Garside monoid groups? (group) Questions about the Our answer: yes for QYBE groups with additional condition ( C ) Garside gps A class of Garside groups Dehornoy’s extension 2014: condition ( C ) can be relaxed the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

QYBE groups with condition ( C ) admit Coxeter-like quotient groups Garside groups and Theorem (F.C and E.Godelle 2013) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided solution Fabienne of the QYBE with structure group G and | X | = n. Assume Chouraqui ( X , S ) satisfies the condition ( C ) . Then there exits a short Definition of a Garside exact sequence: 1 → N → G → W → 1 monoid (group) satisfying Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

QYBE groups with condition ( C ) admit Coxeter-like quotient groups Garside groups and Theorem (F.C and E.Godelle 2013) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided solution Fabienne of the QYBE with structure group G and | X | = n. Assume Chouraqui ( X , S ) satisfies the condition ( C ) . Then there exits a short Definition of a Garside exact sequence: 1 → N → G → W → 1 monoid (group) satisfying Questions N is a normal free abelian group of rank n about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

QYBE groups with condition ( C ) admit Coxeter-like quotient groups Garside groups and Theorem (F.C and E.Godelle 2013) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided solution Fabienne of the QYBE with structure group G and | X | = n. Assume Chouraqui ( X , S ) satisfies the condition ( C ) . Then there exits a short Definition of a Garside exact sequence: 1 → N → G → W → 1 monoid (group) satisfying Questions N is a normal free abelian group of rank n about the Garside gps There exists a bijection between W and Div(∆) A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

QYBE groups with condition ( C ) admit Coxeter-like quotient groups Garside groups and Theorem (F.C and E.Godelle 2013) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided solution Fabienne of the QYBE with structure group G and | X | = n. Assume Chouraqui ( X , S ) satisfies the condition ( C ) . Then there exits a short Definition of a Garside exact sequence: 1 → N → G → W → 1 monoid (group) satisfying Questions N is a normal free abelian group of rank n about the Garside gps There exists a bijection between W and Div(∆) A class of Garside W is a finite group of order 2 n groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

QYBE groups with condition ( C ) admit Coxeter-like quotient groups Garside groups and Theorem (F.C and E.Godelle 2013) some of their properties Let ( X , S ) be a non-degenerate, involutive and braided solution Fabienne of the QYBE with structure group G and | X | = n. Assume Chouraqui ( X , S ) satisfies the condition ( C ) . Then there exits a short Definition of a Garside exact sequence: 1 → N → G → W → 1 monoid (group) satisfying Questions N is a normal free abelian group of rank n about the Garside gps There exists a bijection between W and Div(∆) A class of Garside W is a finite group of order 2 n groups the QYBE groups Coxeter-like What is condition ( C )? quotient groups Let x i , x j ∈ X . If S ( i , j ) = ( i , j ), then f i f j = g i g j = Id X . Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

A remark about: QYBE groups admit Coxeter-like quotient groups Garside groups and some of their properties Theorem (F.C and E.Godelle 2013) Fabienne Chouraqui Let ( X , S ) be a non-degenerate, involutive and braided solution of the QYBE with structure group G and | X | = n. Assume Definition of a Garside ( X , S ) satisfies the condition ( C ) . Then there exits a short monoid (group) exact sequence: 1 → N → G → W → 1 satisfying Questions N is a normal free abelian group of rank n about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

A remark about: QYBE groups admit Coxeter-like quotient groups Garside groups and some of their properties Theorem (F.C and E.Godelle 2013) Fabienne Chouraqui Let ( X , S ) be a non-degenerate, involutive and braided solution of the QYBE with structure group G and | X | = n. Assume Definition of a Garside ( X , S ) satisfies the condition ( C ) . Then there exits a short monoid (group) exact sequence: 1 → N → G → W → 1 satisfying Questions N is a normal free abelian group of rank n about the Garside gps A class of T.Gateva-Ivanova and M. Van den Bergh show G is a Garside groups Bieberbach group (i.e G ≤ Iso( R n )). the QYBE groups E.Jespers and J.Okninski call W a IYB group, but there is no Coxeter-like quotient connection between W and Div(∆). groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside G is bi-orderable monoid (group) if ≺ is invariant under left and right multiplication: Questions about the g ≺ h = ⇒ fgk ≺ fhk , ∀ f , g , h , k ∈ G . Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside G is bi-orderable monoid (group) if ≺ is invariant under left and right multiplication: Questions about the g ≺ h = ⇒ fgk ≺ fhk , ∀ f , g , h , k ∈ G . Garside gps A class of Garside Examples of bi-orderable and left-orderable groups groups the QYBE Bi-orderable: free groups, groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside G is bi-orderable monoid (group) if ≺ is invariant under left and right multiplication: Questions about the g ≺ h = ⇒ fgk ≺ fhk , ∀ f , g , h , k ∈ G . Garside gps A class of Garside Examples of bi-orderable and left-orderable groups groups the QYBE Bi-orderable: free groups,torsion-free abelian groups, groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside G is bi-orderable monoid (group) if ≺ is invariant under left and right multiplication: Questions about the g ≺ h = ⇒ fgk ≺ fhk , ∀ f , g , h , k ∈ G . Garside gps A class of Garside Examples of bi-orderable and left-orderable groups groups the QYBE Bi-orderable: free groups,torsion-free abelian groups,pure braid groups Coxeter-like groups, quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside G is bi-orderable monoid (group) if ≺ is invariant under left and right multiplication: Questions about the g ≺ h = ⇒ fgk ≺ fhk , ∀ f , g , h , k ∈ G . Garside gps A class of Garside Examples of bi-orderable and left-orderable groups groups the QYBE Bi-orderable: free groups,torsion-free abelian groups,pure braid groups Coxeter-like groups, f.g of surfaces except the Klein bottle group and the quotient groups projective plane’s group Orderability Left-orderable: knot groups, of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside G is bi-orderable monoid (group) if ≺ is invariant under left and right multiplication: Questions about the g ≺ h = ⇒ fgk ≺ fhk , ∀ f , g , h , k ∈ G . Garside gps A class of Garside Examples of bi-orderable and left-orderable groups groups the QYBE Bi-orderable: free groups,torsion-free abelian groups,pure braid groups Coxeter-like groups, f.g of surfaces except the Klein bottle group and the quotient groups projective plane’s group Orderability Left-orderable: knot groups, braid groups, of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Orderability of groups Garside A group G is left-orderable groups and some of their if there exists a strict total ordering ≺ of its elements which is properties Fabienne invariant under left multiplication: Chouraqui g ≺ h = ⇒ fg ≺ fh , ∀ f , g , h ∈ G . Definition of a Garside G is bi-orderable monoid (group) if ≺ is invariant under left and right multiplication: Questions about the g ≺ h = ⇒ fgk ≺ fhk , ∀ f , g , h , k ∈ G . Garside gps A class of Garside Examples of bi-orderable and left-orderable groups groups the QYBE Bi-orderable: free groups,torsion-free abelian groups,pure braid groups Coxeter-like groups, f.g of surfaces except the Klein bottle group and the quotient groups projective plane’s group Orderability Left-orderable: knot groups, braid groups, Homeo + ( R ) of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some more definitions Garside A left order ≺ in a countable group G is recurrent if for groups and some of their every g ∈ G and every finite increasing sequence properties h 1 ≺ h 2 ≺ ... ≺ h r with h i ∈ G , there exists n i → ∞ such Fabienne that ∀ i , h 1 g n i ≺ h 2 g n i ≺ ... ≺ h r g n i . Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some more definitions Garside A left order ≺ in a countable group G is recurrent if for groups and some of their every g ∈ G and every finite increasing sequence properties h 1 ≺ h 2 ≺ ... ≺ h r with h i ∈ G , there exists n i → ∞ such Fabienne that ∀ i , h 1 g n i ≺ h 2 g n i ≺ ... ≺ h r g n i . Chouraqui A left order ≺ is Conradian if for any strictly positive Definition of a Garside elements a , b ∈ G , there is a natural number n such that monoid (group) b ≺ ab n . Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some more definitions Garside A left order ≺ in a countable group G is recurrent if for groups and some of their every g ∈ G and every finite increasing sequence properties h 1 ≺ h 2 ≺ ... ≺ h r with h i ∈ G , there exists n i → ∞ such Fabienne that ∀ i , h 1 g n i ≺ h 2 g n i ≺ ... ≺ h r g n i . Chouraqui A left order ≺ is Conradian if for any strictly positive Definition of a Garside elements a , b ∈ G , there is a natural number n such that monoid (group) b ≺ ab n . ≺ recurrent ⇒ Conradian (D. Witte-Morris). Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some more definitions Garside A left order ≺ in a countable group G is recurrent if for groups and some of their every g ∈ G and every finite increasing sequence properties h 1 ≺ h 2 ≺ ... ≺ h r with h i ∈ G , there exists n i → ∞ such Fabienne that ∀ i , h 1 g n i ≺ h 2 g n i ≺ ... ≺ h r g n i . Chouraqui A left order ≺ is Conradian if for any strictly positive Definition of a Garside elements a , b ∈ G , there is a natural number n such that monoid (group) b ≺ ab n . ≺ recurrent ⇒ Conradian (D. Witte-Morris). Questions LO ( G ) is a topological space (compact and totally about the Garside gps disconnected and G acts on LO ( G ) by conjugation A class of (A.Sikora). Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Some more definitions Garside A left order ≺ in a countable group G is recurrent if for groups and some of their every g ∈ G and every finite increasing sequence properties h 1 ≺ h 2 ≺ ... ≺ h r with h i ∈ G , there exists n i → ∞ such Fabienne that ∀ i , h 1 g n i ≺ h 2 g n i ≺ ... ≺ h r g n i . Chouraqui A left order ≺ is Conradian if for any strictly positive Definition of a Garside elements a , b ∈ G , there is a natural number n such that monoid (group) b ≺ ab n . ≺ recurrent ⇒ Conradian (D. Witte-Morris). Questions LO ( G ) is a topological space (compact and totally about the Garside gps disconnected and G acts on LO ( G ) by conjugation A class of (A.Sikora). Garside groups The set LO ( G ) cannot be countably infinite (P. Linnell). the QYBE groups If G is a countable left-orderable group, LO ( G ) is either Coxeter-like finite, or homeomorphic to the Cantor set, or quotient groups homeomorphic to a subspace of the Cantor space with Orderability isolated points. of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

So what if a group is left-orderable? Garside groups and some of their Bi-orderable ⇒ Recurrent left-orderable ⇒ Locally indicable ⇒ properties Left-orderable ⇒ Diffuse ⇒ Unique product ⇒ Torsion-free Fabienne Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

So what if a group is left-orderable? Garside groups and some of their Bi-orderable ⇒ Recurrent left-orderable ⇒ Locally indicable ⇒ properties Left-orderable ⇒ Diffuse ⇒ Unique product ⇒ Torsion-free Fabienne Chouraqui G is diffuse if ∀ F ⊆ G finite, ∃ x ∈ F s.t ∀ g ∈ G \ { 1 } , either Definition of a Garside ga or g − 1 a is not in F . monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

So what if a group is left-orderable? Garside groups and some of their Bi-orderable ⇒ Recurrent left-orderable ⇒ Locally indicable ⇒ properties Left-orderable ⇒ Diffuse ⇒ Unique product ⇒ Torsion-free Fabienne Chouraqui G is diffuse if ∀ F ⊆ G finite, ∃ x ∈ F s.t ∀ g ∈ G \ { 1 } , either Definition of a Garside ga or g − 1 a is not in F . monoid (group) G satisfies the UPP , if for any finite subsets A , B ⊆ G , Questions ∃ x ∈ AB that can be uniquely written as x = ab , a ∈ A , b ∈ B . about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

So what if a group is left-orderable? Garside groups and some of their Bi-orderable ⇒ Recurrent left-orderable ⇒ Locally indicable ⇒ properties Left-orderable ⇒ Diffuse ⇒ Unique product ⇒ Torsion-free Fabienne Chouraqui G is diffuse if ∀ F ⊆ G finite, ∃ x ∈ F s.t ∀ g ∈ G \ { 1 } , either Definition of a Garside ga or g − 1 a is not in F . monoid (group) G satisfies the UPP , if for any finite subsets A , B ⊆ G , Questions ∃ x ∈ AB that can be uniquely written as x = ab , a ∈ A , b ∈ B . about the Garside gps A class of Garside For a torsion free group groups the QYBE Unique product ⇒ Kaplansky’s Unit conjecture satisfied ⇒ groups Coxeter-like Kaplansky’s Zero-divisor conjecture satisfied quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

So what if a group is left-orderable? Garside groups and some of their Bi-orderable ⇒ Recurrent left-orderable ⇒ Locally indicable ⇒ properties Left-orderable ⇒ Diffuse ⇒ Unique product ⇒ Torsion-free Fabienne Chouraqui G is diffuse if ∀ F ⊆ G finite, ∃ x ∈ F s.t ∀ g ∈ G \ { 1 } , either Definition of a Garside ga or g − 1 a is not in F . monoid (group) G satisfies the UPP , if for any finite subsets A , B ⊆ G , Questions ∃ x ∈ AB that can be uniquely written as x = ab , a ∈ A , b ∈ B . about the Garside gps A class of Garside For a torsion free group groups the QYBE Unique product ⇒ Kaplansky’s Unit conjecture satisfied ⇒ groups Coxeter-like Kaplansky’s Zero-divisor conjecture satisfied: there are no zero quotient groups divisors in the group algebra Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

So what if a group is left-orderable? Garside groups and some of their Bi-orderable ⇒ Recurrent left-orderable ⇒ Locally indicable ⇒ properties Left-orderable ⇒ Diffuse ⇒ Unique product ⇒ Torsion-free Fabienne Chouraqui G is diffuse if ∀ F ⊆ G finite, ∃ x ∈ F s.t ∀ g ∈ G \ { 1 } , either Definition of a Garside ga or g − 1 a is not in F . monoid (group) G satisfies the UPP , if for any finite subsets A , B ⊆ G , Questions ∃ x ∈ AB that can be uniquely written as x = ab , a ∈ A , b ∈ B . about the Garside gps A class of Garside For a torsion free group groups the QYBE Unique product ⇒ Kaplansky’s Unit conjecture satisfied ⇒ groups Coxeter-like Kaplansky’s Zero-divisor conjecture satisfied ⇒ Kaplansky’s quotient groups Idempotent conjecture satisfied Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

So what if a group is left-orderable? Garside groups and some of their Bi-orderable ⇒ Recurrent left-orderable ⇒ Locally indicable ⇒ properties Left-orderable ⇒ Diffuse ⇒ Unique product ⇒ Torsion-free Fabienne Chouraqui G is diffuse if ∀ F ⊆ G finite, ∃ x ∈ F s.t ∀ g ∈ G \ { 1 } , either Definition of a Garside ga or g − 1 a is not in F . monoid (group) G satisfies the UPP , if for any finite subsets A , B ⊆ G , Questions ∃ x ∈ AB that can be uniquely written as x = ab , a ∈ A , b ∈ B . about the Garside gps A class of Garside For a torsion free group groups the QYBE Unique product ⇒ Kaplansky’s Unit conjecture satisfied ⇒ groups Coxeter-like Kaplansky’s Zero-divisor conjecture satisfied ⇒ Kaplansky’s quotient groups Idempotent conjecture satisfied: there are no non-trivial Orderability idempotents in the group algebra of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Are all the Garside groups left-orderable? Garside Are all the Garside groups left-orderable? groups and some of their properties Question from book Ordering braids Fabienne of P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest Chouraqui Definition of a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Are all the Garside groups left-orderable? Garside Are all the Garside groups left-orderable? groups and some of their properties Question from book Ordering braids Fabienne of P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest Chouraqui Definition of The short answer is: Not necessarily!! a Garside monoid (group) Questions about the Garside gps A class of Garside groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Are all the Garside groups left-orderable? Garside Are all the Garside groups left-orderable? groups and some of their properties Question from book Ordering braids Fabienne of P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest Chouraqui Definition of The short answer is: Not necessarily!! a Garside monoid (group) The more detailed answer: Questions about the Garside gps There exist Garside groups: A class of Garside with a recurrent left order groups the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to

Are all the Garside groups left-orderable? Garside Are all the Garside groups left-orderable? groups and some of their properties Question from book Ordering braids Fabienne of P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest Chouraqui Definition of The short answer is: Not necessarily!! a Garside monoid (group) The more detailed answer: Questions about the Garside gps There exist Garside groups: A class of Garside with a recurrent left order groups with space of left orders homeomorphic to the Cantor set. the QYBE groups Coxeter-like quotient groups Orderability of groups Remarks and Fabienne Chouraqui Garside groups and some of their properties questions to