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

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 X W = Y W / W (Γ), we have π 1 ( X W ) = 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) � x 2 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),

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 X W = Y W / W (Γ), we have π 1 ( X W ) = 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) � x 2 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 { x 1 , . . . , x n } ) are also Artin groups, and are convex.

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 X W = Y W / W (Γ), we have π 1 ( X W ) = 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) � x 2 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 { x 1 , . . . , x n } ) are also Artin groups, and are convex. Proofs of these use various actions of G (Γ) or a related Artin group.

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 X W = Y W / W (Γ), we have π 1 ( X W ) = 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) � x 2 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 { x 1 , . . . , x n } ) 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.

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

Artin groups of spherical type G (Γ) is of spherical type (aka finite type) when | W (Γ) | < ∞ . Irreducible diagrams are in the following list:- 3 . . . . . . r r r r r A n 1 2 n − 1 n 3 . . . . . . r r r r r C n 1 2 n − 1 n 3 . . . . . . D n r r r r r 1 2 n − 2 n − 1 r n . . . . . . . . . E n ( n = 6 , 7 , 8) r r r r r 1 2 n − 3 n − 2 n − 1 r n m r F 4 G 2 ( m ) r r r r r 1 2 3 4 1 2 5 r 5 r H 3 H 4 r r r r r 1 2 3 1 2 3 4

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

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.

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) , gcd r ( u , v ) = 1 ,

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) , gcd r ( u , v ) = 1 , since for a , b ∈ M (Γ), when lcm r ( a , b ) = a α = b β, lcm l ( a , b ) = γ a = δ b , ab − 1 = G (Γ) γ − 1 δ a − 1 b = G (Γ) αβ − 1 then and

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) , gcd r ( u , v ) = 1 , since for a , b ∈ M (Γ), when lcm r ( a , b ) = a α = b β, lcm l ( a , b ) = γ a = δ b , ab − 1 = G (Γ) γ − 1 δ a − 1 b = G (Γ) αβ − 1 then and So the word problem is soluble: ∃ a uniform solution for all Artin groups of spherical type.

Just beyond spherical Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ ˜ r r r r r A 3 B 3 1 2 1 2 3 4 3 r 4 r r

Just beyond spherical Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ ˜ r r r r r A 3 B 3 1 2 1 2 3 4 3 r 4 r r They are subgroups of Garside groups (McCammond& Sulway, 2013), and hence have soluble word problem (and more).

Just beyond spherical Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ ˜ r r r r r A 3 B 3 1 2 1 2 3 4 3 r 4 r r 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 ∞ r ∞ r ∞ r r r r r ❅ � ❅ � 1 2 1 2 3 1 2 3 ❅ � 4 3 ❅ � r 4 r 4 r r

Just beyond spherical Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ ˜ r r r r r A 3 B 3 1 2 1 2 3 4 3 r 4 r r 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 ∞ r ∞ r ∞ r r r r r ❅ � ❅ � 1 2 1 2 3 1 2 3 ❅ � 4 3 ❅ � r 4 r 4 r r FC includes right-angled groups (RAAGs), (Baudisch 1977, Green 1990).

Just beyond spherical Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ ˜ r r r r r A 3 B 3 1 2 1 2 3 4 3 r 4 r r 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 ∞ r ∞ r ∞ r r r r r ❅ � ❅ � 1 2 1 2 3 1 2 3 ❅ � 4 3 ❅ � r 4 r 4 r r FC includes right-angled groups (RAAGs), (Baudisch 1977, Green 1990). FC groups are amalgamated products of spherical type groups over parabolic subgroups.

Just beyond spherical Euclidean Artin groups are defined by extended Coxeter diagrams such as ˜ ˜ r r r r r A 3 B 3 1 2 1 2 3 4 3 r 4 r r 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 ∞ r ∞ r ∞ r r r r r ❅ � ❅ � 1 2 1 2 3 1 2 3 ❅ � 4 3 ❅ � r 4 r 4 r r 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 (m ij ∈ 2 Z ) are poly-free (Hermiller&˘ Suni´ c,2007, Blasco,Martinez-Perez&Paris 2017).

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , a , b , α, β, γ, δ ∈ M (Γ) , between subwords over generators of parabolic subgroups of spherical type.

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , 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.

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , 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 m ij = 3 , aba → bab , aba − 1 → b − 1 ab are OK, but not ab → baba − 1 .

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , 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 m ij = 3 , aba → bab , aba − 1 → b − 1 ab are OK, but not ab → baba − 1 . We have observed that: Theorem (Dehornoy&Godelle, 2013) Artin groups of FC type satisfy property H.

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , 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 m ij = 3 , aba → bab , aba − 1 → b − 1 ab 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).

Larger type groups seem to need a different approach An Artin group G (Γ) is of large type if m ij ≥ 3. These groups were introduced by Appel&Schupp (1983).

Larger type groups seem to need a different approach An Artin group G (Γ) is of large type if m ij ≥ 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 � x 2 i , ∀ i � free, and much more.

Larger type groups seem to need a different approach An Artin group G (Γ) is of large type if m ij ≥ 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 � x 2 i , ∀ i � free, and much more. Standard presentations for two generator subgroups G ij 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.

Larger type groups seem to need a different approach An Artin group G (Γ) is of large type if m ij ≥ 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 � x 2 i , ∀ i � free, and much more. Standard presentations for two generator subgroups G ij 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 (m ij ≥ 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.

Larger type groups seem to need a different approach An Artin group G (Γ) is of large type if m ij ≥ 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 � x 2 i , ∀ i � free, and much more. Standard presentations for two generator subgroups G ij 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 (m ij ≥ 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 m ij ≥ 7, G is relatively hyperbolic in the sense of Farb, relative to its 2-generator parabolic subgroups.

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.

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).

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.

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.

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 A 1 × A 1 , A 2 , B 2 and G 2 (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.

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 A 1 × A 1 , A 2 , B 2 and G 2 (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 A 2 or B 2 , and hence are biautomatic.

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 A 1 × A 1 , A 2 , B 2 and G 2 (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 A 2 or B 2 , 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.

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.

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 ).

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 2 s in any square) are systolic.

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 2 s in any square) are systolic. Consequences: all Artin groups of almost large type • are biautomatic,

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 2 s in any square) are systolic. Consequences: all Artin groups of almost large type • are biautomatic, • have all virtually soluble subgroups being virtually (cyclic or Z 2 ),

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 2 s in any square) are systolic. Consequences: all Artin groups of almost large type • are biautomatic, • have all virtually soluble subgroups being virtually (cyclic or Z 2 ), • have C G ( t ), for | t | infinite, commensurate with F n × Z or Z ,

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 2 s in any square) are systolic. Consequences: all Artin groups of almost large type • are biautomatic, • have all virtually soluble subgroups being virtually (cyclic or Z 2 ), • have C G ( t ), for | t | infinite, commensurate with F n × Z or Z , • satisfy Novikov, Burghelea and Bass conjectures, and more . . .

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 2 s in any square) are systolic. Consequences: all Artin groups of almost large type • are biautomatic, • have all virtually soluble subgroups being virtually (cyclic or Z 2 ), • have C G ( t ), for | t | infinite, commensurate with F n × Z or Z , • satisfy Novikov, Burghelea and Bass conjectures, and more . . . NB: Brady-McCammond complexes are also systolic.

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 ∞ .

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.

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.

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 m ij ) satisfy p ( w ) + n ( w ) = m ij .

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 m ij ) satisfy p ( w ) + n ( w ) = m ij . That equation ensures that critical words are just geodesic.

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 m ij ) satisfy p ( w ) + n ( w ) = m ij . That equation ensures that critical words are just geodesic. A non-geodesic word in a suff. large group must contain a critical subword.

Critical words and τ -moves: a little more detail We call a word on two generators a := x i and b := x j critical if either w or its reverse has the form n p � �� � � �� � · · · c − 1 d − 1 ) ( aba · · · )( y 1 · · · y k )( where { a , b } = { c , d } , p + n = m ij ;

Critical words and τ -moves: a little more detail We call a word on two generators a := x i and b := x j critical if either w or its reverse has the form n p � �� � � �� � · · · c − 1 d − 1 ) ( aba · · · )( y 1 · · · y k )( where { a , b } = { c , d } , p + n = m ij ; a τ -move swaps the above with its critical friend n p � �� � � �� � b − 1 a − 1 · · · ) δ ( y 1 · · · y k )( · · · cdc ) , (

Critical words and τ -moves: a little more detail We call a word on two generators a := x i and b := x j critical if either w or its reverse has the form n p � �� � � �� � · · · c − 1 d − 1 ) ( aba · · · )( y 1 · · · y k )( where { a , b } = { c , d } , p + n = m ij ; a τ -move swaps the above with its critical friend n p � �� � � �� � b − 1 a − 1 · · · ) δ ( y 1 · · · y k )( · · · cdc ) , ( where the permutation δ () (of order 1 or 2) of both generators and words m ij � �� � is induced by conjugation by the element ∆ ij := aba · · · .

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

� a , b , c , d | aba = bab , aca = cac , bcbc = cbcb , bd = db � ; reducing w = a − 1 baddc − 1 bcaba . a − 1 ba is critical, and bab − 1 its critical friend.

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

� a , b , c , d | aba = bab , aca = cac , bcbc = cbcb , bd = db � ; reducing w = a − 1 baddc − 1 bcaba . a − 1 ba 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. c − 1 a a b b a d d b c a − 1 b − 1 b − 1 b − 1 a − 1 q q q q q q q a d c q q b c − 1 b a b d Using combinations of rightward and leftward sequences we have rapid reduction of any input word to a shortlex geodesic.

Consequences of rewrite system for sufficiently large groups • groups are proved shortlex automatic over standard generating set, any order,

Consequences of rewrite system for sufficiently large groups • groups are proved shortlex automatic over standard generating set, any order, • fast O ( n 2 ) solution to word problem, and O ( n 3 ) solution to conjugacy problem for extra-large type (using Appel-Schupp approach),

Consequences of rewrite system for sufficiently large groups • groups are proved shortlex automatic over standard generating set, any order, • fast O ( n 2 ) solution to word problem, and O ( n 3 ) 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),

Consequences of rewrite system for sufficiently large groups • groups are proved shortlex automatic over standard generating set, any order, • fast O ( n 2 ) solution to word problem, and O ( n 3 ) 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.

Consequences of rewrite system for sufficiently large groups • groups are proved shortlex automatic over standard generating set, any order, • fast O ( n 2 ) solution to word problem, and O ( n 3 ) 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?

Consequences of rewrite system for sufficiently large groups • groups are proved shortlex automatic over standard generating set, any order, • fast O ( n 2 ) solution to word problem, and O ( n 3 ) 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.

Can we find a uniform rewrite system for all Artin groups? And if so, can we solve the word problem?

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.

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , for braid relations a α = b β , γ a = β b , where neither of a or b is empty unless the rule is a direct application of a braid relation or its inverse.

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , for braid relations a α = b β , γ a = β b , 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):

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 ab − 1 → γ − 1 δ, a − 1 b → αβ − 1 , for braid relations a α = b β , γ a = β b , 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.

Dehornoy’s new idea: multifraction reduction Recent work of Dehornoy et al. explores multifractions.

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) , gcd r ( a , b ) = 1 .

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) , gcd r ( a , b ) = 1 . In a general Artin monoid M (Γ), � ∃ the (right) lcms needed to reduce to a single fraction;

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) , gcd r ( 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 . · · · a ( − 1) k − 1 a 1 a − 1 Where F ( M ) := { a 1 / a 2 / · · · / a k := , a j ∈ M (Γ) } , 2 k

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) , gcd r ( 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 . · · · a ( − 1) k − 1 a 1 a − 1 Where F ( M ) := { a 1 / a 2 / · · · / a k := , a j ∈ M (Γ) } , 2 k G (Γ) ∼ ∼ = U ( M ) F ( M ) / ≈ . then = U ( M ) is the enveloping group of M , and 1 ≈ a / a ≈ 1 / a / a ≈ ∅ .

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) , gcd r ( 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 . · · · a ( − 1) k − 1 a 1 a − 1 Where F ( M ) := { a 1 / a 2 / · · · / a k := , a j ∈ M (Γ) } , 2 k G (Γ) ∼ ∼ = U ( M ) F ( M ) / ≈ . then = 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?

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) , gcd r ( 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 . · · · a ( − 1) k − 1 a 1 a − 1 Where F ( M ) := { a 1 / a 2 / · · · / a k := , a j ∈ M (Γ) } , 2 k G (Γ) ∼ ∼ = U ( M ) F ( M ) / ≈ . then = 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.

When M (Γ) isn’t Garside . . . For i = 2 j , if a i and a i +1 don’t have a common right multiple,

When M (Γ) isn’t Garside . . . For i = 2 j , if a i and a i +1 don’t have a common right multiple, maybe some left divisor x of a i +1 has a common right multiple with a i , so that b i − 1 = a i − 1 x ′ ? lcm r ( x , a i ) = xb i = a i x ′ , a i +1 = xb i +1 ,

When M (Γ) isn’t Garside . . . For i = 2 j , if a i and a i +1 don’t have a common right multiple, maybe some left divisor x of a i +1 has a common right multiple with a i , so that b i − 1 = a i − 1 x ′ ? lcm r ( x , a i ) = xb i = a i x ′ , a i +1 = xb i +1 , Then a 1 / · · · / a i − 1 / a i / a i +1 / · · · / a k and a 1 / · · · / b i − 1 / b i / b i +1 / · · · / a k represent the same element.

When M (Γ) isn’t Garside . . . For i = 2 j , if a i and a i +1 don’t have a common right multiple, maybe some left divisor x of a i +1 has a common right multiple with a i , so that b i − 1 = a i − 1 x ′ ? lcm r ( x , a i ) = xb i = a i x ′ , a i +1 = xb i +1 , Then a 1 / · · · / a i − 1 / a i / a i +1 / · · · / a k and a 1 / · · · / b i − 1 / b i / b i +1 / · · · / a k represent the same element. Similary, for i = 2 j + 1, if some right divisor x of a i +1 has a common left multiple with a i , so that lcm r ( a i , x ) = b i x = x ′ a i , b i − 1 = x ′ a i − 1 , a i +1 = b i +1 x ,

When M (Γ) isn’t Garside . . . For i = 2 j , if a i and a i +1 don’t have a common right multiple, maybe some left divisor x of a i +1 has a common right multiple with a i , so that b i − 1 = a i − 1 x ′ ? lcm r ( x , a i ) = xb i = a i x ′ , a i +1 = xb i +1 , Then a 1 / · · · / a i − 1 / a i / a i +1 / · · · / a k and a 1 / · · · / b i − 1 / b i / b i +1 / · · · / a k represent the same element. Similary, for i = 2 j + 1, if some right divisor x of a i +1 has a common left multiple with a i , so that lcm r ( a i , x ) = b i x = x ′ a i , b i − 1 = x ′ a i − 1 , a i +1 = b i +1 x , then a 1 / . . . a i − 1 / a i / a i +1 / · · · / a k and a 1 / . . . b i − 1 / b i / b i +1 / · · · / a k represent the same element.

A system of rewrite rules for F ( M ) We define a rewrite system R to perform rewrites of the type a 1 / a 2 / · · · / a i − 1 / a i / a i +1 / · · · / a k → a 1 / a 2 / · · · / b i − 1 / b i / b i +1 / · · · / a k , given the situations we have just described: a i a i ✛ ✲ ❍❍❍ a i − 1 ✟ a i − 1 a i +1 a i +1 ✯ ✟ ❍ ❨ s s s s ✟✟✟ ✟ ❍ x ′ ✻ ✻ ✟ ❍ ❍ ❥ ✙ ✟ x ′ ❍ x x ❍❍❍ ✟ ✯ s ✟ ❍ ❨ s s s ✟✟✟ ❍ ✟ ❍ ✟ ❄ ❄ b i − 1 ❍ ❥ b i − 1 ❍ ✙ ✟ s ☞ ✛ s ☞ ✲ b i +1 b i +1 s s b i b i

A system of rewrite rules for F ( M ) We define a rewrite system R to perform rewrites of the type a 1 / a 2 / · · · / a i − 1 / a i / a i +1 / · · · / a k → a 1 / a 2 / · · · / b i − 1 / b i / b i +1 / · · · / a k , given the situations we have just described: a i a i ✛ ✲ ❍❍❍ a i − 1 ✟ a i − 1 a i +1 a i +1 ✯ ✟ ❍ ❨ s s s s ✟✟✟ ✟ ❍ x ′ ✻ ✻ ✟ ❍ ❍ ❥ ✙ ✟ x ′ ❍ x x ❍❍❍ ✟ ✯ s ✟ ❍ ❨ s s s ✟✟✟ ❍ ✟ ❍ ✟ ❄ ❄ b i − 1 ❥ ❍ b i − 1 ❍ ✟ ✙ s ☞ ✛ s ☞ ✲ b i +1 b i +1 s s b i b i We also use rules that delete trailing 1s at the right-hand end of a multifraction.