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I Know About Artin–Tits Groups All I know about Artin{Tits groups Or: Why type A is so much easier... Daniel Tubbenhauer The first \published" braid diagram. (Page 283 from Gauß’ handwritten notes, volume seven, 1830). ≤ Joint with David Rose April 2019 Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 1 / 6 Let Γ be a Coxeter graph. Artin 1925, Tits 1961++. The Artin{Tits group and its Coxeter group quotient∼ are given by∼ generators-relations: AT(Γ) = bi bibjbi = bjbibj h j| · · · {z } |··· {z }i mij factors mij factors 2 W(Γ) = σi σi = 1; σiσjσi = σjσiσj h j |··· {z } |··· {z }i mij factors mij factors Artin{Tits groups generalize classical braid groups, Coxeter groups generalize polyhedron groups. Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 2 / 6 Maybe some categorical considerations help? In particular, what can Artin{Tits groups tell you about flavor two? My failure. What I would like to understand, but I do not. Artin{Tits groups come in four main flavors. Question: Why are these special? What happens in general type? Artin{Tits (braid) groups Flavor one. Finite helps helps Flavor three. Map- and affine types ping class groups Example Example Many open problems, e.g. the word problem. Flavor two. Con- ? Flavor four. Right figuration spaces helps helps angled groups Example Example Vanilla fla- vor. ?????. Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 3 / 6 My failure. What I would like to understand, but I do not. Artin{Tits groups come in four main flavors. Question: Why are these special? What happens in general type? Artin{Tits (braid) groups Flavor one. Finite helps helps Flavor three. Map- and affine types ping class groups Example Example Many open problems, e.g. the word problem. Flavor two. Con- ? Flavor four. Right figuration spaces helps helps angled groups Example Example Vanilla fla- vor. ?????. Maybe some categorical considerations help? In particular, what can Artin{Tits groups tell you about flavor two? Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 3 / 6 Faithfulness? Theorem (handlebodyFaithfulness? faithfulness). Rouquier's action is known to be faithful in quite a few cases: For all g; n, Rouquier'sfinite type action (Khovanov{Seidel,gives rise Brav{Thomas), to a family of faithful actions The Hecke action is known to be faithfulJ−K in very few cases, e.g. for Γ of rank 1; 2. affine type A (Gadbled{Thiel{Wagner), affine type C (handlebody). But there is \no way"b toq prove this in general. ThereB mightr(g; n) bey hope( to prove(Γ)); b this inb general.M. K S 7! J K Example (seems to work). Hecke distinguishes the braids where Burau failed: Example (the whole point). Zigzag already distinguishes braids: Theorem (handlebody HOMFLYPT homology). This action extends to a HOMFLYPT invariant for handlebody links. Mnemonic: M M k M M k b = & b M = & b = J K J KH2 M M k M M k Please stop! Let Γ be a Coxeter graph. The following commuting diagram exists in any type: AT(Γ) AT(Γ) [ ] J−K decat. − b( q(Γ)) q(Γ) K S H decat. b( q(Γ)) q(Γ) K Z B Question. How does this help to study Artin{Tits groups? Here (killing idempotents for the last row): q b q I Hecke algebra (Γ), homotopy category of Soergel bimodules ( (Γ)). H K S I Hecke action [ ], Rouquier complex . − q J−K I Burau representation (Γ), homotopy category of representations of zigzag algebras b( q(Γ)). B K Z Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 4 / 6 Faithfulness? Theorem (handlebody faithfulness). Rouquier's action is known to be faithful in quite a few cases: For all g; n, Rouquier'sfinite type action (Khovanov{Seidel,gives rise Brav{Thomas), to a family of faithful actions J−K affine type A (Gadbled{Thiel{Wagner), affine type C (handlebody). b q ThereB mightr(g; n) bey hope( to prove(Γ)); b this inb general.M. K S 7! J K Example (the whole point). Zigzag already distinguishes braids: Theorem (handlebody HOMFLYPT homology). This action extends to a HOMFLYPT invariant for handlebody links. Mnemonic: M M k M M k b = & b M = & b = J K J KH2 M M k M M k Please stop! Let Γ be a Coxeter graph. The following commuting diagram exists in any type: AT(Γ) AT(Γ) Faithfulness? [ ] J−K decat. − The Hecke action is knownb to( beq(Γ)) faithful in very fewq cases,(Γ) e.g. for Γ of rank 1; 2. But thereK isS \no way" to prove thisH in general. decat. Example (seems to work).b( Heckeq(Γ)) distinguishes theq(Γ) braids where Burau failed: K Z B Question. How does this help to study Artin{Tits groups? Here (killing idempotents for the last row): q b q I Hecke algebra (Γ), homotopy category of Soergel bimodules ( (Γ)). H K S I Hecke action [ ], Rouquier complex . − q J−K I Burau representation (Γ), homotopy category of representations of zigzag algebras b( q(Γ)). B K Z Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 4 / 6 Theorem (handlebodyFaithfulness? faithfulness). For all g; n, Rouquier's action gives rise to a family of faithful actions The Hecke action is known to be faithfulJ−K in very few cases, e.g. for Γ of rank 1; 2. But there is \no way"b toq prove this in general. Br(g; n) y ( (Γ)); b b M. K S 7! J K Example (seems to work). Hecke distinguishes the braids where Burau failed: Theorem (handlebody HOMFLYPT homology). This action extends to a HOMFLYPT invariant for handlebody links. Mnemonic: M M k M M k b = & b M = & b = J K J KH2 M M k M M k Please stop! Let Γ be a Coxeter graph. The following commuting diagram exists in any type: AT(Γ)Faithfulness? AT(Γ) Rouquier's action is known to be faithful in quite[ ] a few cases: J−K decat. − finite typeb( (Khovanov{Seidel,q(Γ)) Brav{Thomas),q(Γ) affine type A (Gadbled{Thiel{Wagner),K S affineH type C (handlebody). There might be hope to prove this in general. decat. b( q(Γ)) q(Γ) Example (the wholeK Z point). Zigzag alreadyB distinguishes braids: Question. How does this help to study Artin{Tits groups? Here (killing idempotents for the last row): q b q I Hecke algebra (Γ), homotopy category of Soergel bimodules ( (Γ)). H K S I Hecke action [ ], Rouquier complex . − q J−K I Burau representation (Γ), homotopy category of representations of zigzag algebras b( q(Γ)). B K Z Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 4 / 6 Faithfulness? Faithfulness? Rouquier's action is known to be faithful in quite a few cases: The Hecke action isfinite known type to (Khovanov{Seidel, be faithful in very Brav{Thomas), few cases, e.g. for Γ of rank 1; 2. affine typeBut A (Gadbled{Thiel{Wagner), there is \no way" to prove affine this type in general. C (handlebody). There might be hope to prove this in general. Example (seems to work). Hecke distinguishes the braids where Burau failed: Example (the whole point). Zigzag already distinguishes braids: Let Γ be a Coxeter graph. The following commuting diagram exists in any type: AT(Γ) AT(Γ) Theorem (handlebody faithfulness). [ ] J−K − For all g; n, Rouquier's actionq givesdecat. rise to aq family of faithful actions b( (Γ))− (Γ) K J K H S b q Br(g; n) y ( (Γ)); b b . K 7! M Sdecat. J K b( q(Γ)) q(Γ) K Z B Theorem (handlebody HOMFLYPT homology). Question. How does this help to study Artin{Tits groups? This action extends to a HOMFLYPT invariant for handlebody links. Mnemonic: Here (killing idempotents for the last row): q M M k b q I Hecke algebra (Γ), homotopy category of Soergel bimodules ( (Γ)). H M M k K S I Hecke action [ ], Rouquier complex . b = − &q b = J−K & b = I Burau representation (Γ),M homotopy category ofH representations2 of zigzag b q B J K J K algebras ( (Γ)). M M k K Z M M k Please stop! Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 4 / 6 Example (type A). Braid cobordismsTheorem are movies (handlebody of braids. functoriality). E.g. some generators are group monoid birth For all g; n, Rouquier's action gives rise to a family of functorial actions J−K b q Bcob(g; n) y ( (Γ)); b b M ; bcob isotopybcob M. ; & ; K & 7!; J K : 7! J K S −−−! −−−−−−! (Bcob(g; n) is the 2-category of handlebody braid cobordisms.) invertible invertible non-invertible Invertible ones encode isotopies, non-invertible ones \more interesting" topology. Final observation. Theorem (well-known?). In all (non-trivial) cases I know \faithful functorial". The Rouquier complex, is functorial in types A, B = C and affine C. Is there a general statement? b q Rouquier 2004. The 2-braid group (Γ) is im( ) ( s (Γ)). ∼ AT J−K ⊂ K S If you have a configuration space picture for AT(Γ) one can define the category of braid cobordisms Bcob(Γ) in four space. Fact (well-known?). For Γ of type A, B = C or affine type C we have (Γ) = inv(B (Γ)): AT cob Corollary (strictness). We have a categorical action b q inv(Bcob(g; n)) y ( (Γ)); b b ; bcob bcob : K S 7! J K 7! J K Question (functoriality). Can we lift to a categorical action J−K b q Bcob(g; n) y ( (Γ))? K S Daniel Tubbenhauer All I know about Artin{Tits groups April 2019 5 / 6 Theorem (handlebody functoriality).
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