The K ( , 1 ) conjecture for affine Artin groups Giovanni Paolini - - PowerPoint PPT Presentation

The K(π, 1) conjecture for affine Artin groups

Giovanni Paolini

AWS & Caltech

Joint work with Mario Salvetti (UniPi)

Seminar on Combinatorics, Lie Theory, and Topology April 28, 2020

Reflection groups

A reflectiongroup is a discrete group generated by orthogonal reflections in a Euclidean space Rn. To every reflection group W is associated a hyperplanearrangement: the set

• f hyperplanes H such that the reflection with respect to H is an element
• f W.

Arrangement of a finite reflection group Arrangement of an infinite (affine) reflection group

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The symmetric group Sn

Sn is the group generated by the reflections w.r.t. the hyperplanes {xi = xj} in Rn: these correspond to the transpositions (i j).

Example: S3

x1 = x3 x2 = x3 x1 = x2

The arrangement of S3 in {x1 + x2 + x3 = 0} ⊆ R3

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Coxeter groups

A Coxetergroup is a group presented as follows: W = S | s2 = 1 ∀ s ∈ S, sts · · ·

ms,t factors

= tst · · ·

ms,t factors

∀ s = t . Reflection groups are particular instances of Coxeter groups: the set S is given by the reflections with respect to the walls of a fundamentalchamber; the angle between two walls is

π ms,t .

Example: S3

x1 = x3 x2 = x3 x1 = x2 a b

π/3

The reflections a and b yield the following Coxeter presentation: S3 = a, b | a2 = b2 = 1, aba = bab

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Artin groups

Removing the relations s2 = 1 in the presentation of a Coxeter group, we

• btain the corresponding Artingroup:

GW = S | sts · · ·

ms,t factors

= tst · · ·

ms,t factors

∀ s = t .

Example: the braid group on 3 strands

If W = S3 is the symmetric group on 3 letters, the corresponding Artin group is B3 = a, b | aba = bab.

Other examples

◮ Free groups (all ms,t = ∞) ◮ Free abelian groups (all ms,t = 2) ◮ Right-angled Artin groups (all ms,t ∈ {2, ∞})

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Artin groups (2)

Topologically, an Artin group GW is the fundamental group of the configurationspace YW =  Cn \

• H∈AW

HC   / W.

Example (continued): the braid group on 3 strands

YW = {(x1, x2, x3) ∈ C3 | xi = xj}/S3 x1 = x3 x2 = x3 x1 = x2

The (real) arrangement

C C

t = 0 t = 1

Loops in YW are “braids”

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Open problems on Artin groups

(1) Artin groups are torsion-free. (2) Determine the center. (3) Solve the word problem. (4) K(π, 1) conjecture (Brieskorn-Arnol'd-Pham-Thom ’60s): the configuration space YW is a classifying space for GW.

π1(YW) = GW, and the higher homotopy groups are trivial (equivalently, the univer- sal cover of YW is contractible).

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Open problems on Artin groups

(1) Artin groups are torsion-free. (2) Determine the center. (3) Solve the word problem. (4) K(π, 1) conjecture (Brieskorn-Arnol'd-Pham-Thom ’60s): the configuration space YW is a classifying space for GW.

π1(YW) = GW, and the higher homotopy groups are trivial (equivalently, the univer- sal cover of YW is contractible).

Solved for spherical Artin groups (Brieskorn-Saito 1971, Deligne 1972). (1)-(3) solved for affine Artin groups (McCammond-Sulway 2017). (4) solved for some affine Artin groups (Okonek 1979, Callegaro-Moroni-Salvetti 2010).

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Theorem (P .-Salvetti 2019)

The K(π, 1) conjecture holds for all affine Artin groups.

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Interval groups

G group, R generating set, g ∈ G. Let [1, g]G be the interval between 1 and g in the (right) Cayley graph of G.

Definition (Interval group)

Let Gg be the group generated by R, with the relations visible in [1, g]G.

Example

If G = W (a finite Coxeter group), R = S, and g = δ (the longest element), then Gg is the spherical Artin group GW. ba δ ab a 1 b W = a, b | a2 = b2 = 1, aba = bab δ = aba = bab

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Interval groups

G group, R generating set, g ∈ G. Let [1, g]G be the interval between 1 and g in the (right) Cayley graph of G.

Definition (Interval group)

Let Gg be the group generated by R, with the relations visible in [1, g]G.

Example

If G = W (a finite Coxeter group), R = S, and g = δ (the longest element), then Gg is the spherical Artin group GW. ba δ ab a 1 b b a b a b a W = a, b | a2 = b2 = 1, aba = bab δ = aba = bab Wδ = a, b | aba = bab

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Garside groups

Theorem (Garside 1969, Dehornoy 2002, Bessis 2003)

If [1, g]G is a balancedlattice, then Gg is a Garsidegroup, and: ◮ the elements of Gg have a normal form gmx1 · · · xk, with xi ∈ [1, g]G; ◮ the complex KG = ∆([1, g]G)/labeling is a classifying space for Gg. ba δ ab a 1 b b a b a b a

The balanced lattice [1, δ]W a b a δ ab ba b a b ba ab The associated classifying space KW

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Spherical Artin groups as Garside groups

There are two natural ways to realize spherical Artin groups as Garside groups, from finite Coxeter groups.

Standard Garside structure

R = S (simple system) g = δ (longest element)

Dual Garside structure

R = {all reflections} g = w (Coxeter element)

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Spherical Artin groups as Garside groups

There are two natural ways to realize spherical Artin groups as Garside groups, from finite Coxeter groups. We show the S3 example: W = a, b | a2 = b2 = 1, aba = bab.

Standard Garside structure

R = S = {a, b} (simple system) g = δ = aba (longest element) Wδ = a, b | aba = bab = GW ba δ ab a 1 b b a b a b a (weak Bruhat order)

Dual Garside structure

R = {all reflections} = {a, b, c} g = w = ab (Coxeter element) Ww = a, b, c | ab = bc = ca ∼ = GW 1 a b c w a b c b c a (noncrossing partition lattice)

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The lattice property

Theorem (Bessis 2003, Brady-Watt 2008)

If W is a finite Coxeter group, the associated noncrossing partition poset is a lattice.

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The lattice property

Theorem (Bessis 2003, Brady-Watt 2008)

If W is a finite Coxeter group, the associated noncrossing partition poset is a lattice.

Theorem (McCammond 2015)

If W is an affine Coxeter group, the associated noncrossing partition poset is not a lattice in general.

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Proof of the K(π, 1) conjecture for affine Artin groups

KW is a classifying space

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Proof of the K(π, 1) conjecture for affine Artin groups

KW is a classifying space A new CW model XW ≃ YW with XW ⊆ KW

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Proof of the K(π, 1) conjecture for affine Artin groups

KW is a classifying space A new CW model XW ≃ YW with XW ⊆ KW Deformation retraction KW ց XW

(via discrete Morse theory)

Factorizations of affine Coxeter elements Shellability of [1, w]W

Topology Geometry Combinatorics

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The road goes ever on

◮ Arbitrary Coxeter groups (or other families, e.g. hyperbolic)

◮ Is KW a classifying space? ◮ What can we say about the factorizations of Coxeter elements? ◮ Is [1, w]W shellable? ◮ Does KW deformation retract onto XW?

◮ Affine complex reflection groups ◮ Simplicial arrangements of affine (real) hyperplanes

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Thanks!

paolini@caltech.edu

• G. Paolini and M. Salvetti, ProofoftheK(π, 1)conjectureforaffineArtingroups,

arXiv preprint 1907.11795 (2019)

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