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 R n . To every reflection group W is associated a hyperplanearrangement : the set of hyperplanes H such that the reflection with respect to H is an element of W . Arrangement of a finite Arrangement of an infinite reflection group ( affine ) reflection group 2 / 21
The symmetric group S n S n is the group generated by the reflections w.r.t. the hyperplanes { x i = x j } in R n : these correspond to the transpositions ( i j ) . Example: S 3 x 1 = x 2 x 2 = x 3 x 1 = x 3 The arrangement of S 3 in { x 1 + x 2 + x 3 = 0 } ⊆ R 3 3 / 21
Coxeter groups A Coxetergroup is a group presented as follows: W = � S | s 2 = 1 ∀ s ∈ S , sts · · · = tst · · · ∀ s � = t � . � �� � � �� � m s , t factors m s , t factors Reflection groups are particular instances of Coxeter groups: the set S is given by the reflections with respect to the walls of a fundamentalchamber ; π the angle between two walls is m s , t . Example: S 3 x 1 = x 2 x 2 = x 3 The reflections a and b yield the following Coxeter presentation: x 1 = x 3 S 3 = � a , b | a 2 = b 2 = 1 , aba = bab � π/ 3 a b 4 / 21
Artin groups Removing the relations s 2 = 1 in the presentation of a Coxeter group, we obtain the corresponding Artingroup : G W = � S | sts · · · = tst · · · ∀ s � = t � . � �� � � �� � m s , t factors m s , t factors Example: the braid group on 3 strands If W = S 3 is the symmetric group on 3 letters, the corresponding Artin group is B 3 = � a , b | aba = bab � . Other examples ◮ Free groups (all m s , t = ∞ ) ◮ Free abelian groups (all m s , t = 2) ◮ Right-angled Artin groups (all m s , t ∈ { 2 , ∞} ) 5 / 21
Artin groups (2) Topologically, an Artin group G W is the fundamental group of the configurationspace � C n \ / W . Y W = H C H ∈A W Example (continued): the braid group on 3 strands Y W = { ( x 1 , x 2 , x 3 ) ∈ C 3 | x i � = x j } / S 3 C x 1 = x 2 x 2 = x 3 t = 0 x 1 = x 3 C t = 1 The (real) arrangement Loops in Y W are “braids” 6 / 21
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 Y W is a classifying space for G W . π 1 ( Y W ) = G W , and the higher homotopy groups are trivial (equivalently, the univer- sal cover of Y W is contractible). 7 / 21
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 Y W is a classifying space for G W . π 1 ( Y W ) = G W , and the higher homotopy groups are trivial (equivalently, the univer- sal cover of Y W 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). 8 / 21
Theorem (P .-Salvetti 2019) The K ( π, 1 ) conjecture holds for all affine Artin groups. 9 / 21
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 G g 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 G g is the spherical Artin group G W . δ W = � a , b | a 2 = b 2 = 1 , aba = bab � ab ba δ = aba = bab a b 1 10 / 21
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 G g 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 G g is the spherical Artin group G W . δ a b W = � a , b | a 2 = b 2 = 1 , aba = bab � ab ba δ = aba = bab a b a b W δ = � a , b | aba = bab � a b 1 11 / 21
Garside groups Theorem (Garside 1969, Dehornoy 2002, Bessis 2003) If [ 1 , g ] G is a balancedlattice , then G g is a Garsidegroup , and: ◮ the elements of G g have a normal form g m x 1 · · · x k , with x i ∈ [ 1 , g ] G ; ◮ the complex K G = ∆([ 1 , g ] G ) / labeling is a classifying space for G g . a b δ a b ab ba ba ab a a b b δ ab ba a b a b 1 a b The balanced lattice [ 1 , δ ] W The associated classifying space K W 12 / 21
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 Dual Garside structure R = S R = { all reflections } (simple system) g = δ (longest element) g = w (Coxeter element) 13 / 21
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 S 3 example: W = � a , b | a 2 = b 2 = 1 , aba = bab � . Standard Garside structure Dual Garside structure R = S = { a , b } (simple system) R = { all reflections } = { a , b , c } g = δ = aba (longest element) g = w = ab (Coxeter element) W w = � a , b , c | ab = bc = ca � ∼ W δ = � a , b | aba = bab � = G W = G W δ w a b a b ab ba c a b a c b a b b a c a b 1 1 (weak Bruhat order) (noncrossing partition lattice) 14 / 21
The lattice property Theorem (Bessis 2003, Brady-Watt 2008) If W is a finite Coxeter group, the associated noncrossing partition poset is a lattice. 15 / 21
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. 16 / 21
Proof of the K ( π, 1 ) conjecture for affine Artin groups K W is a classifying space 17 / 21
Proof of the K ( π, 1 ) conjecture for affine Artin groups K W is a classifying space A new CW model X W ≃ Y W with X W ⊆ K W 18 / 21
Proof of the K ( π, 1 ) conjecture for affine Artin groups K W is a classifying space Factorizations of affine Coxeter elements Geometry Deformation retraction K W ց X W Topology (via discrete Morse theory) Combinatorics A new CW model X W ≃ Y W Shellability of [ 1 , w ] W with X W ⊆ K W 19 / 21
The road goes ever on ◮ Arbitrary Coxeter groups (or other families, e.g. hyperbolic) ◮ Is K W a classifying space? ◮ What can we say about the factorizations of Coxeter elements? ◮ Is [ 1 , w ] W shellable? ◮ Does K W deformation retract onto X W ? ◮ Affine complex reflection groups ◮ Simplicial arrangements of affine (real) hyperplanes 20 / 21
Thanks! paolini@caltech.edu G. Paolini and M. Salvetti, ProofoftheK ( π, 1 ) conjectureforaffineArtingroups , arXiv preprint 1907.11795 (2019) 21 / 21
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