Non-positively curved aspects of Artin groups of finite type

Abstract

Artin groups of finite type are not as well understood as braid groups. This is due to the additional geometric properties of braid groups coming from their close connection to mapping class groups. For each Artin group of finite type, we construct a space (simplicial complex) analogous to Teichmueller space that satisfies a weak nonpositive curvature condition and also a space "at infinity" analogous to the space of projective measured laminations. Using these constructs, we deduce several group-theoretic properties of Artin groups of finite type that are well-known in the case of braid groups.

34 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper12.abs.html

Cited by in corpus (27)

• Garside categories, periodic loops and cyclic sets
• Large-type Artin groups are systolic
• Garside groups are strongly translation discrete
• Lattices in finite real reflection groups
• Braid groups of imprimitive complex reflection groups
• Translation numbers in a Garside group are rational with uniformly bounded denominators
• Configuration Lie groupoids and orbifold braid groups
• Some power of an element in a Garside group is conjugate to a periodically geodesic element
• Determining the action dimension of an Artin group by using its complex of abelian subgroups
• Abelian subgroups of Garside groups
• Two-dimensional Artin groups with CAT(0) dimension three
• Periodic elements in Garside groups
• Topology of complex reflection arrangements
• Finite transducers for divisibility monoids
• Asymptotic combinatorics of Artin-Tits monoids and of some other monoids
• Uniform measures on braid monoids and dual braid monoids
• Poly-freeness of Artin groups and the Farrell-Jones Conjecture
• Conjugacy classes of periodic braids
• Morse elements in Garside groups are strongly contracting
• Dual structures on Coxeter and Artin groups of rank three
• On homology torsion growth
• Notes on periodic elements of Garside groups
• Some virtually poly-free Artin groups
• Generalization of the Dehornoy-Lafont order complex to categories. Application to exceptional braid groups
• The Farrell-Jones Conjecture for normally poly-free groups
• The affine Artin group of type ${\tilde{B}}_n$ is virtually poly-free
• A representation of generalized braid group in classical braid group