arXiv subjects
Publications and source records attributed to Daan Krammer.
This text consists of additions to the book "Foundations of Garside Theory", EMS Tracts in Mathematics, vol. 22 (2015) -- see introduction and table of contents in arXiv:1309.0796 -- namely skipped proofs and solutions to selected exercises.
This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication. This manuscript develops an extension of Garside's approach to braid groups and provides a unified treatment for the various algebraic structures that appear in this context. The complete text can be found at http://www.math.unicaen.fr/~garside/Garside.pdf.
We propose a slight weakening of the definitions of Artin monoids and Coxeter monoids. We study one `infinite series' in detail.
We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes and simplicial chambers. The generators of K_n are given by 2-element subsets of {0, .., n}. We give some easy combinatorial results on the finite residues of K_n.
We define pseudo-Garside groups and prove a theorem about them parallel to Garside's result on the word problem for the usual braid groups. The main novelty is that the set of simple elements can be infinite. We introduce a group B=B(Z^n) which we call the braid group of Z^n, and which bears some vague resemblance to mapping class groups. It is to GL(n,Z) what the braid group is to the symmetric group S_n. We prove that B is a pseudo-Garside group. We give a small presentation for B(Z^n) assuming one for B(Z^3) is given.
We construct a class of Garside groupoid structures on the pure braid groups, one for each function (called labelling) from the punctures to the integers greater than 1. The object set of the groupoid is the set of ball decompositions of the punctured disk; the labels are the perimeters of the regions. Our construction generalises Garside's original Garside structure, but not the one by Birman-Ko-Lee. As a consequence, we generalise the Tamari lattice ordering on the set of vertices of the associahedron.
For any tangle $T$ (up to isotopy) and integer $k\geq 1$ we construct a group $F(T)$ (up to isomorphism). It is the fundamental group of the configuration space of $k$ points in a horizontal plane avoiding the tangle, provided the tangle is in what we call Heegaard position. This is analogous to the first half of Lawrence's homology construction of braid group representations. We briefly discuss the second half: homology groups of $F(T)$.
In a previous work [11], the author considered a representation of the braid group ρ: B_n\to GL_m(\Bbb Z[q^{\pm 1},t^{\pm 1}]) (m=n(n-1)/2), and proved it to be faithful for n=4. Bigelow [3] then proved the same representation to be faithful for all n by a beautiful topological argument. The present paper gives a different proof of the faithfulness for all n. We establish a relation between the Charney length in the braid group and exponents of t. A certain B_n-invariant subset of the module is constructed whose properties resemble those of convex cones. We relate line segments in this set with the Thurston normal form of a braid.