Mathematics Institute

University of Warwick

Coventry CV4 7AL

United Kingdom

telephone: +44-24-7652 3488

fax: +44-24-7652 4182

e-mail: D.Krammer@warwick.ac.uk

office: B1.17

Teaching

Some of my preprints and publications:

[pdf] An asymmetric generalisation of Artin monoids, version 1, 23 November 2012, 24 pages.

Abstract: We propose a slight weakening of the definitions of Artin monoids and Coxeter monoids. We study one `infinite series' in detail.

[pdf] Generalisations of the Tits representation, version 3, 29 August 2008, 28 pages, 8 figures.

Abstract: We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes, simplicial chambers and a Tits cone. The generators of K_n are given by 2-element subsets of {0,...,n}. We provide some generalities to deal with groups like these. We give some easy combinatorial results on the finite residues of K_n, which are equivalent to certain simplicial real central hyperplane arrangements.

[pdf] The braid group of Z^n, 31 January 2007, 21 pages.

Abstract: 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.

[pdf] A class of Garside groupoid structures on the pure braid group. Final version (27 March 2006), 32 pages, to appear in Transactions AMS.

Abstract: 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.

[pdf] Horizontal configurations of points in link complements, in Proc. European Congress Math. 2004, EMS (2005), 233--245.

Abstract: For any tangle T (up to isotopy) and integer k>0 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).

[pdf] Braid groups are linear, Annals of Math. 155, No. 1, 131--156 (2002).

Abstract: In a previous work [11], the author considered a representation of the braid group rho: B_n\to GL_m(Z[q,1/q,t,1/t]) (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.

[dvi] The conjugacy problem for Coxeter groups (my thesis, 1994).

[pdf] Final version, 18 June 2008. To appear in Geometry, Groups and Dynamics.