15 research outputs found
Image of the Burau Representation at -th Roots of unity
We prove that the image of the Full braid group on strands
under the Burau representation, evaluated at a primitive -th root of unity
is arithmetic provided .Comment: To appear in Annals of Mathematics. arXiv admin note: text overlap
with arXiv:1204.477
Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue
This article surveys many standard results about the braid group with
emphasis on simplifying the usual algebraic proofs.
We use van der Waerden's trick to illuminate the Artin-Magnus proof of the
classic presentation of the algebraic mapping-class group of a punctured disc.
We give a simple, new proof of the Dehornoy-Larue braid-group trichotomy,
and, hence, recover the Dehornoy right-ordering of the braid group.
We then turn to the Birman-Hilden theorem concerning braid-group actions on
free products of cyclic groups, and the consequences derived by Perron-Vannier,
and the connections with the Wada representations. We recall the very simple
Crisp-Paris proof of the Birman-Hilden theorem that uses the Larue-Shpilrain
technique. Studying ends of free groups permits a deeper understanding of the
braid group; this gives us a generalization of the Birman-Hilden theorem.
Studying Jordan curves in the punctured disc permits a still deeper
understanding of the braid group; this gave Larue, in his PhD thesis,
correspondingly deeper results, and, in an appendix, we recall the essence of
Larue's thesis, giving simpler combinatorial proofs.Comment: 51`pages, 13 figure
Graphs and Reflection Groups
It is shown that graphs that generalize the ADE Dynkin diagrams and have
appeared in various contexts of two-dimensional field theory may be regarded in
a natural way as encoding the geometry of a root system. After recalling what
are the conditions satisfied by these graphs, we define a bilinear form on a
root system in terms of the adjacency matrices of these graphs and undertake
the study of the group generated by the reflections in the hyperplanes
orthogonal to these roots. Some ``non integrally laced " graphs are shown to be
associated with subgroups of these reflection groups. The empirical relevance
of these graphs in the classification of conformal field theories or in the
construction of integrable lattice models is recalled, and the connections with
recent developments in the context of supersymmetric theories and
topological field theories are discussed.Comment: 42 pages TEX file, harvmac and epsf macros, AMS fonts optional,
uuencoded, 8 figures include