Factorability, Discrete Morse Theory and a Reformulation of K(π, 1)-conjecture

Abstract

The first aim of this thesis is to study factorable groups and monoids. We give a new family of examples for factorability structures, provided by Garside theory, in particular, we provide a factorability structure on braid groups. Furthermore, we investigate the connection between factorability structures and rewriting systems, and give conditions under which a factorability structure yields a complete rewriting system on a monoid. Moreover, we exhibit a factorability structure on the orthogonal group O(n) and the induced factorability structure on the reflection subgroup of type B(n). <br /> Another aim of this thesis is the study of Artin groups and monoids. We exhibit several chain complexes computing the homology of an Artin monoid. Moreover, we give a new proof for Dobrinskaya's Theorem which states a reformulation of the K(π,1)-conjecture for Artin groups.