The braid group of ℤ n

Abstract

We define pseudo-Garside groups and prove a theorem 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. A group B = B(ℤn) called the braid group of ℤn and which resembles mapping class groups is introduced. It is to GL(n,ℤ) what the braid group is to the symmetric group Sn. We prove that B is a pseudo-Garside group. We give a small presentation for B(ℤn) assuming one for B(ℤ3) is given.