 
Summary: Research Statement Robert Sulway Autumn 2009
My primary research area is geometric group theory, with a secondary interest
in lowdimensional Topology. And, as a geometric group theorist, I attempt to under
stand groups by understanding the geometry of the spaces on which the groups act. In
particular, I am seeking to understand the class of groups known as affinetype Artin
groups, by constructing the groups according to a procedure that also provides a space
on which the groups act. This relatively new procedure is called the `pullingapart'
process, and it constructs braided versions of previously known groups. For example,
applying the process to a symmetric group yields the corresponding braid group. In
my dissertation I apply this process to alternating groups, crystallographic groups
(particularly the 17 wallpaper groups) and 3dimensional affine Coxeter groups, com
piling detailed structural information about the resulting groups and spaces. In Sec
tion 1 I describe the process of pulling groups apart and in Section 2 I outline my
own contributions.
1 Coxeter Groups and Artin Groups
A Coxeter group is any group with presentation of the form
r1, . . . , rn  (rirj)mij
= 1 ,
where mij N {}, mii = 1 and mij = mji for all i and j and when mij = this
indicates that the presentation lacks a relation involving ri and rj. Instead of using
