Topics in Self-Similar Groups

The self-similarity of these groups, fittingly, appears in a number

of different ways, corresponding to the different constructions that give rise to

them. We will explore three such constructions, each highlighting a particular way

in which self-similarity manifests in these groups; iterative calculations, branch

constructions, and holomorphic dynamics. The study of these groups globally is

quite delicate, and there are no apparent general techniques to deal with all self-

similar groups. However, this hints that they can be applied to a huge number of

problems occurring across mathematics, whenever self-similarity enters the picture.