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Abstract
This dissertation comprises three papers. In Chapter 1 we will introduce the basic setup, definitions, and notation common to these papers, including the braid group $B_n$ on $n$ strands, and the configuration spaces $\mathrm{UConf}_n\mathbb{C}$ of $n$ unordered points in the complex plane. In Chapter 2 we consider the two related problems of classifying the homomorphisms $B_n\to B_m$ between braid groups, and classifying the holomorphic maps $\mathrm{UConf}_n\mathbb{C}\to\mathrm{UConf}_m\mathbb{C}$. As a consequence of our work, we complete the classification of holomorphic maps $\mathrm{UConf}_n\mathbb{C}\to\mathrm{UConf}_m\mathbb{C}$ for $2\leq m\leq n$. In particular, the map $R\colon\mathrm{UConf}_4\mathbb{C}\to\mathrm{UConf}_3\mathbb{C}$ that appears in Ferrari's solution to the quartic equation is characterized as the the unique holomorphic map, up to an appropriate equivalence relation, whose image is not contained in a single orbit of the affine group. The contents of this chapter are joint work with Jeroen Schillewaert, first published in Mathematische Annalen 391.3 (2025), pp. 4409--4440. In Chapter 3 we consider certain subgroups $B_n[m]$ of $B_n$, called level $m$ congruence subgroups, associated to the integral Burau representation $\rho_n\colon B_n\to\mathrm{GL}_n(\mathbb{Z})$. For $n\geq5$ we prove that $B_n[m]$ is generated by $\ker(\rho_n)$ and the normal closure of a single $m$th power of a half twist. As a consequence, we find that $B_n[m]$ is normally generated in $B_n$ by just three elements for $n\geq5$, each of which admits a simple topological description. The results of this chapter are joint work with Ishan Banerjee. In Chapter 4 we work with a stratification of the configuration space $\mathrm{UConf}_n\mathbb{C}$, called the equicritical stratification, indexed by partitions $\kappa$ of $n-1$. The fundamental group of a stratum $\mathrm{Poly}_n(\mathbb{C})[\kappa]$ is called a stratified braid group $\mathscr{B}_n[\kappa]$. If $r=|\kappa|$ is the number of parts in the partition, then there is a monodromy homomorphism $\mathscr{B}_n[\kappa]\to B_{n+r}$ which arises from a natural map $\mathrm{Poly}_n(\mathbb{C})[\kappa]\to\mathrm{UConf}_{n+r}(\mathbb{C})$. We prove that this monodromy homomorphism is not injective when $r=2$. The results of this chapter are joint work with Nick Salter.