Real algebraic links in $S^3$ and braid group actions on the set of $n$-adic integers

Abstract

We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct non-zero points in the complex plane. This results in an action of the braid group ${\mathbb{B}}_n$ on the set of $n$-adic integers ${\mathbb{Z}}_n$ for all natural numbers $n\ge 2$. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid $B$ an infinite sequence of braids, whose braid types are invariants of $B$. We present computations for the cases of $n=2$ and $n=3$ and use these to show that an infinite family of braids close to real algebraic links, i.e., links of isolated singularities of real polynomials ${\mathbb{R}}^4\to {\mathbb{R}}^2$.

32 pages, 5 figures, Changes from v1: Definition of the action simplified. Calculations in Section 5 redone with different base points. The proof of Prop 6.1 simplified and Section 8 extended. Changed the title from v1 (Braid group actions on the n-adic integers) to reflect the stronger emphasis on real alg. links in this version and to clarify that this is not a homomorphism from B_n to Aut(Z_n)