Residually finite rationally $p$ groups

Abstract

In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in $3$-manifold topology enjoy this residual property. We then prove a combination theorem for RFR$p$ groups, which we use to study the boundary manifolds of algebraic curves $\mathbb{CP}^2$ and in $\mathbb{C}^2$. We show that boundary manifolds of a large class of curves in $\mathbb{C}^2$ (which includes all line arrangements) have RFR$p$ fundamental groups, whereas boundary manifolds of curves in $\mathbb{CP}^2$ may fail to do so.