Residually finite rationally $p$ groups

TL;DR

This paper develops the theory of residually finite rationally p (RFRp) groups, establishing their structural properties, identifying natural examples, and applying the theory to boundary manifolds in algebraic geometry and topology.

Contribution

It introduces the RFRp group concept, proves structural theorems, and applies the theory to boundary manifolds of algebraic curves, including line arrangements.

Findings

RFRp groups are torsion-free and satisfy a Tits alternative.

Many naturally occurring groups in geometry and topology are RFRp.

Boundary manifolds of certain algebraic curves have RFRp fundamental 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.