Profinite genus of fundamental groups of compact flat manifolds with holonomy group of prime order

TL;DR

This paper computes the profinite genus of fundamental groups of certain flat manifolds with prime order holonomy, showing that for dimensions up to 21, the manifold is uniquely determined by this algebraic invariant.

Contribution

It provides explicit calculations of the profinite genus for these manifolds and characterizes the isomorphism class of their fundamental groups' profinite completions.

Findings

Manifolds of dimension ≤ 21 are distinguished by their fundamental groups' profinite completions.

The isomorphism class of the profinite completion is characterized by the representation genus of the holonomy group.

Explicit formulas for the profinite genus in the prime order holonomy case.

Abstract

In this article we calculate the profinite genus of the fundamental group of a $n$-dimensional compact flat manifold $X$ with holonomy group of prime order. As consequence, we prove that if $n\leq 21$, then $X$ is determined among all $n$-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of profinite completion of the fundamental group of $X$ in terms of the representation genus of its holonomy group.