A central limit theorem for connected components of random coverings of manifolds with nilpotent fundamental groups

TL;DR

This paper establishes a central limit theorem for the number of connected components in random coverings of manifolds with nilpotent fundamental groups, extending previous abelian cases to nonabelian groups.

Contribution

It generalizes earlier results by proving a CLT for connected components in random coverings with nilpotent fundamental groups, using subgroup growth zeta functions and Tauberian theorems.

Findings

Proves a CLT for connected components in nilpotent fundamental group coverings.

Extends abelian case results to nonabelian nilpotent groups.

Utilizes subgroup growth zeta functions and Tauberian theorems in proof.

Abstract

There is a well understood way of generating random coverings of a fixed manifold by sampling homomorphisms from the fundamental group of this manifold into the symmetric group. We prove a central limit theorem for the number of connected components of these random coverings when the fundamental group is nilpotent. This provides a nonabelian generalization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least two. Our result relies on the work of du Sautoy and Grunewald on the subgroup growth zeta functions of nilpotent groups, and on Delange's generalization of the Wiener-Ikehara Tauberian theorem.