Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Abdelmalek Abdesselam

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.

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

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.

Paper Structure

This paper contains 10 sections, 11 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topology
  4. Combinatorics
  5. Group theory
  6. Tauberian theory
  7. Setting up the saddle point analysis
  8. Major and minor arc estimates
  9. Completion of the proof of the main theorem
  10. Outlook

Key Result

Theorem 1.1

Let $G$ be an infinite $\mathscr{T}$-group, then $\alpha_G$ is a rational number, and there exists $\delta>0$ such that $\zeta_G(s)$ admits a meromorphic continuation to the domain ${\rm Re}(s)>\alpha_G-\delta$. Moreover, $\zeta_G(s)$ has a pole at $s=\alpha_G$, and no other singularity on the line

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.2
  • Lemma 2.3
  • ...and 1 more