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On the fundamental group schemes of certain quotient varieties

Indranil Biswas, Phùng Hô Hai, João Pedro dos Santos

Abstract

In \cite{armstrong}, M. Armstrong proved a beautiful result describing fundamental groups of quotient spaces. In this paper we prove an analogue of Armstrong's theorem in the setting of $F$-divided \cite{dS07} and essentially finite \cite{Nori76} fundamental group schemes.

On the fundamental group schemes of certain quotient varieties

Abstract

In \cite{armstrong}, M. Armstrong proved a beautiful result describing fundamental groups of quotient spaces. In this paper we prove an analogue of Armstrong's theorem in the setting of -divided \cite{dS07} and essentially finite \cite{Nori76} fundamental group schemes.

Paper Structure

This paper contains 9 sections, 29 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Tannakian duality
  3. Preliminaries on genuinely ramified finite morphisms
  4. Preliminaries on $F$-divided sheaves
  5. The $F$-divided fundamental group scheme of a quotient: the case of a free action
  6. The $F$-divided group scheme of a quotient
  7. Extending the Tannakian interpretation of the essentially finite fundamental group
  8. The essentially finite fundamental group scheme of a quotient: the case of a free action
  9. The essentially finite fundamental group of a quotient

Key Result

Theorem 1.1

Let $X$ be a path connected, simply connected, locally compact metric space. Given a group $G$ acting discontinuously on $X$, the fundamental group of the quotient space $G\backslash X$ is isomorphic with the quotient $G/I$, where $I< G$ is the (necessarily normal) subgroup generated by all elements

Theorems & Definitions (66)

  • Theorem 1.1: armstrong
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 3.1: Balaji-Parameswaran
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 56 more