# Free Proalgebraic Groups

## TL;DR

This paper introduces free proalgebraic groups, extending the concept of free profinite groups into algebraic geometry, and characterizes them via embedding problems, motivated by a differential analog of Shafarevic's conjecture.

## Contribution

It defines and characterizes free proalgebraic groups, providing an algebraic-geometric framework analogous to free profinite groups, inspired by a differential version of Shafarevic's conjecture.

## Key findings

- Introduced the concept of free proalgebraic groups.
- Characterized free proalgebraic groups through embedding problems.
- Linked the theory to a differential analog of Shafarevic's conjecture.

## Abstract

Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic.