# Free quotients of fundamental groups of smooth quasi-projective varieties

## TL;DR

This paper investigates the structure of certain curve classes on projective surfaces, focusing on when their complements' fundamental groups admit large free quotients, and establishes finiteness results under specific conditions.

## Contribution

It provides new insights into the relationship between curve classes on surfaces and the free quotients of their fundamental groups, including finiteness results for large ranks.

## Key findings

- Finiteness results for classes with large free quotients
- Characterization of curves with fundamental groups admitting free quotients
- Conditions under which free quotients of fundamental groups are constrained

## Abstract

We consider the structure of classes of curves on a projective simply connected surface for which fundamental groups of the complements admit free quotients having rank greater than one with irreducible components belonging to a selected subset the effective cone of the surface. In particular we show a finiteness result for such classes if the ranks of free quotients of the fundamental groups with components in the subset of effective cone are sufficiently large.