Finding simple curves in surface covers is undecidable

TL;DR

This paper proves that determining the existence of simple closed curves in certain surface group subgroups is an undecidable problem, highlighting fundamental limits in geometric group theory.

Contribution

It establishes the undecidability of questions regarding simple closed curves in normal subgroups of surface groups, a novel result in the field.

Findings

Questions about simple closed curves in surface group subgroups are undecidable.

Undecidability results apply to normal subgroups of surface groups.

The work reveals fundamental computational limits in surface topology.

Abstract

It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.