Characterizing covers via simple closed curves

TL;DR

This paper characterizes when two finite covers of a surface are equivalent by examining which curves lift to simple closed curves, using Teichmüller theory and the complex of curves.

Contribution

It provides a new criterion for cover equivalence based on lifting properties of curves, including a characterization for abelian covers.

Findings

Two regular covers are equivalent if they lift the same simple closed curves.

For abelian covers, equivalence is characterized by which powers of simple curves lift.

The approach uses Teichmüller theory and the complex of curves to establish these results.

Abstract

Given two finite covers $p: X \to S$ and $q: Y \to S$ of a connected, oriented, closed surface $S$ of genus at least $2$, we attempt to characterize the equivalence of $p$ and $q$ in terms of which curves lift to simple curves. Using Teichm\"uller theory and the complex of curves, we show that two regular covers $p$ and $q$ are equivalent if for any closed curve $\gamma \subset S$, $\gamma$ lifts to a simple closed curve on $X$ if and only if it does to $Y$. When the covers are abelian, we also give a characterization of equivalence in terms of which powers of simple closed curves lift to closed curves.