A note on an effective characterization of covers with an application to higher rank representations

TL;DR

This paper provides an effective method to determine when two surface covers are isomorphic based on simple curve elevations, and applies this to distinguish non-isomorphic covers via traces in high-rank representations.

Contribution

It introduces a new characterization of surface covers using bounded self-intersection curves and applies it to representation theory for distinguishing covers.

Findings

Characterization of surface covers using bounded self-intersection curves

Distinguishing non-isomorphic covers via traces in SL(N, R) representations for large N

Effective criteria for surface cover isomorphism

Abstract

In this note we prove an effective characterization of when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations, weakening the hypotheses to consider curves with explicitly bounded self-intersection number. As an application we show that for sufficiently large N, the set of unmarked traces associated to simple closed curves in a generically chosen representation to SL(N, R) distinguishes between pairs of non-isomorphic covers.