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

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 \mathrm{SL}_{N}(\mathbb{R}) distinguishes between pairs of non-isomorphic covers.