2013/09/14 by Rebecca R. Winarski, Winarski, Rebecca R.
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.GT
paper · pdf · doi:10.48550/arxiv.1309.3650
arxiv created 2013/09/14 · openalex publication_date 2013/09/14 · arxiv updated 2013/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We say that a cover of surfaces S -> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a cover has this property. We give new explicit examples of irregular branched covers that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.