2024/12/10 by Neeraj K. Dhanwani, Pankaj Kapari, Dhanwani, Neeraj K. +5 · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Rings, Modules, and Algebras #Coding theory and cryptography
paper · pdf · doi:10.48550/arxiv.2412.07319
Let Sg be the closed oriented surface of genus g ≥ 0, and let Mod(Sg) be the mapping class group of Sg. For g≥ 2, we develop an algorithm to obtain a finite generating set for the liftable mapping class group LModp(Sg) of a regular abelian cover p of Sg. A key ingredient of our method is a result that provides a generating set of a group G acting on a connected graph X such that the quotient graph X/G is finite. As an application of our algorithm, when k is prime, we provide a finite generating set for LModpk(S2) for cyclic cover pk:Sk+1→ S2. Using the Birman-Hilden theory, when k=2,3 and g=2, we also obtain a finite generating set for the normalizer of the Deck transformation group of pk in Mod(Sk+1). We conclude the paper with an application of our algorithm that gives a finite generating set for LModp(S2), where p:S5→ S2 is a cover with deck transformation group isomorphic to ℤ2⊕ ℤ2.