Liftable mapping class groups of regular cyclic covers
Nikita Agarwal, Soumya Dey, Neeraj K. Dhanwani, Kashyap, Rajeevsarathy

TL;DR
This paper studies liftable mapping class groups of cyclic covers of surfaces, characterizing their structure, criteria for liftability, and providing generators, with implications for understanding surface symmetries and congruence subgroups.
Contribution
It introduces a symplectic criterion for liftability, describes the structure of liftable groups as stabilizers, and generalizes known series of congruence subgroups.
Findings
Liftable groups are stabilizers of vectors in homology.
A symplectic criterion determines liftability of mapping classes.
A finite generating set for liftable groups is provided.
Abstract
Let be the mapping class group of the closed orientable surface of genus . For , we consider the standard -sheeted regular cover , and analyze the liftable mapping class group associated with the cover . In particular, we show that is the stabilizer subgroup of with respect to a collection of vectors in , and also derive a symplectic criterion for the liftability of a given mapping class under . As an application of this criterion, we obtain a normal series of , which generalizes a well known normal series of congruence subgroups in . Among other applications, we describe a procedure for obtaining a finite generating set for and…
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Taxonomy
TopicsGeometric and Algebraic Topology · Algebraic Geometry and Number Theory · Mathematical Dynamics and Fractals
