Compositions of Belyi Maps and their Extended Monodromy Groups
Jacob Bond

TL;DR
This paper investigates the monodromy groups of composed Belyi maps, revealing their structure via extended monodromy groups and providing a method to determine these groups using monodromy representations.
Contribution
It introduces a framework for understanding the monodromy groups of composed Belyi maps through extended monodromy groups and a new approach to compute them using monodromy representations.
Findings
Monodromy group of composition is a subgroup of a wreath product.
Method to determine monodromy of composition from individual monodromies.
Extended monodromy groups facilitate analysis of Belyi map compositions.
Abstract
Given a composition of Bely\u{\i} maps , paths between edges of are extended to form loops, then lifted by . These liftings are then studied to understand how loops in act on edges of , demonstrating the group operation in . Abstracting away the specific Bely\u{\i} map and finding the image of in instead allows subsequently determining , for any , using only the monodromy representation of .
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Taxonomy
TopicsMathematics and Applications · Advanced Topics in Algebra
