Compositions of Belyi Maps and their Extended Monodromy Groups

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 $\beta \circ \gamma: X \rightarrow Z$, paths between edges of $\beta$ are extended to form loops, then lifted by $\gamma$. These liftings are then studied to understand how loops in $Z$ act on edges of $\beta \circ \gamma$, demonstrating the group operation in $\mathop{\mathrm{Mon}} \beta \circ \gamma \unlhd \mathop{\mathrm{Mon}} \gamma \wr \mathop{\mathrm{Mon}} \beta$. Abstracting away the specific Bely\u{\i} map $\gamma$ and finding the image of $\pi_{1}(Z)$ in $\pi_{1}(Y) \wr \mathop{\mathrm{Mon}} \beta$ instead allows subsequently determining $\mathop{\mathrm{Mon}} \beta \circ \gamma$, for any $\gamma$, using only the monodromy representation of $\gamma$.