Compositions of Belyi Maps and their Extended Monodromy Groups

Abstract

Given a composition of Bely\uı maps $\beta \circ \gamma :X\to Z$, 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 $\mathrm{Mon}\beta \circ \gamma ⊴\mathrm{Mon}\gamma \wr \mathrm{Mon}\beta$. Abstracting away the specific Bely\uı map $\gamma$ and finding the image of ${\pi }_1(Z)$ in ${\pi }_1(Y)\wr \mathrm{Mon}\beta$ instead allows subsequently determining $\mathrm{Mon}\beta \circ \gamma$, for any $\gamma$, using only the monodromy representation of $\gamma$.

17 pages, 8 figures, 1 appendix; v2 adjusted scaling of figures