Compositions of Belyi Maps and their Extended Monodromy Groups

Given a composition of Bely\u{\i} 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 $\mathop{$\mathrm{M}\mathrm{o}\mathrm{n}$} \beta \circ \gamma \unlhd \mathop{$\mathrm{M}\mathrm{o}\mathrm{n}$} \gamma \wr \mathop{$\mathrm{M}\mathrm{o}\mathrm{n}$} \beta$\mathrm{.}AbstractingawaythespecificBely\breve{\text{i}}map$\gamma$andfindingtheimageof$\pi_{1}(Z)$in$\pi_{1}(Y) \wr \mathop{$\mathrm{M}\mathrm{o}\mathrm{n}$} \beta$insteadallowssubsequentlydetermining$\mathop{$\mathrm{M}\mathrm{o}\mathrm{n}$} \beta \circ \gamma$,forany$\gamma$, using only the monodromy representation of $\gamma$.