Universal braids for elliptic fibrations: from character varieties to Coxeter's factor groups

Abstract

Let $π: M \to B$ be an elliptic fibration over $B = D^2$ or $B = S^2$ with $n$ nodal fibers over $Δ\subseteq B$. We study the universal liftable braids for $π$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B,Δ)$. When $B = S^2$, we show that nontrivial universal braids do not exist by proving a Zariski-density theorem on the $\mathrm{SL}_2$-character variety for $(S^2,Δ)$. When $B = D^2$ we classify when the subgroup of universal braids has finite index in the braid group $B_n = \mathrm{Mod}(D^2,Δ)$, and relate these examples to Coxeter's factor groups of braid groups, which in turn are related to the platonic solids. Finally, we generalize the results derived in the finite-index cases by considering a canonical family of branched covers of the base $B$ associated to any elliptic fibration. The generalization naturally connects the universal braids to the integral Burau representation reduced modulo 3.

Disclosure

“https://github.com/FayeAlephNil/solomon (††) or, upon reasonable request, from the author. Gemini Pro was used to improve the figure in Figure 1 from a hand-drawn picture into professional TikZ code. Gemini Pro was also used to perform literature searches for references before being hand-checked by the author. Claude Opus was used in the editing process to detect typos and check proofs. Acknowledgements. The author would like to thank Benson”

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2026-08-22T18:18:40.548888Z

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