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

Abstract

Let $\pi :M\to B$ be an elliptic fibration over $B=D^2$ or $B=S^2$ with $n$ nodal fibers over $\Delta \subseteq B$. We study the universal liftable braids for $\pi$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B,\Delta )$. 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 $(S^2,\Delta )$. When $B=D^2$ we classify when the subgroup of universal braids has finite index in the braid group $B_n=\mathrm{Mod}(D^2,\Delta )$, 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.

30 pages, 6 figures