Small Cancellation for Random Branched Covers of Groups
Hyeran Cho, Jean-François Lafont, Rachel Skipper
TL;DR
The paper develops a random model for n-fold branched coverings of finite acceptable polygonal 2-complexes and proves that, for any $\lambda>0$, the random cover $X(\sigma)$ is asymptotically almost surely homotopy equivalent to a 2-complex with geometric $C'(\lambda)$-small cancellation as $n\to\infty$. The approach hinges on encoding covers via random permutations, analyzing overlaps and disk indices, and collapsing small disks to enforce small cancellation, thereby deriving geometric and group-theoretic consequences. In the $\lambda=1/6$ case, the results yield aspherical, Gromov-hyperbolic, torsion-free, cubulable fundamental groups with cohomological dimension at most 2, linking random topology with hyperbolic group theory. The work also clarifies connectedness criteria, the behavior of overlaps under branched covers, and provides a framework for extending the results to general acceptable 2-complexes and to explorations of non-uniform measures on permutation spaces.
Abstract
We construct a random model for an $n$-fold branched cover of a finite acceptable $2$-complex $X$. This includes presentation $2$-complexes for finitely presented groups satisfying some mild conditions. For any $λ>0$, we show that as $n$ goes to infinity, a random branched cover asymptotically almost surely is homotopy equivalent to a $2$-complex satisfying geometric small cancellation $C'(λ)$. As a consequence the fundamental group of a random branched cover is asymptotically almost surely Gromov hyperbolic and has small cohomological dimension.