Small Cancellation for Random Branched Covers of Groups

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 $\hat{I}»>0$, 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 $C^{\prime }(\hat{I}»)$. As a consequence the fundamental group of a random branched cover is asymptotically almost surely Gromov hyperbolic and has small cohomological dimension.

28 pages, 5 figures