Small Cancellation for Random Branched Covers of Groups

TL;DR

This paper introduces a probabilistic model for random branched covers of groups, demonstrating that such covers typically satisfy small cancellation conditions and have hyperbolic fundamental groups with low cohomological dimension.

Contribution

It develops a new random model for branched covers of groups and proves they almost surely satisfy small cancellation properties and are Gromov hyperbolic.

Findings

Random branched covers satisfy small cancellation conditions asymptotically.

Fundamental groups of these covers are Gromov hyperbolic.

Covers have low cohomological dimension.

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 $\lambda >0$, 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 $C'(\lambda)$. As a consequence the fundamental group of a random branched cover is asymptotically almost surely Gromov hyperbolic and has small cohomological dimension.