In April, I blogged about the space of small disks in a box. One question I mentioned there was the following: if C(n,r) is the space of configurations of n non-overlapping disks of radius r in a box of sidelength 1, what kind of upper bounds on r assure that C(n,r) is connected? A recent preprint of Matthew Kahle gives some insight into this question: he produces configurations of disks which are *stable* (each disk is hemmed in by its neighbors) with r on order of 1/n. (In particular, the density of such configurations goes to 0 as n goes to infinity.) Note that Kahle’s configurations are not obviously isolated points in C(n,r); it could be, and Kahle suggests it is likely to be, that his configurations can be deformed by moving several disks at once.

Also appearing in Kahle’s paper is the stable 5-disk configuration at left; this one is in fact an isolated point in C(5,r).

More Kahle: another recent paper, with Babson and Hoffman, features the theorem that a random 2-complex on n vertices, where the edges are all present and each 2-face appears with probability p, transitions from non-simply-connected to simply connected when p crosses n^{-1/2}. This is in sharp contrast with the H_1 of the complex with Z/ ell Z coefficients, which disappears almost surely once p exceeds 2 log n / n, by a result of Meshulam and Wallach. So in some huge range, the fundamental group is almost surely a big group with no nontrivial abelian quotient! (I guess this doesn’t formally follow from Meshulam-Wallach unless you have some reasonable uniformity in ell…)

One naturally wonders: Let pi_1(n,p) be the fundamental group of a random 2-complex on n vertices with facial probability p. If G is a finite simple group, what is the expected number of surjections from pi_1(n,p) to G? Does it sharply transition from nonzero to zero? Is there a range of p in which pi_1(n,p) is almost certainly an infinite group with no finite quotients?

### Like this:

Like Loading...