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?

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By a modification of the Meshulam-Wallach argument, one can apparently check that for any fixed finite group $H$, there will be no nontrivial maps G(n,p) –> H, once p >> log n /n.

The question might still be interesting, but you need to make your finite group target grow with n. The problem is exactly what you suggest — no one has any reasonable uniformity over ell. So in particular there are still no bounds on the threshold for H_1(Y,Z), other than what follows from MW on one side (n^{-1}) and BHK on the other(n^{-1/2}). Just improving the exponent would be an important step, and we seem to need new tools…

See also Dunfield & Thurston’s recent article where they make random 3-manifolds by gluing handlebodies with a map chosen by a random walk on he mapping class group. I think these things are rational homology spheres but pi_1 is an interesting story, and they consider maps to other finite simple groups — in particular I know they look at the case of A_n.

Yeah, I have thought a lot about that paper of Dunfield and Thurston and that was partially what prompted the question! Their “random 3-manifold” and your “random 2-complex on a complete graph” both have fundamental groups which are in some sense models for a random finitely generated group. One difference is that models like theirs don’t really have any analogue of the parameter p.

Yes, the Dunfield-Thurston manifolds have trivial rational first homology with probability converging (exponentially fast) to 1; on the other hand, the size of the torsion is, with high probability, almost exponentially large, and is divisible by “many” primes. (The asymptotic parameter being the length of the random walk).

(These are fairly elementary applications of sieve methods to the action of the random walk on homology, quantifying various parts of the Dunfield-Thurston analysis).