## Homotopy of random simplicial complexes

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A random graph on $n$ vertices is defined by selectiung the edges according to some probability distribution, the simplest case being the one where the edge between any two vertices exists with probability $p = \frac{1}{2}$. I believe this is the Erdős–Rényi model $G(n,p)$ for generating random graphs.

Similarly, in higher dimensions we can construct random simplicial complexes on $n$ vertices in many ways. One such method is as follows: fix a top dimension $d$, and now define the random simplicial model $S_d(n,p)$ where each $d$ simplex spanning any $d+1$ vertices exists with probability $p$. Some work has been done investigating the homology of such complexes in limiting cases, see for example this paper.

What is known about the properties of the fundamental group (or higher homotopy groups) of random simplicial complexes?

If there is a good reference, that would be enough. I can not find one on google. Thank you for your time.

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@jc: I would suggest you make your comment an answer, since that paper is the state of the art in the field. – Igor Rivin Jul 8 at 4:42

## 1 Answer

Babson, Hoffman, and Kahle have written a paper on fundamental groups of random 2-complexes. They worked with the Linial-Meshulam model whereby you begin with a complete graph on $n$ vertices and then add independently uniformly random 2-simplices.

Babson has just written a paper on the fundamental groups of clique complexes of Erdős–Rényi random graphs using similar techniques.

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Thank you for the answer. – Pinying Jul 8 at 16:40

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If I recall Babson's results, depending on your biases in generation of 2-complexes you get a landscape of results of the form: generically the fundamental group tends to be either trivial, hyperbolic or free. Other types of groups tend to be rare. – Ryan Budney Jul 23 at 2:00
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