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### The fundamental group of random 2-dimensional simplicial complexes

**Geometry/Topology**

Speaker: | Matt Kahle, University of Washington |

Location: | 2112 MSB |

Start time: | Wed, Oct 18 2006, 4:10PM |

In 1959, Erdos and Renyi found a sharp threshold function for the connectivity of the random graph. In other words, they gave a sharp estimate for the number of edges one needs to randomly add to n isolated points before the graph becomes connected. More recently, Nati Linial and Roy Meshulam viewed this as a topological statement and gave a homological analogue of the Erdos-Renyi theorem for random 2-dimensional simplicial complexes.

In this talk, we'll prove some facts about the expected properties of Linial-Meshulam complexes, and show in particular that the fundamental group must have a different threshold for vanishing than the first homology group. The main technique is a local-to-global theorem for isoperimetric inequalities due to Gromov, and we will especially emphasize connections to random groups and hyperbolicity. This is joint work with Chris Hoffman.