The fundamental group of random 2-complexes

Geometry & Topology
Event time: 
Monday, February 21, 2011 - 11:30am to Sunday, February 20, 2011 - 7:00pm
431 DL
Matthew Kahle
Speaker affiliation: 
Event description: 

Erdős and Rényi initiated the study of the random graph
G(n,p), where n is the number of vertices and where each edge has
probability p. A celebrated theorem is that p = log n / n is a sharp
threshold for connectivity. In 2006 Linial and Meshulam defined
2-dimensional random simplicial complexes Y(n,p) and exhibited a
homological analogue of the Erdos-Renyi theorem.

Babson, Hoffman, and I studied fundamental group of these complexes
and found a very different story. To find a sharp exponent for the
vanishing threshold we combined techniques from combinatorial homotopy
theory and geometric group theory. Time permitting, I will discuss
more recent work showing that these random complexes satisfy certain
“expansion” properties, making them higher-dimensional analogues of
expander graphs.