April 10, 2002

Yair Glasner

Title: Ramanujan graphs and their girth

Abstract:

Both the Ramanujan property and the "high girth" property are measures for a finite graph being similar to its universal cover". The first property is defined in spectral terms whereas the second definition says that the graph looks locally like the tree.

There are connections between the spectral properties of the graph and the existence of short circuits, but in general none of the above properties implies the other one.

I will show an example for a sequence of Ramanujan graphs, covering each other

....--> X_n --> X_n-1 --> ... --> X_1

where all the graphs contain small circles.

What can be the minimal common covering of such a tower of Ramanujan graphs? I will show how one can obtain a tree or a graph with one circle. I will explain why it is not possible, using the known families of Ramanujan graphs (i.e. these coming from arithmetic lattices in p-adic groups), to obtain a minimal common covering with a non Abelian fundamental group.