May 12, 2003

Beno Eckmann

Title: Discrete Subgroups of Lie Groups


This is a survey on some new aspects of lattices. A lattice in a connected Lie group $L$ is a discrete subgroup $G$ such that $L/G$ is compact, or more generally has finite volume. A lattice $G$ is finitely presented. It is proved that the deficiency of $G$ (the maximum difference of the numbers of generators and of defining relations) is "in general" $\le 1$, i.e. except for special elementary cases. This is due to the fact that in general the first $\ell_2$-Betti number of $G$ vanishes. The case where the deficiency is $=1$ is examined; it occurs in very few Lie groups only. In particular this helps deciding which classical knot groups are lattices and which are not.