Kazhdan–Margulis theorem


In Lie theory, an area of mathematics, the Kazhdan–Margulis theorem is a statement asserting that a discrete subgroup in semisimple Lie groups cannot be too dense in the group. More precisely, in any such Lie group there is a uniform neighbourhood of the identity element such that every lattice in the group has a conjugate whose intersection with this neighbourhood contains only the identity. This result was proven in the sixties by David Kazhdan and Grigori Margulis.

Statement and remarks

The formal statement of the Kazhdan–Margulis theorem is as follows.
Note that in general Lie groups this statement is far from being true; in particular, in a nilpotent Lie group, for any neighbourhood of the identity there exists a lattice in the group which is generated by its intersection with the neighbourhood: for example, in, the lattice satisfies this property for small enough.

Proof

The main technical result of Kazhdan–Margulis, which is interesting in its own right and from which the better-known statement above follows immediately, is the following.
The neighbourhood is obtained as a Zassenhaus neighbourhood of the identity in : the theorem then follows by standard Lie-theoretic arguments.
There also exist other proofs, more geometric in nature and which can give more information.

Applications

Selberg's hypothesis

One of the motivations of Kazhdan–Margulis was to prove the following statement, known at the time as Selberg's hypothesis :
This result follows from the more technical version of the Kazhdan–Margulis theorem and the fact that only unipotent elements can be conjugated arbitrarily close to the identity.

Volumes of locally symmetric spaces

A corollary of the theorem is that the locally symmetric spaces and orbifolds associated to lattices in a semisimple Lie group cannot have arbitrarily small volume.
For hyperbolic surfaces this is due to Siegel, and there is an explicit lower bound of for the smallest covolume of a quotient of the hyperbolic plane by a lattice in . For hyperbolic three-manifolds the lattice of minimal volume is known and its covolume is about 0.0390. In higher dimensions the problem of finding the lattice of minimal volume is still open, though it has been solved when restricting to the subclass of arithmetic groups.

Wang's finiteness theorem

Together with local rigidity and finite generation of lattices the Kazhdan-Marguilis theorem is an important ingredient in the proof of Wang's finiteness theorem.