Kazhdan–Margulis theorem

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.[2]

Given a semisimple Lie group without compact factors ${\displaystyle G}$ endowed with a norm ${\displaystyle |\cdot |}$, there exists ${\displaystyle c>1}$, a neighbourhood ${\displaystyle U_{0}}$ of ${\displaystyle e}$ in ${\displaystyle G}$, a compact subset ${\displaystyle E\subset G}$ such that, for any discrete subgroup ${\displaystyle \Gamma \subset G}$ there exists a ${\displaystyle g\in E}$ such that ${\displaystyle |g\gamma g^{-1}|\geq c|\gamma |}$ for all ${\displaystyle \gamma \in \Gamma \cap U_{0}}$.

The neighbourhood ${\displaystyle U_{0}}$ is obtained as a Zassenhaus neighbourhood of the identity in ${\displaystyle G}$: the theorem then follows by standard Lie-theoretic arguments.

There also exist other proofs, more geometric in nature and which can give more information. [3]

One of the motivations of Kazhdan–Margulis was to prove the following statement, known at the time as Selberg's hypothesis (recall that a lattice is called uniform if its quotient space is compact):

A lattice in a semisimple Lie group is non-uniform if and only if it contains a unipotent element.

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 (for a given element) to the identity.

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 (given a normalisation for the Haar measure).

For hyperbolic surfaces this is due to Siegel, and there is an explicit lower bound of ${\displaystyle \pi /21}$ for the smallest covolume of a quotient of the hyperbolic plane by a lattice in ${\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}$ (see Hurwitz's automorphisms theorem). For hyperbolic three-manifolds the lattice of minimal volume is known and its covolume is about 0.0390.[4] 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.[5]

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.

If ${\displaystyle G}$ is a simple Lie group not locally isomorphic to ${\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}$ or ${\displaystyle \mathrm {SL} _{2}(\mathbb {C} )}$ with a fixed Haar measure and ${\displaystyle v>0}$ there are only finitely many lattices in ${\displaystyle G}$ of covolume less than ${\displaystyle v}$.