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 1960s by David Kazhdan and Grigory Margulis. ^[1]

Statement and remarks

The formal statement of the Kazhdan–Margulis theorem is as follows.

Let ${\displaystyle G}$ be a semisimple Lie group: there exists an open neighbourhood ${\displaystyle U}$ of the identity ${\displaystyle e}$ in ${\displaystyle G}$ such that for any discrete subgroup ${\displaystyle \Gamma \subset G}$ there is an element ${\displaystyle g\in G}$ satisfying ${\displaystyle g\Gamma g^{-1}\cap U=\{e\}}$.

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 ${\displaystyle \mathbb {R} ^{n}}$, the lattice ${\displaystyle \varepsilon \mathbb {Z} ^{n}}$ satisfies this property for ${\displaystyle \varepsilon >0}$ 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. ^[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. There is one proof which is more geometric in nature and which can give more information, ^{[3] [4]} and there is a third proof, relying on the notion of invariant random subgroups, which is considerably shorter. ^[5]

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 (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.

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 (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. ^[6] 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. ^[7]

Wang's finiteness theorem

Together with local rigidity and finite generation of lattices the Kazhdan-Margulis theorem is an important ingredient in the proof of Wang's finiteness theorem. ^[8]

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 ${\displaystyle v}$.

See also

• Margulis lemma

Notes

References

• Gelander, Tsachik (2014). "Lectures on lattices and locally symmetric spaces". In Bestvina, Mladen; Sageev, Michah; Vogtmann, Karen (eds.). Geometric group theory. pp. 249–282. arXiv: 1402.0962. Bibcode: 2014arXiv1402.0962G.
• Raghunathan, M. S. (1972). Discrete subgroups of Lie groups. Ergebnisse de Mathematik und ihrer Grenzgebiete. Springer-Verlag. MR  0507234.