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.^[1]

Statement and remarks

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

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

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 $\mathbb {R} ^{n}$ , the lattice $\varepsilon \mathbb {Z} ^{n}$ satisfies this property for $\varepsilon >0$ 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 $G$ endowed with a norm $|\cdot |$ , there exists $c > 1$ , a neighbourhood $U_{0}$ of $e$ in $G$ , a compact subset $E\subset G$ such that, for any discrete subgroup $\Gamma \subset G$ there exists a $g\in E$ such that $|g\gamma g^{-1}|\geq c|\gamma |$ for all $\gamma \in \Gamma \cap U_{0}$ .

The neighbourhood $U_{0}$ is obtained as a Zassenhaus neighbourhood of the identity in $G$ : 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]

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 $\pi /21$ for the smallest covolume of a quotient of the hyperbolic plane by a lattice in $\mathrm {PSL} _{2}(\mathbb {R} )$ (see Hurwitz's automorphisms theorem). For hyperbolic three-manifolds the lattice of minimal colume 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]

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.

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

Notes

↑ Kazhdan, David; Margulis, Grigori (1968). "A proof of Selberg’s hypothesis". Math. Sbornik. (N.S.) (in Russian). 75: 162–168. MR 0223487.
↑ Raghunatan 1972, Theorem 11.7.
↑ Gelander 2012, Remark 3.16.
↑ Marshall, Timothy H.; Martin, Gaven J. (2012). "Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group". Ann. Math. 176: 261–301. MR 2925384.
↑ Belolipetsky, Mikhail; Emery, Vincent (2014). "Hyperbolic manifolds of small volume". Documenta Math. 19: 801–814.

References

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

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