Kazhdan–Margulis theorem explained

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

G</matH> be a semisimple Lie group: there exists an open neighbourhood <math>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

R ^{n} , the lattice \varepsilon 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.

Given a semisimple Lie group without compact factors

G endowed with a norm | ⋅ | , 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} | \ge 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. There is one proof which is more geometric in nature and which can give more information,^{[2] [3]} and there is a third proof, relying on the notion of invariant random subgroups, which is considerably shorter.^[4]

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 PSL _{2(R)} (see Hurwitz's automorphisms theorem). For hyperbolic three-manifolds the lattice of minimal volume is known and its covolume is about 0.0390.^[5] 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.^[6]

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.^[7]

If

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

See also

• Margulis lemma

References

• Book: Gelander, Tsachik . Bestvina . Mladen . Sageev . Michah . Vogtmann . Karen . Geometric group theory . Lectures on lattices and locally symmetric spaces . 2014 . 249–282 . 1402.0962. 2014arXiv1402.0962G .
• Book: Raghunathan, M. S. . Discrete subgroups of Lie groups . . Ergebnisse de Mathematik und ihrer Grenzgebiete . 1972 . 0507234.

Notes and References

1. Kazhdan . David . David Kazhdan . Margulis . Grigory . Grigory Margulis . A proof of Selberg's hypothesis . . 4 . 147–152 . 1968 . 10.1070/SM1968v004n01ABEH002782 . en. 0223487 . Z. Skalsky .
2. Gelander . Tsachik . Volume versus rank of lattices. Journal für die reine und angewandte Mathematik. 2011. 2011. 661 . 237–248. 10.1515/CRELLE.2011.085. 1102.3574. 122888051 .
3. Book: Ballmann . Werner. Gromov . Mikhael . Mikhael Gromov (mathematician). Schroeder . Viktor. Manifolds of nonpositive curvature. Progress in Mathematics. 61. Birkhäuser Boston, Inc., Boston, MA. 1985. 10.1007/978-1-4684-9159-3 . 978-1-4684-9161-6. free.
4. Gelander . Tsachik . Kazhdan-Margulis theorem for invariant random subgroups. Advances in Mathematics. 327. 2018. 47–51. 1510.05423. 10.1016/j.aim.2017.06.011. 119314646 .
5. Marshall . Timothy H. . Martin . Gaven J. . Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group . Annals of Mathematics . 176 . 2012 . 261–301 . 2925384 . 10.4007/annals.2012.176.1.4. free .
6. Belolipetsky . Mikhail . Emery . Vincent . Hyperbolic manifolds of small volume . . 19 . 2014 . 801–814 . 10.4171/dm/464 . 1310.2270 . 303659 .
7. Theorem 8.1 in