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 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

1. Kazhdan, David; Margulis, Grigory (1968). "A proof of Selberg's hypothesis". Math. USSR Sbornik. 4. Translated by Z. Skalsky: 147–152. doi:10.1070/SM1968v004n01ABEH002782. MR 0223487.
2. Raghunathan 1972, Theorem 11.7.
3. Gelander, Tsachik (2011). "Volume versus rank of lattices". Journal für die reine und angewandte Mathematik. 2011 (661): 237–248. arXiv:1102.3574. doi:10.1515/CRELLE.2011.085. S2CID 122888051.
4. Ballmann, Werner; Gromov, Mikhael; Schroeder, Viktor (1985). Manifolds of nonpositive curvature. Progress in Mathematics. Vol. 61. Birkhäuser Boston, Inc., Boston, MA. doi:10.1007/978-1-4684-9159-3. ISBN 978-1-4684-9161-6.
5. Gelander, Tsachik (2018). "Kazhdan-Margulis theorem for invariant random subgroups". Advances in Mathematics. 327: 47–51. arXiv:1510.05423. doi:10.1016/j.aim.2017.06.011. S2CID 119314646.
6. Marshall, Timothy H.; Martin, Gaven J. (2012). "Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group". Annals of Mathematics. 176: 261–301. doi:10.4007/annals.2012.176.1.4. MR 2925384.
7. Belolipetsky, Mikhail; Emery, Vincent (2014). "Hyperbolic manifolds of small volume" (PDF). Documenta Mathematica. 19: 801–814. arXiv:1310.2270. doi:10.4171/dm/464. S2CID 303659.
8. Theorem 8.1 in Wang, Hsien-Chung (1972), "Topics on totally discontinuous groups", in Boothby, William M.; Weiss, Guido L. (eds.), Symmetric Spaces, short Courses presented at Washington Univ., Pure and Applied Mathematics., vol. 1, Marcel Dekker, pp. 459–487, Zbl 0232.22018

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.