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 be a semisimple Lie group: there exists an open neighbourhood of the identity in such that for any discrete subgroup there is an element satisfying .

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 , the lattice satisfies this property for 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 endowed with a norm , there exists , a neighbourhood of in , a compact subset such that, for any discrete subgroup there exists a such that for all .

The neighbourhood is obtained as a Zassenhaus neighbourhood of the identity in : 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 for the smallest covolume of a quotient of the hyperbolic plane by a lattice in (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]

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 is a simple Lie group not locally isomorphic to or with a fixed Haar measure and there are only finitely many lattices in of covolume less than .

See also

Notes

  1. Kazhdan, David; Margulis, Grigori (1968). "A proof of Selberg's hypothesis". Mat. Sbornik (N.S.) (in Russian). 75: 162–168. MR 0223487.CS1 maint: ref=harv (link)
  2. Raghunatan 1972, Theorem 11.7.
  3. Gelander 2012, Remark 3.16.
  4. Marshall, Timothy H.; Martin, Gaven J. (2012). "Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group". Ann. Math. 176: 261–301. doi:10.4007/annals.2012.176.1.4. MR 2925384.CS1 maint: ref=harv (link)
  5. Belolipetsky, Mikhail; Emery, Vincent (2014). "Hyperbolic manifolds of small volume". Documenta Math. 19: 801–814.CS1 maint: ref=harv (link)

References

  • Gelander, Tsachik. "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.CS1 maint: ref=harv (link)
  • Raghunathan, M. S. (1972). Discrete subgroups of Lie groups. Ergebnisse de Mathematik und ihrer Grenzgebiete. Springer-Verlag. MR 0507234.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.