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.
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Statement and remarks
The formal statement of the Kazhdan–Margulis theorem is as follows.
LetNote 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
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 factorsThe neighbourhood
There also exist other proofs, more geometric in nature and which can give more information.
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
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