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]
The formal statement of the Kazhdan–Margulis theorem is as follows.
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.
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]
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. 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]
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):
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.
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. [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]
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]
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]
The formal statement of the Kazhdan–Margulis theorem is as follows.
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.
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]
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. 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]
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):
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.
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. [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]
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]