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.

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 ${\displaystyle \mathbb {R} ^{n}}$, the lattice ${\displaystyle \varepsilon \mathbb {Z} ^{n}}$ satisfies this property for ${\displaystyle \varepsilon >0}$ small enough.

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