Editing Discrete geometry (section)

===Lattices and discrete groups===
{{main|Lattice (group)|discrete group}}
A '''discrete group''' is a [[Group (mathematics)|group]] ''G'' equipped with the [[discrete topology]]. With this topology, ''G'' becomes a [[topological group]]. A '''discrete subgroup''' of a topological group ''G'' is a [[subgroup]] ''H'' whose [[relative topology]] is the discrete one. For example, the [[integer]]s, '''Z''', form a discrete subgroup of the [[real numbers|reals]], '''R''' (with the standard [[Metric space|metric topology]]), but the [[rational number]]s, '''Q''', do not.

A '''lattice''' in a [[locally compact]] [[topological group]] is a [[discrete subgroup]] with the property that the [[Quotient space (topology)|quotient space]] has finite [[invariant measure]]. In the special case of subgroups of '''R'''<sup>''n''</sup>, this amounts to the usual geometric notion of a [[lattice (group)|lattice]], and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of [[Armand Borel|Borel]], [[Harish-Chandra]], [[George Mostow|Mostow]], [[Tsuneo Tamagawa|Tamagawa]], [[M. S. Raghunathan]], [[Grigory Margulis|Margulis]], [[Robert Zimmer (mathematician)|Zimmer]] obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of [[nilpotent group|nilpotent]] [[Lie group]]s and [[semisimple algebraic group]]s over a [[local field]]. In the 1990s, [[Hyman Bass|Bass]] and [[Alexander Lubotzky|Lubotzky]] initiated the study of ''tree lattices'', which remains an active research area.

Topics in this area include:
*[[Reflection group]]s
*[[Triangle group]]s