In mathematics, in the theory of

discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...

s, superrigidity is a concept designed to show how a

linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...

ρ of a discrete group Γ inside an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

''G'' can, under some circumstances, be as good as a representation of ''G'' itself. That this phenomenon happens for certain broadly defined classes of

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...

s inside

semisimple group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

s was the discovery of

Grigory Margulis Grigory Aleksandrovich Margulis (russian: Григо́рий Алекса́ндрович Маргу́лис, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on ...

, who proved some fundamental results in this direction. There is more than one result that goes by the name of ''Margulis superrigidity''. One simplified statement is this: take ''G'' to be a simply connected semisimple real algebraic group in ''GL''_''n'', such that the Lie group of its real points has real rank at least 2 and no compact factors. Suppose Γ is an irreducible lattice in G. For a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...

''F'' and ρ a linear representation of the lattice Γ of the Lie group, into ''GL''_''n'' (''F''), assume the image ρ(Γ) is not

relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sin ...

(in the topology arising from ''F'') and such that its closure in the

Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...

is connected. Then ''F'' is the real numbers or the complex numbers, and there is a

rational representation In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map In mathematics, in particu ...

of ''G'' giving rise to ρ by restriction.

Notes

References

* * Gromov, M.; Pansu, P
''Rigidity of lattices: an introduction.''
Geometric topology: recent developments (Montecatini Terme, 1990), 39–137, Lecture Notes in Math., 1504, Springer, Berlin, 1991. doi:10.1007/BFb0094289 * Gromov, Mikhail; Schoen, Richard. ''Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one.'' Inst. Hautes Études Sci. Publ. Math. No. 76 (1992), 165–246. * Ji, Lizhen
''A summary of the work of Gregory Margulis.''
Pure Appl. Math. Q. 4 (2008), no. 1, Special Issue: In honor of Grigory Margulis. Part 2, 1–69. ages 17-19* Jost, Jürgen; Yau, Shing-Tung. ''Applications of quasilinear PDE to algebraic geometry and arithmetic lattices.'' Algebraic geometry and related topics (Inchon, 1992), 169–193, Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993. * {{cite book , last=Margulis , first=G.A. , title=Discrete subgroups of semisimple lie groups , series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, publisher=Springer-Verlag , year=1991 , isbn=3-540-12179-X , oclc=471802846 , mr=1090825 *Tits, Jacques. ''Travaux de Margulis sur les sous-groupes discrets de groupes de Lie.'' Séminaire Bourbaki, 28ème année (1975/76), Exp. No. 482, pp. 174–190. Lecture Notes in Math., Vol. 567, Springer, Berlin, 1977. Discrete groups

See also

Notes

References