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 ...

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 essen ...

ρ of a discrete group Γ inside an

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

''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 or ...

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 that has a finite kernel and is a ...

s was the discovery of

Grigory Margulis Grigory Aleksandrovich Margulis (, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on lattices in Lie groups, and the introduction of methods from ergodic the ...

, 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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

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 metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...

''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 (sinc ...

(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 defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...

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 group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebrai ...

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