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 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.
...
''G'' can, under some circumstances, be as good as a representation of ''G'' itself. That this phenomenon happens for certain broadly defined classes of
lattices 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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
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 compa ...
''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 is connected. Then ''F'' is the real numbers or the complex numbers, and there is a
rational representation of ''G'' giving rise to ρ by restriction.
See also
*
Mostow rigidity theorem Mostow may refer to: People
* George Mostow (1923–2017), American mathematician
** Mostow rigidity theorem
* Jonathan Mostow (born 1961), American movie and television director
Places
* Mostów
Mostów is a village in the administrative dist ...
*
Local rigidity
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