Super rigidity theorem

In the mathematics describing Super rigidity theorem of Margulis (ger .: Margulis superrigidity theorem ) the representations of lattices in Lie groups of higher rank. One consequence of the superstiffness theorem is the arithmeticity of these grids.

motivation

Representations of groups are of great importance in mathematics and physics. Therefore one would like to classify their representations for given groups, for example according to or also into other Lie groups . ${\displaystyle GL(n,\mathbb {C} ),GL(n,\mathbb {R} )}$

Margulis' superstiffness theorem tries to do this for groups that are already a lattice in a Lie group (of rank ), for example as a lattice in . (In order for the condition to be fulfilled, must be in this example .) For such grids, the inclusion gives an obvious representation in the Lie group and, moreover, every representation of the Lie group in another Lie group also provides a representation of in . Since the finite-dimensional representations of Lie groups can be completely classified, the question then remains whether the grid has other representations beyond that. ${\displaystyle \Gamma }$ ${\displaystyle G}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \geq 2}$ ${\displaystyle SL(n,\mathbb {Z} )}$ ${\displaystyle SL(n,\mathbb {R} )}$ ${\displaystyle \mathbb {R} -rk(G)\geq 2}$ ${\displaystyle n\geq 3}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle \Gamma }$ ${\displaystyle H}$

Lattices in Lie groups usually have numerous homomorphisms on finite groups . For example, has surjective homomorphisms for every natural number . If such a finite group occurs as a subgroup in a Lie group , then the homomorphism provides a representation of in the Lie group . ${\displaystyle SL(n,\mathbb {Z} )}$ ${\displaystyle SL(n,\mathbb {Z} /N\mathbb {Z} )}$ ${\displaystyle N}$ ${\displaystyle H}$ ${\displaystyle \Gamma }$ ${\displaystyle H}$

The theorem of superstigness states that these are the only two possible ways of representing in . ${\displaystyle \Gamma }$ ${\displaystyle H}$

The super rigidity theorem does not apply grid in and . For example, surface groups are lattices in , but Mostow's theorem of rigidity does not apply to their representations , and they also have numerous faithful representations in which are not limitations of representations , see Quasifuchs group and Cannon-Thurston maps . Although similar applies for grid in the Mostowsche rigidity theorem, however, some grid can in deform without these deformations to representations could continue. ${\displaystyle SO(n,1)}$ ${\displaystyle SU(n,1)}$ ${\displaystyle PSL(2,\mathbb {R} )=SO(2,1)}$ ${\displaystyle SO(2,1)}$ ${\displaystyle PSL(2,\mathbb {C} )=SO(3,1)_{0}}$ ${\displaystyle PSL(2,\mathbb {R} )\to PSL(2,\mathbb {C} )}$ ${\displaystyle n>2}$ ${\displaystyle SO(n,1)}$ ${\displaystyle \Gamma \subset SO(n,1)}$ ${\displaystyle SO(n+1,1)}$ ${\displaystyle SO(n,1)\to SO(n+1,1)}$

Statement of the super rigidity theorem

Let be a non-compact simple Lie group that is not locally isomorphic to or and be a lattice in . ${\displaystyle G}$ ${\displaystyle O(n,1)=Isom(H^{n})}$ ${\displaystyle U(n,1)=Isom(H_{\mathbb {C} }^{n})}$ ${\displaystyle \Gamma }$ ${\displaystyle G}$

Then every representation is with tsariski - a dense picture ${\displaystyle \rho \colon \Gamma \to GL(n,\mathbb {C} )}$

• either the restriction of a continuous homomorphism ${\displaystyle h\colon G\to GL(n,\mathbb {C} )}$
• or it has a precompact picture .

Generalizations

The above formulation is not the most general possible. For example, the statement still applies if the image of the representation is instead of a simple Lie group with a trivial center or if only a semi-simple Lie group , but then an irreducible grid and the image Zariski - close in . ${\displaystyle GL(n,\mathbb {C} )}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle \rho (\Gamma )}$ ${\displaystyle H}$

An even more general formulation in the context of algebraic groups is the following.

Be

• ${\displaystyle G}$be a connected , semi-simple, real algebraic group without a compact factor and be . ${\displaystyle \mathbb {R} -rk(G)\geq 2}$
• ${\displaystyle \Gamma }$an irreducible lattice in . ${\displaystyle G(\mathbb {R} )}$
• ${\displaystyle k}$a local body of characteristic 0, i.e. H. or a finite extension of ${\displaystyle k=\mathbb {R} ,\mathbb {C} }$ ${\displaystyle \mathbb {Q} _{p}}$

and be a simple, connected, algebraic group. Be a homomorphism with a Zariski-dense image. Then: ${\displaystyle H}$ ${\displaystyle k}$ ${\displaystyle \pi \colon \Gamma \to H(k)}$

1. If and is not compact, then one can continue to a rational homomorphism defined by (i.e. inducing a homomorphism ). ${\displaystyle k=\mathbb {R} }$ ${\displaystyle H(\mathbb {R} )}$ ${\displaystyle \pi }$ ${\displaystyle G\to H}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle G(\mathbb {R} )\to H(\mathbb {R} )}$
2. If is, then either is compact, or can be continued to a rational homomorphism . ${\displaystyle k=\mathbb {C} }$ ${\displaystyle \pi (\Gamma )}$ ${\displaystyle \pi }$ ${\displaystyle G\to H}$
3. If is totally disjointed , then is compact. ${\displaystyle k}$ ${\displaystyle \pi (\Gamma )}$

literature

• Michael Gromow , Pierre Pansu : Rigidity of lattices: an introduction. Geometric topology: recent developments (Montecatini Terme, 1990), 39–137, Lecture Notes in Math., 1504, Springer, Berlin, 1991. Online (pdf)
• GA Margulis : Discrete subgroups of semisimple lie groups. Results of mathematics and their border areas (3), 17. Springer-Verlag, Berlin, 1991. ISBN 3-540-12179-X

Web links

• Furman: Introduction to super-rigidity
• Fisher: Superrigidity, arithmeticity, normal subgroups: results, ramifications and directions

Individual evidence

1. ↑ Dennis Johnson ; John Millson : Deformation spaces associated to compact hyperbolic manifolds. Discrete groups in geometry and analysis (New Haven, Conn., 1984), 48-106, Progr. Math., 67, Birkhäuser Boston, Boston, MA, 1987. online (pdf)
2. ^ Theorem 5.6 in Margulis, op.cit.