Boundaries, Weyl groups, and Superrigidity

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2012, 19: 41-48. doi: 10.3934/era.2012.19.41

Boundaries, Weyl groups, and Superrigidity

Uri Bader ^1, and Alex Furman ^2,

1.	Mathematics Department, The Technion, 32000 Haifa, Israel
2.	Mathematics, Statistics and Computer Science Department, University of Illinois at Chicago, Chicago, 851 S. Morgan St., Illinois, 60607, United States

Received September 2011 Published March 2012

Abstract
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This note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group $\Gamma$ and a boundary action $\Gamma$ ↷ $B$ we associate a certain generalized Weyl group $W_{{\Gamma}{B}}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $\rho:\Gamma\to \bf{H}$, defines a special homomorphism $W_{{\Gamma}{B}}\to Weyl_{\bf H}$. This general fact allows the deduction of the aforementioned superrigidity results.

Keywords: superrigidity, lattices., Weyl groups.

Mathematics Subject Classification: Primary: 20E4.

Citation: Uri Bader, Alex Furman. Boundaries, Weyl groups, and Superrigidity. Electronic Research Announcements, 2012, 19: 41-48. doi: 10.3934/era.2012.19.41

References:

[1]	U. Bader and A. Furman, Superrigidity via Weyl groups: Hyperbolic-like targets, preprint.
[2]	U. Bader, A. Furman and A. Shaker, Superrigidity via Weyl groups: Actions on the circle, preprint.
[3]	U. Bader and Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math., 163 (2006), 415-454. doi: 10.1007/s00222-005-0469-5.
[4]	M. Burger and N. Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., 12 (2002), 219-280. doi: 10.1007/s00039-002-8245-9.
[5]	M. Burger and S. Mozes, $CAT$(-$1$)-spaces, divergence groups and their commensurators, J. Amer. Math. Soc., 9 (1996), 57-93. doi: 10.1090/S0894-0347-96-00196-8.
[6]	M. Burger, S. Mozes and R. J. Zimmer, Linear representations and arithmeticity of lattices in products of trees, in "Essays in Geometric Group Theory," Ramanujan Math. Soc. Lect. Notes Ser., 9, Ramanujan Math. Soc., Mysore, (2009), 1-25.
[7]	H. Furstenberg, A note on Borel's density theorem, Proc. Amer. Math. Soc., 55 (1976), 209-212.
[8]	T. Gelander, A. Karlsson and G. A. Margulis, Superrigidity, generalized harmonic maps and uniformly convex spaces, Geom. Funct. Anal., 17 (2008), 1524-1550. doi: 10.1007/s00039-007-0639-2.
[9]	V. A. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, Geom. Funct. Anal., 13 (2003), 852-861. doi: 10.1007/s00039-003-0433-8.
[10]	G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, in "Proceedings of the International Congress of Mathematicians" (Vancouver, B.C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., (1975), 21-34 (Russian).
[11]	_____, Finiteness of quotient groups of discrete groups, Funkts. Anal. Prilozh., 13 (1979), 28-39.
[12]	_____, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991.
[13]	N. Monod, Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc., 19 (2006), 781-814. doi: 10.1090/S0894-0347-06-00525-X.
[14]	_____, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, Vol. 1758, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80626.
[15]	_____, Arithmeticity vs. nonlinearity for irreducible lattices, Geom. Dedicata, 112 (2005), 225-237. doi: 10.1007/s10711-004-6162-9.
[16]	, Y. Shalom and T. Steger, unpublished.
[17]	Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis, 27 (1978), 350-372. doi: 10.1016/0022-1236(78)90013-7.
[18]	R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2), 112 (1980), 511-529. doi: 10.2307/1971090.
[19]	_____, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, Vol. 81, Birkhäuser Verlag, Basel, 1984.

show all references

References:

[1]	U. Bader and A. Furman, Superrigidity via Weyl groups: Hyperbolic-like targets, preprint.
[2]	U. Bader, A. Furman and A. Shaker, Superrigidity via Weyl groups: Actions on the circle, preprint.
[3]	U. Bader and Y. Shalom, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math., 163 (2006), 415-454. doi: 10.1007/s00222-005-0469-5.
[4]	M. Burger and N. Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., 12 (2002), 219-280. doi: 10.1007/s00039-002-8245-9.
[5]	M. Burger and S. Mozes, $CAT$(-$1$)-spaces, divergence groups and their commensurators, J. Amer. Math. Soc., 9 (1996), 57-93. doi: 10.1090/S0894-0347-96-00196-8.
[6]	M. Burger, S. Mozes and R. J. Zimmer, Linear representations and arithmeticity of lattices in products of trees, in "Essays in Geometric Group Theory," Ramanujan Math. Soc. Lect. Notes Ser., 9, Ramanujan Math. Soc., Mysore, (2009), 1-25.
[7]	H. Furstenberg, A note on Borel's density theorem, Proc. Amer. Math. Soc., 55 (1976), 209-212.
[8]	T. Gelander, A. Karlsson and G. A. Margulis, Superrigidity, generalized harmonic maps and uniformly convex spaces, Geom. Funct. Anal., 17 (2008), 1524-1550. doi: 10.1007/s00039-007-0639-2.
[9]	V. A. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, Geom. Funct. Anal., 13 (2003), 852-861. doi: 10.1007/s00039-003-0433-8.
[10]	G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, in "Proceedings of the International Congress of Mathematicians" (Vancouver, B.C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., (1975), 21-34 (Russian).
[11]	_____, Finiteness of quotient groups of discrete groups, Funkts. Anal. Prilozh., 13 (1979), 28-39.
[12]	_____, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991.
[13]	N. Monod, Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc., 19 (2006), 781-814. doi: 10.1090/S0894-0347-06-00525-X.
[14]	_____, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, Vol. 1758, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80626.
[15]	_____, Arithmeticity vs. nonlinearity for irreducible lattices, Geom. Dedicata, 112 (2005), 225-237. doi: 10.1007/s10711-004-6162-9.
[16]	, Y. Shalom and T. Steger, unpublished.
[17]	Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis, 27 (1978), 350-372. doi: 10.1016/0022-1236(78)90013-7.
[18]	R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups, Ann. of Math. (2), 112 (1980), 511-529. doi: 10.2307/1971090.
[19]	_____, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, Vol. 81, Birkhäuser Verlag, Basel, 1984.

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