# American Institute of Mathematical Sciences

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

## Boundaries, Weyl groups, and Superrigidity

 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

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.
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., ().   Google Scholar [2] U. Bader, A. Furman and A. Shaker, Superrigidity via Weyl groups: Actions on the circle,, preprint., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] H. Furstenberg, A note on Borel's density theorem, Proc. Amer. Math. Soc., 55 (1976), 209-212.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).  Google Scholar [11] _____, Finiteness of quotient groups of discrete groups, Funkts. Anal. Prilozh., 13 (1979), 28-39. Google Scholar [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.  Google Scholar [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.  Google Scholar [14] _____, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, Vol. 1758, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80626.  Google Scholar [15] _____, Arithmeticity vs. nonlinearity for irreducible lattices, Geom. Dedicata, 112 (2005), 225-237. doi: 10.1007/s10711-004-6162-9.  Google Scholar [16] , Y. Shalom and T. Steger,, unpublished., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [19] _____, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, Vol. 81, Birkhäuser Verlag, Basel, 1984.  Google Scholar

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##### References:
 [1] U. Bader and A. Furman, Superrigidity via Weyl groups: Hyperbolic-like targets,, preprint., ().   Google Scholar [2] U. Bader, A. Furman and A. Shaker, Superrigidity via Weyl groups: Actions on the circle,, preprint., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] H. Furstenberg, A note on Borel's density theorem, Proc. Amer. Math. Soc., 55 (1976), 209-212.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).  Google Scholar [11] _____, Finiteness of quotient groups of discrete groups, Funkts. Anal. Prilozh., 13 (1979), 28-39. Google Scholar [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.  Google Scholar [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.  Google Scholar [14] _____, "Continuous Bounded Cohomology of Locally Compact Groups," Lecture Notes in Mathematics, Vol. 1758, Springer-Verlag, Berlin, 2001. doi: 10.1007/b80626.  Google Scholar [15] _____, Arithmeticity vs. nonlinearity for irreducible lattices, Geom. Dedicata, 112 (2005), 225-237. doi: 10.1007/s10711-004-6162-9.  Google Scholar [16] , Y. Shalom and T. Steger,, unpublished., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [19] _____, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, Vol. 81, Birkhäuser Verlag, Basel, 1984.  Google Scholar
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