January  2011, 5(1): 49-69. doi: 10.3934/jmd.2011.5.49

Boundary unitary representations-irreducibility and rigidity

1. 

Mathematics Department, The Technion - Israel Institute of Technology Haifa, 32000, Israel

2. 

Department of Mathematics and Computer Science, Lehman College, CUNY, 2500 Johnson Avenue Bronx, NY 10463, United States

Received  January 2010 Revised  December 2010 Published  April 2011

Let $M$ be compact negatively curved manifold, $\Gamma =\pi_1(M)$ and $M$ be its universal cover. Denote by $B =\partial M$ the geodesic boundary of $M$ and by $\nu$ the Patterson-Sullivan measure on $X$. In this note we prove that the associated unitary representation of $\Gamma$ on $L^2(B,\nu)$ is irreducible. We also establish a new rigidity phenomenon: we show that some of the geometry of $M$, namely its marked length spectrum, is reflected in this $L^2$-representations.
Citation: Uri Bader, Roman Muchnik. Boundary unitary representations-irreducibility and rigidity. Journal of Modern Dynamics, 2011, 5 (1) : 49-69. doi: 10.3934/jmd.2011.5.49
References:
[1]

M. E. B. Bekka and M. Cowling, Some irreducible unitary representations of $G(K)$ for a simple algebraic group $G$ over an algebraic number field $K$,, Math. Z., 241 (2002), 731.  doi: 10.1007/s00209-002-0442-6.  Google Scholar

[2]

Marc Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace,, Enseign. Math., 2 (1995), 63.   Google Scholar

[3]

Marc Burger and Pierre de la Harpe, Constructing irreducible representations of discrete groups,, Proc. Indian Acad. Sci. Math. Sci., 107 (1997), 223.  doi: 10.1007/BF02867253.  Google Scholar

[4]

M. Burger and S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators,, J. Amer. Math. Soc., 9 (1996), 57.  doi: 10.1090/S0894-0347-96-00196-8.  Google Scholar

[5]

Chris Connell and Roman Muchnik, Harmonicity of quasiconformal measures and poisson boundaries of hyperbolic spaces,, to appear in GAFA., ().   Google Scholar

[6]

M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices,, J. Reine Angew. Math., 420 (1991), 85.   Google Scholar

[7]

Alessandro Figà-Talamanca and Massimo A. Picardello, "Harmonic Analysis on Free Groups,", Lecture Notes in Pure and Applied Mathematics, 87 (1983).   Google Scholar

[8]

Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees,, Mem. Amer. Math. Soc., 110 (1994).   Google Scholar

[9]

Alex Furman, Rigidity of group actions on infinite volume homogeneous spaces, II,, preprint., ().   Google Scholar

[10]

George W. Mackey, "The Theory of Unitary Group Representations,", University of Chicago Press, (1976).   Google Scholar

[11]

Grigoriy A. Margulis, "On Some Aspects of the Theory of Anosov Systems,", Springer Monographs in Mathematics, (2004).   Google Scholar

[12]

Chengbo Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature,, Trans. Amer. Math. Soc., 348 (1996), 4965.  doi: 10.1090/S0002-9947-96-01614-5.  Google Scholar

show all references

References:
[1]

M. E. B. Bekka and M. Cowling, Some irreducible unitary representations of $G(K)$ for a simple algebraic group $G$ over an algebraic number field $K$,, Math. Z., 241 (2002), 731.  doi: 10.1007/s00209-002-0442-6.  Google Scholar

[2]

Marc Bourdon, Structure conforme au bord et flot géodésique d'un CAT(-1)-espace,, Enseign. Math., 2 (1995), 63.   Google Scholar

[3]

Marc Burger and Pierre de la Harpe, Constructing irreducible representations of discrete groups,, Proc. Indian Acad. Sci. Math. Sci., 107 (1997), 223.  doi: 10.1007/BF02867253.  Google Scholar

[4]

M. Burger and S. Mozes, CAT(-1)-spaces, divergence groups and their commensurators,, J. Amer. Math. Soc., 9 (1996), 57.  doi: 10.1090/S0894-0347-96-00196-8.  Google Scholar

[5]

Chris Connell and Roman Muchnik, Harmonicity of quasiconformal measures and poisson boundaries of hyperbolic spaces,, to appear in GAFA., ().   Google Scholar

[6]

M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices,, J. Reine Angew. Math., 420 (1991), 85.   Google Scholar

[7]

Alessandro Figà-Talamanca and Massimo A. Picardello, "Harmonic Analysis on Free Groups,", Lecture Notes in Pure and Applied Mathematics, 87 (1983).   Google Scholar

[8]

Alessandro Figà-Talamanca and Tim Steger, Harmonic analysis for anisotropic random walks on homogeneous trees,, Mem. Amer. Math. Soc., 110 (1994).   Google Scholar

[9]

Alex Furman, Rigidity of group actions on infinite volume homogeneous spaces, II,, preprint., ().   Google Scholar

[10]

George W. Mackey, "The Theory of Unitary Group Representations,", University of Chicago Press, (1976).   Google Scholar

[11]

Grigoriy A. Margulis, "On Some Aspects of the Theory of Anosov Systems,", Springer Monographs in Mathematics, (2004).   Google Scholar

[12]

Chengbo Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature,, Trans. Amer. Math. Soc., 348 (1996), 4965.  doi: 10.1090/S0002-9947-96-01614-5.  Google Scholar

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