Boundary unitary representations-irreducibility and rigidity

Uri Bader; Roman Muchnik

doi:10.3934/jmd.2011.5.49

Article Contents

2011, Volume 5, Issue 1: 49-69. Doi: 10.3934/jmd.2011.5.49

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Boundary unitary representations-irreducibility and rigidity

Uri Bader^1, and
Roman Muchnik^2,

1.
Mathematics Department, The Technion - Israel Institute of Technology Haifa, 32000
2.
Department of Mathematics and Computer Science, Lehman College, CUNY, 2500 Johnson Avenue Bronx, NY 10463

Received: January 2010

Revised: December 2010

Published: April 2011

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: Primary: 20C07, 22D40; Secondary: 37A30, 47A35.

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References

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