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A rigidity phenomenon for germs of actions of R 2

Aubin Arroyo, Adolfo Guillot (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

We study germs of Lie algebras generated by two commuting vector fields in manifolds that are maximal in the sense of Palais (those which do not present any evident obstruction to be the local model of an action of  R 2 ). We study three particular pairs of homogeneous quadratic commuting vector fields (in  R 2 , R 3 and  R 4 ) and study the maximal Lie algebras generated by commuting vector fields whose 2-jets at the origin are the given homogeneous ones. In the first case we prove that the quadratic algebra...

A spectral gap property for subgroups of finite covolume in Lie groups

Bachir Bekka, Yves Cornulier (2010)

Colloquium Mathematicae

Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation λ G / H of G on L²(G/H) has a spectral gap, that is, the restriction of λ G / H to the orthogonal complement of the constants in L²(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

A spectral gap theorem in SU ( d )

Jean Bourgain, Alex Gamburd (2012)

Journal of the European Mathematical Society

We establish the spectral gap property for dense subgroups of SU ( d ) ( d ≥ 2 ) , generated by finitely many elements with algebraic entries; this result was announced...

A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on ℂ n

E. K. Narayanan, S. Thangavelu (2006)

Annales de l’institut Fourier

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on ℂ n . If f ( z ) e 1 4 | z | 2 is a Schwartz class function we show that f is supported in a ball of radius B in ℂ n if and only if f × μ r ( z ) = 0 for r > B + | z | for all z ∈ ℂ n . This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When n = 1 we show that the two conditions f × μ r ( z ) = μ r × f ( z ) = 0 for r > B + | z | imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter...

A theorem on generic intersections in an o-minimal structure

Krzysztof Jan Nowak (2014)

Fundamenta Mathematicae

Consider a transitive definable action of a Lie group G on a definable manifold M. Given two (locally) definable subsets A and B of M, we prove that the dimension of the intersection σ(A) ∩ B is not greater than the expected one for a generic σ ∈ G.

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