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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.

Addendum to: On volumes of arithmetic quotients of SO (1, n)

Mikhail Belolipetsky (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.

An extension of Mahler's theorem to simply connected nilpotent groups

Martin Moskowitz (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

This Note gives an extension of Mahler's theorem on lattices in R n to simply connected nilpotent groups with a Q -structure. From this one gets an application to groups of Heisenberg type and a generalization of Hermite's inequality.

An extension of the Khinchin-Groshev theorem

Anish Ghosh, Robert Royals (2015)

Acta Arithmetica

We prove a version of the Khinchin-Groshev theorem in Diophantine approximation for quadratic extensions of function fields in positive characteristic.

An extention of Nomizu’s Theorem –A user’s guide–

Hisashi Kasuya (2016)

Complex Manifolds

For a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ, C) of the solvmanifold Γ. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ, C).

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