### A better proof of the Goldman-Parker conjecture.

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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 ${\lambda}_{G/H}$ of G on L²(G/H) has a spectral gap, that is, the restriction of ${\lambda}_{G/H}$ 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.

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.

This Note gives an extension of Mahler's theorem on lattices in ${\mathbb{R}}^{n}$ 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.

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

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).