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We prove that the first reduced cohomology with values in a mixing $L^{p}$ -representation, $1 < p < \infty$ , vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced $ℓ^{p}$ -cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced $L^{p}$ -cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also...

Vanishing theorems and conjectures for the 2-homology of right-angled Coxeter groups.

Davis, Michael W., Okun, Boris (2001)

Geometry & Topology

Variational problems for riemannian functionals and arithmetic groups

Alexander Nabutovsky, Shmuel Weinberger (2000)

Publications Mathématiques de l'IHÉS

Varieties of half lattice-ordered groups of monotonic permutations of chains

Michèle Giraudet, Jiří Rachůnek (1999)

Czechoslovak Mathematical Journal

Veech Groups of Loch Ness Monsters

Piotr Przytycki, Gabriela Schmithüsen, Ferrán Valdez (2011)

Annales de l’institut Fourier

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $G L_{+} (2, R$ ) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surface is either countable, or one of three specific types.

Virtual and welded links and their invariants.

Bardakov, V.G. (2005)

Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]

Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Romain Tessera (2007)

Bulletin de la Société Mathématique de France

Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then ${(U^{n})}_{n \in ℕ}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $\frac{μ (U^{n + 1} ∖ U^{n})}{μ (U^{n})} .$ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a $L^{p}$ -pointwise...

Volumes of discrete groups and topological complexity of homology spheres.

Alexander Reznikov (1996)

Mathematische Annalen

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