On coarse embeddability into ℓp\ell_p-spaces and a conjecture of Dranishnikov

We show that the Hilbert space is coarsely embeddable into any $\ell_p$ for $1\le p<\infty$. In particular, this yields new characterizations of embeddability of separable metric spaces into the Hilbert space.Comment: 5 page

Controlled coarse homology and isoperimetric inequalities

We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize existence of primitives of the volume form with prescribed growth and show that coarse homology classes obstruct weighted Poincare inequalities.Comment: Final version, to appear in the Journal of Topolog

Aperiodic tilings of manifolds of intermediate growth

We give a homological construction of aperiodic tiles for certain open Riemannian surfaces admitting actions of Grigorchuk groups of intermediate growth.Comment: Accepted for publication in Groups, Geometry and Dynamic

Eilenberg swindles and higher large scale homology of products of trees

We show that uniformly finite homology of products of $n$ trees vanishes in all degrees except degree $n$, where it is infinite dimensional. Our method is geometric and applies to several large scale homology theories, including almost equivariant homology and controlled coarse homology. As an application we determine group homology with $\ell_{\infty}$-coefficients of lattices in products of trees. We also show a characterization of amenability in terms of 1-homology and construct aperiodic tilings using higher homology.Comment: Final version, to appear in Groups, Geometry & Dynamic