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

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

The superconducting and magnetic states in RuSr2GdCu2O8, based on the magnetic, transport and magneto-caloric characteristics

The article discusses selected properties of the non- and superconducting polycrystalline samples of RuSr2GdCu2O8 and comments the consequences of introducing insignificant sub-stoichiometry of Ru into the nominal formula. The magneto-resistive and the magnetic characteristics are interpreted in favour of the formation of the intrinsically inhomogeneous superconducting phase, which seems to be stabilized along with the structural modifications likely enhanced with the modification of starting stoichiometry. The specific heat data reveals the shift of temperature of the magnetic ordering T_{m}, suggesting the dilution in magnetic sublattice of the Ru moments. The measurements of the magnetic field dependences of the isothermal magnetocaloric coefficient M_{T} show that there is no gain in magnetic entropy in a broad range of the accessed fields and temperatures. Whereas the multi-component character of the probed magnetic system precludes from concluding on the ground state for the Ru ordering, the maximum in M_{T}(H) which occurs at weak magnetic fields for temperature vicinity of T_{m} may reflect dominance of the ferromagnetic type interactions with a constrained correlation range. The literature explored models for the Ru magnetic ordering and possible phase separation in the RuSr2GdCu2O8 are brought into the discussion.Comment: Most of the discussed results conform to the first author's presentations given at XIV Polish National School for Superconductivity (XIV KSN) in October 2009, and at M2S-IX Conference in September 2009. Submitted to the proceedings of XIV KSN

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

Hilbert C*-modules and amenable actions

We study actions of discrete groups on Hilbert $C^*$-modules induced from topological actions on compact Hausdorff spaces. We show non-amenability of actions of non-amenable and non-a-T-menable groups, provided there exists a quasi-invariant probability measure which is sufficiently close to being invariant.Comment: Final version, to appear in Studia Mathematic