Coarse equivalence and topological couplings of locally compact groups
M. Gromov has shown that any two finitely generated groups $$\Gamma $$Γ and $$\Lambda $$Λ are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions $$\Gamma \curvearrowright X \curvearrowleft \Lambda $$Γ↷X↶Λ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second-countable groups.
On sublinear bilipschitz equivalence of groups
We discuss the notion of sublinearly bilipschitz equivalences (SBE), which generalize quasi-isometries, allowing some additional terms that behave sublinearly with respect to the distance from the origin. Such maps were originally motivated by the fact they induce bilipschitz homeomorphisms between asymptotic cones. We prove here that for hyperbolic groups, they also induce Holder homeomorphisms between the boundaries. This yields many basic examples of hyperbolic groups that are pairwise non-SBE. Besides, we check that subexponential growth is an SBE-invariant. 
The central part of the paper addresses nilpotent groups. While classification up to sublinearly bilipschitz equivalence is known in this case as a consequence of Pansu's theorems, its quantitative version is not. We introduce a computable algebraic invariant $e=e_G<1$ for every such group $G$, and check that $G$ is $O(r^e)$-bilipschitz equivalent to its associated Carnot group. Here $r\mapsto r^e$ is a quantitive sublinear bound. 
Finally, we define the notion of large-scale contractable and large-scale homothetic metric spaces. We check that these notions imply polynomial growth under general hypotheses, and formulate conjectures about groups with these properties.
Comparison of model prediction with measurements of galactic background noise at L-band
The spectral window at L-band (1.413 GHz) is important for passive remote sensing of surface parameters such as soil moisture and sea surface salinity that are needed to understand the hydrological cycle and ocean circulation. Radiation from celestial sources (mostly galactic) is strong in this window, and an accurate accounting of this background radiation is often needed for calibration. This paper presents a comparison of the background radiation predicted by a model developed from modern radio astronomy measurements with measurements made with several modern L-band remote sensing radiometers. The comparison validates the model and illustrates the magnitude of the correction necessary in remote sensing applications.