Strengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

The purpose of the present expository paper is to give an account of the recent progress and present status of the classification of solvable Lie groups admitting an Einstein left invariant Riemannian metric, the only known examples so far of noncompact Einstein homogeneous manifolds. The problem turns to be equivalent to the classification of Ricci soliton left invariant metrics on nilpotent L...

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R o S of 3D-positions and orientations naturally embedded in the group SE(3) of 3Drigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossingpreserving fiber enhancement on probability densities defined on...

We explore the cosmological viability of a theory gravity defined by Lagrangian $f(\mathcal{R})=\mathcal{R}^{n\left(\mathcal{R}\right)}$ in Palatini formalism, where $n\left(\mathcal{R}\right)$ is dimensionless function scalar curvature $\mathcal{R}$ that interpolates between general relativity when $n\left(\mathcal{R}\right)=1$ and locally scale-invariant superficially renormalizable $n\left(\...