In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to R⋉Rn where the semidirect product is defined by a diagonalizable matrix of determinant one with ...

Bielliptic and quasi-bielliptic surfaces form one of the four classes minimal smooth projective Kodaira dimension $0$. In this article, we determine automorphism schemes these over algebraically closed fields arbitrary characteristic, generalizing work Bennett Miranda complex numbers; also find some cases that are missing from classification groups bielliptic in characteristic

Quantum theory of self-organization based on the idea that quantum jump serves as the basic step of self-organization, is represented. The notion of self and the identification of self as the fundamental statistical ensemble gives totally new meaning for the concept of selforganization as a generation of hierarchies of selves. Zero modes of the configuration space geometry, whose existence deri...

Introduction Gromov's Polynomial Growth Theorem [Gro81] characterizes the class of virtually nilpotent groups by their asymptotic geometry. Since Gromov's theorem it has been a major open question (see, e.g. [GH91]) to find an appropriate generalization for solvable groups. This paper gives the first step in that direction. One fundamental class of examples of finitely-generated solvable groups...