Gromov’s Polynomial Growth Theorem [Gro81] states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups. Gromov’s theorem inspired the more general problem (see, e.g. [GdlH91]) of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure—in short, are solvable...

‎given an integer $n$‎, ‎we denote by $mathfrak b_n$ and $mathfrak c_n$ the classes of all groups $g$ for which the map $phi_{n}:gmapsto g^n$ is a monomorphism and an epimorphism of $g$‎, ‎respectively‎. ‎in this paper we give a characterization for groups in $mathfrak b_n$ and for groups in $mathfrak c_n$‎. ‎we also obtain an arithmetic description of the set of all integers $n$ such that a gr...

It has long been known that every quasi-homogeneous normal complex surface singularity with Q–homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called “splice type singularities,” which generalize Brieskorn complete intersections....

Abelian family symmetries provide a predictive framework for neutrino mass models. In seesaw models based on an abelian family symmetry, the structures of the Dirac and the Majorana matrices are derived from the symmetry, and the neutrino masses and mixing angles are determined by the lepton charges under the family symmetry. Such models can lead to mass degeneracies and large mixing angles as ...

In 1967 Gurevich [3] published a proof that the class of divisible Axchimedean lat t ice-ordered abelian groups such that the lattice of carriers is an atomic Boolean algebra has a hereditarily undecidable first-order theory. (He essentially showed the reduct of this class to lattices has a hereditarily undecidable first-order theory: on p. 49 of his paper change z ~ u + v to z ~ u v v in the d...

We investigate the relationship between the geometric Bieri–Neumann– Strebel–Renz invariants of a space (or of a group), and the jump loci for homology with coefficients in rank 1 local systems over a field. We give computable upper bounds for the geometric invariants, in terms of the exponential tangent cones to the jump loci over the complex numbers. Under suitable hypotheses, these bounds ca...