TEST MAPS AND DISCRETE GROUPS IN SL(2, ℂ) II

In this paper we present a new discreteness criterion for a non-elementary subgroup G of SL(2, ℂ) containing elliptic elements by using a loxodromic (resp. an elliptic) transformation as a test map that need not be in G.

Geometry and dynamics of admissible metrics in measure spaces

We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

Magnetic Schrödinger Operators with Discrete Spectra on Non-Compact Kähler Manifolds

We identify a class of magnetic Schrödinger operators on Käler manifolds which exhibit pure point spectrum. To this end we embed the Schröinger problem into a Dirac-type problem via a parallel spinor and use a Bochner-Weitzenböck argument to prove our spectral discreteness criterion

Discreteness criteria for subgroups in SL(2, C)

In this paper a new discreteness criterion for subgroups in SL(2, C) is established, which implies all known discreteness criteria by two-generator subgroup and is the best possible. And it is proved that every non-elementary and non-discrete subgroup of SL(2, C) has a non-elementary and non-discrete subgroup generated by two loxodromic elements.