21 research outputs found
On the number of lines in the limit set for discrete subgroups of
Given a discret subgroup \Gamma\subset PSL(3,\C), we determine the number
of complex lines and complex lines in general position lying in the complement
of: maximal regions on which acts properly discontinuously, the
Kularni's limit set of and the equicontinuity set of . We also
provide sufficient conditions to ensure that the equicontinuity region agrees
with the Kulkarni's discontinuity region and is the largest set where the group
acts properly discontinuously and we provide a description of he respective
limit set in terms of the elements of the group
Egerv\'ary's theorems for harmonic trinomials
In this manuscript, we study the arrangements of the roots in the complex
plane for the lacunary harmonic polynomials called harmonic trinomials. We
provide necessary and sufficient conditions so that two general harmonic
trinomials have the same set of roots up to a rotation around the origin in the
complex plane, a reflection over the real axis, or a composition of the
previous both transformations. This extends the results of J. Egerv\'ary 1930
for the setting of trinomials to the setting of harmonic trinomials.Comment: 17 page
The stability region for Schur stable trinomials with general complex coefficients
In this paper, we characterize the stability region for trinomials of the
form , , where ,
and are non-zero complex numbers and with .
More precisely, we provide necessary and sufficient conditions on the
coefficients , and in order that all the roots of the trinomial
belongs to the open unit disc in the complex plane. The proof is based on
Bohl's Theorem introduced in 1908.Comment: 20 pages and 1 figur
On the number of roots for harmonic trinomials
In this manuscript we study the counting problem for harmonic trinomials of the form a zeta(n) + b (zeta) over bar (m) + c, where n, m is an element of N, n > m, and a, b and c are non-zero complex numbers. As a consequence, we obtain the Fundamental Theorem of Algebra and the Wilmshurst conjecture for harmonic trinomials. The proof of the counting problem relies on the Bohl method introduced in [1]. (C) 2022 The Author(s). Published by Elsevier Inc.Peer reviewe
Purely parabolic discrete subgroups of \PSL(3, \Bbb{C})
While for \PSL(2,\C) every purely parabolic subgroup is Abelian and acts on
with limit set a single point, the case of \PSL(3,\C) is far more
subtle and intriguing. We show that there are twelve classes of purely
parabolic discrete groups in \PSL(3,\C), and we classify them. We use first
the Tits Alternative and Borel's fixed point theorem to show that all purely
parabolic discrete groups in \PSL(3,\C) are virtually triangularizable; this
extends a Theorem by Lie-Kolchin. Then we prove that purely parabolic groups in
\PSL(3,\C) are virtually solvable and cocyclic, hence finitely presented. We
then prove a Tits-inspired alternative for these groups: they are either
virtually unipotent or else Abelian of rank 2 and of a very special type. All
the virtually unipotent ones turn out to be conjugate to subgroups of the
Heisenberg group {\rm Heis}(3,\C). We classify these using the obstructor
dimension introduced by Bestvina, Kapovich and Kleiner.
We find that their Kulkarni limit set is either a projective line, a cone of
lines with base an Euclidean circle, or else the whole . We determine its
relation with the Conze-Givarc'h limit set of the action on the dual projective
space \dP^2. We show that in all cases the Kulkarni region of discontinuity
is the largest open set where the group acts properly discontinuously, and that
in all cases but one, this set coincides with the equicontinuity region.Comment: Minor chances in the new versio
Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds
In this article we introduce a notion of normalized angle for Lorentzian
pre-length spaces. This concept allows us to prove some equivalences to the
definition of timelike curvature bounds from below for Lorentzian pre-length
spaces. Specifically, we establish some comparison theorems known as the local
Lorentzian version of the Toponogov theorem and the Alexandrov convexity
property. Finally, as an application we obtain a first variation Formula for
non-negatively curved globally hyperbolic Lorentzian length spaces
Elementary groups in \PSL(3,\C)
In this paper, we give a classification of the subgroups of that act on in such a way that their
Kulkarni limit set has finitely many lines in general position lines. These are
the elementary groups