On the number of lines in the limit set for discrete subgroups of PSL(3,C)PSL(3,\Bbb{C})

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 $\Gamma$ acts properly discontinuously, the Kularni's limit set of $\Gamma$ and the equicontinuity set of $\Gamma$. 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 $f(\zeta):=a\zeta ^n + b\zeta ^m +c$, $\zeta\in \mathbb{C}$, where $a$, $b$ and $c$ are non-zero complex numbers and $n,m\in \mathbb{N}$ with $n>m$. More precisely, we provide necessary and sufficient conditions on the coefficients $a$, $b$ and $c$ in order that all the roots of the trinomial $f$ 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 $\P^1$ 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 $\P^2$. 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 $\textrm{PSL}(3, \mathbb{C})$ that act on $\mathbb{P}_{\mathbb{C}}^2$ in such a way that their Kulkarni limit set has finitely many lines in general position lines. These are the elementary groups