arXiv subjects

İlker S. Yüce

Publications and source records attributed to İlker S. Yüce.

4 recordsLinked to original sources

arXiv2023

Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities

In this paper, we prove a generalization of a discreteness criteria for a large class of subgroups of PSL$_2(\mathbb{C})$. In particular, we show that for a given finitely generated, purely loxodromic, free Kleinian group $Γ=\langleξ_1,ξ_2,\dots,ξ_n\rangle$ for $n\geq 2$, the inequality $$\left|\text{trace}^2(ξ_i)-4\right|+\left|\text{trace}(ξ_iξ_jξ_i^{-1}ξ_j^{-1})-2\right|\geq 2\sinh^2\left(\frac{1}{4}\logα_n\right)$$ holds for some $ξ_i$ and $ξ_j$ for $i\neq j$ in $Γ$ provided that certain conditions on the hyperbolic displacements given by $ξ_i$, $ξ_j$ and their length $3$ conjugates formed by the generators are satisfied. Above, the constant $α_n$ turns out to be the real root strictly larger than $(2n-1)^2$ of a fourth degree, integer coefficient polynomial obtained by solving a family of optimization problems via Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work.

math.GT↗

arXiv2017

Jorgensen's Inequality and Purely Loxodromic 2-Generator Free Kleinian Groups

Let $ξ$ and $η$ be two non--commuting isometries of the hyperbolic $3$--space $\mathbb{H}^3$ so that $Γ=\langleξ,η\rangle$ is a purely loxodromic free Kleinian group. For $γ\inΓ$ and $z\in\mathbb{H}^3$, let $d_γz$ denote the distance between $z$ and $γ\cdot z$. Let $z_1$ and $z_2$ be the mid-points of the shortest geodesic segments connecting the axes of $ξ$, $ηξη^{-1}$ and $η^{-1}ξη$, respectively. In this manuscript it is proved that if $d_γz_2<1.6068...$ for every $γ\in\{η, ξ^{-1}ηξ, ξηξ^{-1}\}$ and $d_{ηξη^{-1}}z_2\leq d_{ηξη^{-1}}z_1$, then \[ |\text{trace}^2(ξ)-4|+|\text{trace}(ξηξ^{-1}η^{-1})-2|\geq 2\sinh^2\left(\tfrac{1}{4}\logα\right) = 1.5937.... \] Above $α=24.8692...$ is the unique real root of the polynomial $21 x^4 - 496 x^3 - 654 x^2 + 24 x + 81$ that is greater than $9$. Also generalisations of this inequality for finitely generated purely loxodromic free Kleinian groups are conjectured.

math.GT↗

arXiv2015

Symmetric Decompositions of Free Kleinian Groups and Hyperbolic Displacements

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least $0.5\log\left(12\cdot 3^{k-1}-3\right)$ by one of the isometries of length at most $k\geq 2$ in a 2-generator Klenian group $Γ$ which is torsion-free, not co-compact and contains no parabolic. Also some lower bounds for the maximum of hyperbolic displacements given by symmetric subsets of isometries in purely loxodromic finitely generated free Kleinian groups are conjectured.

math.GT↗

arXiv2014

Two-Generator Free Kleinian Groups and Hyperbolic Displacements

The $\log 3$ Theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space is moved a distance at least $\log 3$ by one of the non-commuting isometries $ξ$ or $η$ provided that $ξ$ and $η$ generate a torsion-free, discrete group which is not co-compact and contains no parabolic. This theorem lies in the foundation of many techniques that provide lower estimates for the volumes of orientable, closed hyperbolic 3-manifolds whose fundamental group has no 2-generator subgroup of finite index and, as a consequence, gives insights into the topological properties of these manifolds. In this paper, we show that every point in the hyperbolic 3-space is moved a distance at least $(1/2)\log(5+3\sqrt{2})$ by one of the isometries in $\{ξ,η,ξη\}$ when $ξ$ and $η$ satisfy the conditions given in the $\log 3$ Theorem.

math.GT↗