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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2309.15716 |
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| _version_ | 1866910371359490048 |
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| author | Narman, A. Nedim Yüce, İlker S. |
| author_facet | Narman, A. Nedim Yüce, İlker S. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15716 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities Narman, A. Nedim Yüce, İlker S. Geometric Topology 30F40, 20F65, 46N10 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. |
| title | Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities |
| topic | Geometric Topology 30F40, 20F65, 46N10 |
| url | https://arxiv.org/abs/2309.15716 |