A trace inequality for loxodromics of length ≤ 2 in free kleinian groups

Let Γ = ⟨𝜉,𝜂⟩ denote a purely loxodromic free Kleinian group. For any 𝛾 ∈Γ and any 𝑧 in the hyperbolic 3-space H3, let dist(𝑧,𝛾(𝑧))denote the hyperbolic distance between 𝑧 and the image 𝛾(𝑧) of 𝑧 under 𝛾. Let Ψ2 = {𝛾 ∈Γ : ℓ(𝛾) ≤2}, where ℓ(𝛾) denotes the length of 𝛾 as a reduced word in Γ. Firstly, in this thesis, it is proved that max{dist(𝑧,𝛾(𝑧)) : 𝛾 ∈Ψ𝑐6}≥3.36611... for any 𝑧∈H^3 forΨ_6^𝑐 = {𝜙𝜓𝜙−1 : 𝜙,𝜓 ∈Ψ_2}. Secondly, as a consequence, it is shown that a version of the Jørgensen's inequality expressed as |trace^2(𝜙_0)−4|+|trace2(𝜙_0^(-1)𝜓_0𝜙_0^(−1)𝜓_0^(−1))−2|≥13.5 is satisfied for some distinct 𝜙_0,𝜓_0 ∈Ψ^2 assuming the inequalities dist(𝑧_2,𝜙_0𝜓_0𝜙_0^(−1)(𝑧_2)) ≤dist(𝑧_1,𝜙_0𝜓_0𝜙_0^(−1)(𝑧1)) and dist(𝑧_2,𝛾(𝑧_2)) <3.36611... holdforevery𝛾 ∈Ψ𝑐6 −{𝜙0,𝜙−10 ,𝜓−10 𝜙0𝜓0,𝜓−10 𝜙−10 𝜓0,𝜓0𝜙0𝜓−10 ,𝜓0𝜙−10 𝜓−10 }. Above, 𝑧_1 and 𝑧_2 are the midpoints of the shortest geodesic segments connecting the axis of 𝜙_0 to the axes of 𝜓_0𝜙_0𝜓_0^(−1) and 𝜓_0^(−1)𝜙_0𝜓_0, respectively.