On two-generator discrete groups of Möbius transformations

Tan, De Lin

doi:10.1090/S0002-9939-1989-0969527-3

On two-generator discrete groups of Möbius transformations
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Proc. Amer. Math. Soc. 106 (1989), 763-770 Request permission

Abstract:

Assume that Möbius transformations $f$ and $g$ generate a discrete group. We obtain the following generalizations of Jørgensen’s inequalities. If ${\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) \ne 1$, then $\left | {{\text {t}}{{\text {r}}^2}(f) - 2} \right | + \left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right | \geq 1$. If ${\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1$, then either ${\text {t}}{{\text {r}}^2}(f) = 2{\text {ort}}{{\text {r}}^2}(f) = 1{\text {or}}\left | {{\text {t}}{{\text {r}}^2}(f) - 2} \right |\frac {1}{2}$ and $\left | {{\text {t}}{{\text {r}}^2}(f) - 1} \right | > \frac {1}{2}$. If ${\text {t}}{{\text {r}}^2}(f) \ne 1$, then $\left | {{\text {t}}{{\text {r}}^2}(f) - 1} \right | + \left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}})} \right | \geq 1$. If ${\text {t}}{{\text {r}}^2}(f) = 1$ then either ${\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 0{\text {ortr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1$ and $\left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right |\frac {1}{2}$.

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On inequalities for discrete groups of Möbius transformations