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Some trace inequalities for discrete groups of Möbius transformations


Author: Chun Cao
Journal: Proc. Amer. Math. Soc. 123 (1995), 3807-3815
MSC: Primary 30F40
DOI: https://doi.org/10.1090/S0002-9939-1995-1301013-6
MathSciNet review: 1301013
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Abstract: We show that if $ \langle A,B\rangle $ is discrete where A, $ B \in \mathrm{SL}(2, \mathbb{C})$ and if $ {\text{tr}}(AB{A^{ - 1}}{B^{ - 1}}) \ne 2,{\text{tr}}(ABA{B^{ - 1}}) \ne 2$, and $ {\text{\vert tr}^2}(A) - 4\vert \leq 2(\cos (2\pi /7) + \cos (\pi /7) - 1) = 1.0489 \ldots $ , then

$\displaystyle \vert{\text{tr}}(AB{A^{ - 1}}{B^{ - 1}}) - 2\vert \geq 2 - 2\cos (\pi /7) = 0.198 \ldots .$

It follows from above that if $ \langle X,Y\rangle $ is discrete with $ {\text{tr}}(X) = {\text{tr}}(Y) \ne 0$ and $ {\text{tr}}(XY{X^{ - 1}}{Y^{ - 1}}) \ne 2$, then

$\displaystyle \vert{\text{tr}}(XY{X^{ - 1}}{Y^{ - 1}}) - 2\vert \geq 2 - 2\cos (\pi /7) = 0.198 \ldots .$

Both inequalities are sharp.

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DOI: https://doi.org/10.1090/S0002-9939-1995-1301013-6
Article copyright: © Copyright 1995 American Mathematical Society

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