Some trace inequalities for discrete groups of Möbius transformations

Cao, Chun

doi:10.1090/S0002-9939-1995-1301013-6

Some trace inequalities for discrete groups of Möbius transformations
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Proc. Amer. Math. Soc. 123 (1995), 3807-3815 Request permission

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 ${\text {|tr}^2}(A) - 4| \leq 2(\cos (2\pi /7) + \cos (\pi /7) - 1) = 1.0489 \ldots$ , then \[ |{\text {tr}}(AB{A^{ - 1}}{B^{ - 1}}) - 2| \geq 2 - 2\cos (\pi /7) = 0.198 \ldots .\] 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 \[ |{\text {tr}}(XY{X^{ - 1}}{Y^{ - 1}}) - 2| \geq 2 - 2\cos (\pi /7) = 0.198 \ldots .\] Both inequalities are sharp.

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