= Rank-one convergence of divergent Möbius transformations
A divergent sequence of <Möbius transformations> has a <subsequence> converging uniformly on compact subsets of the <Riemann sphere> outside one exceptional point to a constant. Represent the transformations by determinant-one $2\times2$ <matrices> and divide each by its matrix norm. A convergent subsequence has a nonzero rank-one limit: its determinant tends to zero because the original norms tend to infinity. The kernel line is the exceptional point, and the image line is the constant limit. This converts the <limit set of a Kleinian group> into a tractable boundary dynamics problem.
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