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The paper initiates a systematic study of Moebius structures and Ptolemyspaces. We conjecture that every compact Ptolemy space with circles and manyspace inversions is Moebius equivalent to the boundary at infinity of a rankone symmetric space of noncompact type. We prove this conjecture for the classof complex hyperbolic spaces as our main result.

Moebius structures and Ptolemy spaces: boundary at infinity of complex hyperbolic spaces