It is in Chapter 10 that Sayuri is told that Hatsumomo ran another geisha out of the district.
It is Mameha who reveals this information to Sayuri. Sayuri tells Mameha that she cannot understand why Hatsumomo hates her so, and Mameha explains that the answer is simply because Hatsumomo cannot bear to have rivals. Mameha tells Sayuri the story of Hatsuoki, an attractive young lady who was trained by the same geisha that trained Hatsumomo. When Hatsuoki and Hatsumomo both became apprentices, Hatsumomo was consumed by jealousy for Hatsuoki, whom she looked upon as a rival. She began to spread a rumor around Gion "that Hatsuoki had been caught in a public alleyway one night doing something very improper with a young policeman". There was "no truth" to the story, and had Hatsumomo simply persisted in telling it, eventually no one would have believed her, so instead, whenever she "came upon someone very drunk...she whispered the story about Hatsuoki in such a way that the next day the person who'd heard it didn't remember that Hatsumomo had been the source". Soon, Hatsuoki's reputation "was so damaged, it was an easy matter for Hatsumomo to put a few more of her little tricks to use and drive her out (of the district)" (Chapter 10).