The plot of the movie Eastern Promises depends on Russian gangsters and prostitution things.
I didn't find the word "promise" in any expression of the movie's characters.
Why is it called "Eastern Promises"?
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I don't think there's definitive word; this blog post speculates:
The title could cover a dozen meanings, one being the promises told to poor young Eastern girls about a better life in the West.
This is somewhat corroborated by this interview with Cronenberg:
I remember when I first picked up the script and saw the title, Eastern Promises. I thought, this sounds like a cheap perfume, especially since in North America, “Eastern” doesn’t mean Russia, whereas in England it does.