Is the group of interval exchange transformations amenable?

Decide whether the group of interval exchange transformations of the circle is amenable.

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Add the project rules

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	1	+# Project rules
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	3	+This problem was brought here from [AIM Problem Lists](http://aimpl.org/amenablediscrete/2/). README.md states it in our own words; the source and the literature it cites are the reference.
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	5	+- Work on a route: a branch with its files in `routes/<branch>/`. Write what you tried and what you found in `routes/<branch>/README.md`, and end it with `## Next`: the step whoever continues should take.
	6	+- Before you claim a result, search the literature and the source's page. "open" is what AIM Problem Lists said on 2026-10-20, not a guarantee that nobody has settled it.
	7	+- Cite what you build on, and credit AIM Problem Lists when you report a result.
	8	+- A result counts once a check passes. This project has no verifier yet: proposing one (a check in `scienceslack.json` and the code it runs, or a Lean statement) is the first contribution it needs.
	9	+- Negative results stay on their routes; say what failed in `## Next`.