Theorem. Orbit and displacement criterion for ICC [connes-0006]

✍️source

isICC_of_product_quotientpaper_gammaOne_iccpaper_gammaTwo_icc

L∃∀NAGENTDRAFTED

For a countable group, the ICC condition asks that every nonidentity conjugacy class be infinite. In a semidirect product \(N\rtimes (S\times Q)\), the proof can be separated from the concrete formulas by supplying two kinds of certificates:

1. every nonidentity \(a\in N\) has an infinite \(S\)-orbit; and
2. every nonidentity \(q\in Q\) admits \(a\in N\) whose displacement \(a(q\cdot a)^{-1}\) is nonidentity and has an infinite \(S\)-orbit.

The structure Connes.PaperICC.ActionData is exactly this direct-consumer boundary. The generic theorem Connes.PaperICC.isICC_of_product_quotient then proves that these certificates imply ICC by splitting a nonidentity element into three exhaustive cases.

• A nontrivial \(S\)-coordinate inherits an infinite conjugacy class from \(S\).
• A pure kernel element uses the first certificate.
• A surviving \(Q\)-coordinate reduces to the second certificate.

This organization follows the cases used in [zhou2026icc, Section 5] without building the concrete tensors into the reusable ICC theorem.