bibkey: ribeszalesskii2010profinite authors: Luis Ribes; Pavel Zalesskii year: 2010 title: Profinite Groups doi: 10.1007/978-3-642-01642-4 claim: Profinite completion is modeled by compatible finite quotient readings, and the canonical image of the original group is dense in that completion. strata_touched:

• D5/S1/Dynamics/ProfiniteIntegers license: citation-only triage: anchor

Profinite Groups

Ribes and Zalesskii’s monograph is a standard reference for profinite groups and their completions. This note anchors only the classical background used by D5/S1/Dynamics/ProfiniteIntegers: a profinite completion is assembled from compatible finite quotients, and the canonical group image is dense.

The repository theorem is more specific. It models the profinite integers by compatible readings in ZMod m and proves that the image of the natural numbers, not only the additive integers, is both injective and dense. Its finite-window proof chooses one representative at a common product modulus. That strengthening and its Lean proof are repository-derived; this note does not attribute them to a numbered result in the monograph.

Search log

• 2026-08-11: Searched the pinned Mathlib checkout for profinite completion, DenseRange, natural-number casts, and compatible residue families. ProfiniteGrp.ProfiniteCompletion.denseRange proves density of the canonical image for an arbitrary group. No natural-number-density theorem for the profinite completion of the integers was found.
• 2026-08-11: Queried Crossref by DOI. The returned metadata verified DOI 10.1007/978-3-642-01642-4, title Profinite Groups, authors Luis Ribes and Pavel Zalesskii, publisher Springer Berlin Heidelberg, and publication year 2010. The full text was not inspected, so no theorem number or page-level locator is claimed.

Verified locator

• DOI: https://doi.org/10.1007/978-3-642-01642-4