Approximately Free Finitely Generated Groups

Definitions and mathematical statement

A group $G$ is finitely generated if $G=\langle g_1,\ldots,g_d\rangle$ for some finite list of elements. A free group on a set $B$ is a group with the universal property that every map from $B$ to an arbitrary group extends uniquely to a homomorphism. The source asks: what natural class of finitely generated groups should be regarded as approximately free groups?

This is a request to identify a useful concept or class, not a conjecture about a defined predicate. The word "approximately" is not assigned a metric, topology, family of tests, logical fragment or homomorphism condition. Nor are any axioms given for "natural." Consequently there is no specified class $\mathcal C$ for which one can state or verify a membership characterization.

The supplied literature assessment mentions limit groups and elementary free groups as well-studied related notions. These names must not be treated as a source-selected answer. For example, two distinct possible comparison frameworks are finite-set separation by homomorphisms to free groups and agreement in first-order group theory. The former might ask that, for every finite subset $S\subseteq G$, there exist a free group $F$ and a homomorphism $G\to F$ injective on $S$; this spells out a fully residually free type of condition. The latter might ask that $G$ satisfy the same first-order sentences as a specified free group. Neither condition is imposed by the question, and no equivalence, classification or choice between them is asserted. Local or geometric notions of approximation would require still different data.

A definite mathematical reformulation therefore needs a selected approximation criterion, its quantifiers and allowed free groups, and the structural or algorithmic purpose that the proposed class is to serve. The finitely generated hypothesis is explicit and is retained throughout; it is not replaced by finite presentation.

The literature triage checked 2026-08-21 says that no unique settled classification answers the broad question as written, despite the developed theories it mentions. Its open label concerns this curation of a conceptual question, not an assertion that all the related well-defined classes are unclassified. The verification field reports confirmation without textual change.

Short English statement

The source asks which finitely generated groups ought to count as approximately free, without defining approximation or naturalness. Related homomorphism and logical notions illustrate the choices but are not selected as the answer. A precise classification problem requires one such criterion first.

Sources

• Local UnsolvedMath registry, permanent ID 30000328, source code catalogue entry, "Approximately Free Finitely Generated Groups," including the unchanged original question and exact-statement verification.
• Low-Dimensional Manifolds (2006), Oberwolfach Reports, pp. 2519--2570, DOI 10.4171/owr/2005/45: https://doi.org/10.4171/owr/2005/45. This is the sole supplied literature reference, labelled "original Oberwolfach report." The citation uses 2006, while the DOI and proposed-year metadata use 2005.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.