Classifying and Ubiquitous Infinite Group Actions

Definitions and mathematical statement

The source introduces two properties for infinite finitely presented groups through an unspecified operation of transforming actions. A free $G$-set has trivial point stabilizers and is a disjoint union of regular orbits, each isomorphic to $G$ acting by left multiplication. Under the usual semifree convention, stabilizers are either trivial or all of $G$, so every orbit is regular or a fixed point. The source-specific action category and semifree convention require confirmation.

Write $X\Rightarrow_{\mathcal T}Y$ for the intended allowed transformation, which is not defined in the record. If $r(X)$ denotes the number of regular orbits, the classifying property has the pattern \[ X,Y\text{ free},\ r(X)\ne r(Y) \quad\Longrightarrow\quad \neg(X\Rightarrow_{\mathcal T}Y), \] for every allowed ordered pair of actions. Ubiquity has the pattern \[ \forall X\text{ semifree}\ \exists Y\text{ free}:\quad X\Rightarrow_{\mathcal T}Y. \] The source does not restrict orbit counts to finite cardinals or specify underlying-set cardinalities, topology, measures or allowed modifications. These are part of the missing transformation framework.

The two questions are whether an infinite finitely presented nonclassifying group exists, or at least a finitely generated one, and whether every finitely presented amenable group in the intended domain is ubiquitous. Finitely presented means admitting finite generating and relation sets; finitely generated drops the finite-relation requirement. Amenable means admitting an invariant mean on bounded functions. The weaker finitely generated question requires the properties/transformation relation to be defined outside their initially stated finitely presented domain; likewise the scope of infiniteness in the second question must be fixed.

"Transformed" cannot silently mean an ordinary equivariant bijection: such a bijection always preserves orbit counts and cannot remove fixed points, so it would trivialize one property and obstruct the stated ubiquitous behavior. Other equivalences, modifications, limits or action operations give different questions, and none is guessed from the terminology.

The literature triage checked 21 August 2026 reports no authoritative complete resolution found. It supplies no transformation definition or precise theorem beyond the report; that dated open assessment is retained as metadata, not an independent current-status conclusion.

Short English statement

Do nonclassifying finitely presented or finitely generated groups exist, and are finitely presented amenable groups ubiquitous? Both properties depend on an omitted notion of transforming free or semifree actions. Ordinary action isomorphism is not an adequate silent replacement, and the permitted cardinalities/category must be specified.

Sources

• Local UnsolvedMath registry, permanent ID 30000675, exact source code catalogue entry; Classifying and Ubiquitous Infinite Group Actions. Proposed year 2007, status open; corrected_verified statement retains both action-property definitions and both questions.
• Model Theory and Groups (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/02, pp. 83--138, https://doi.org/10.4171/owr/2007/02.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2007/02, and reports no authoritative complete resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no action-transformation definition or separate theorem is supplied.