Typical Properties of Marked Grigorchuk Groups

Definitions and mathematical statement

Let $G_p$ denote the compact family of marked Grigorchuk groups described in the supplied record as parametrized by \[ \Omega_p=\{0,1,\ldots,p\}^{{\mathbb{N}}}. \] The admissible values of $p$, the construction of the groups and their markings, and the parametrization $\Omega_p\to G_p$ are not supplied. Thus $G_p$ is retained as a source-dependent family, rather than given an independently complete construction here.

For a group-theoretic property $\mathcal P$, let \[ X_{\mathcal P}=\{(G,S)\in G_p:G\text{ has property }\mathcal P\}. \] Here $S$ denotes the finite generating marking. For a probability measure $\mu$ on $G_p$, call $\mathcal P$ $\mu$-typical when $X_{\mathcal P}$ is $\mu$-measurable and \[ \mu(X_{\mathcal P})=1. \] Call $\mathcal P$ Baire-typical when $G_p\setminus X_{\mathcal P}$ is meagre in the topology intended on $G_p$. A meagre set is a countable union of nowhere dense sets; a set is nowhere dense when its closure has empty interior.

To express the growth aspect, for $(G,S)\in G_p$ and $n\in{\mathbb{Z}}_{\geq0}$, define \[ B_S(n)=\{g\in G:g\text{ is represented by a word of length at most }n \text{ in }S\cup S^{-1}\},\qquad \gamma_{G,S}(n)=|B_S(n)|. \] In particular, $B_S(0)=\{e_G\}$.

Problem as supplied. Determine which group-theoretic properties of members of $G_p$, especially properties of the asymptotic growth of $\gamma_{G,S}$, are typical in the invariant-measure sense and which are Baire-typical.

This is a research question, not a claim that every property is typical in either sense. A fully specified measure-theoretic formulation requires the intended invariant probability measure or class of measures, the action or transformation under which invariance is required, and whether the assertion is to hold for one, each, or almost every measure in that class. These are not specified. The topology on $G_p$, the precise family construction, the range of $p$, and the intended comparison of growth rates also require confirmation. No particular choices for these missing data are asserted here.

Short English statement

For the supplied compact family $G_p$ of marked Grigorchuk groups, identify the group properties, especially growth behavior, that hold with probability one for the intended invariant measures and those that hold on a comeagre set. The family construction, parameter range, topology, invariant measures, and measure quantifiers are insufficiently specified for a unique fully formal statement.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002391$, title Typical Properties of Marked Grigorchuk Groups, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. The record gives proposed year 2013 and identifies no proposer.
• Supplied report citation: Group Theory, Measure, and Asymptotic Invariants (2014), Oberwolfach Reports, pp. 2375--2422, DOI 10.4171/owr/2013/42. Source URL and supplied literature source labelled "original OWR report": https://doi.org/10.4171/owr/2013/42. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No specific page, contribution, or individual author is supplied for this question.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem "partially solved" and states that the report establishes measure-typical and Baire-typical growth results in $G_p$, while the broader question about all typical group-theoretic properties is not exhausted. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and supplies a claim of manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. That supplied review label does not resolve the missing mathematical specifications identified in this draft.