Asymptotic Growth Under Group Automorphisms

Definitions and mathematical statement

Let $G$ be a finitely generated group, $\alpha\in\operatorname{Aut}(G)$, and $g\in G$. Define $g_n=\alpha^n(g)$ for integers $n\geq0$. The source asks how $g_n$ grows as $n\to\infty$, and whether this sequence has a limit in a suitable sense.

For example, after choosing a finite symmetric generating set $S$, word length is \[ |h|_S=\min\{r:h=s_1\cdots s_r,\ s_i\in S\}, \] with the empty product allowed. This would give the numerical sequence $|\alpha^n(g)|_S$, but the extraction does not say that word length, rather than conjugacy length or another quantity, is its intended growth observable. Nor does it specify a normalization of lengths or a topology or compactification in which the group elements should converge. The statement therefore remains a growth and convergence question requiring these choices. No assertion of convergence for all finitely generated groups, and no restriction to free or hyperbolic groups, is inserted.

Short English statement

Describe iterates of an element under a group automorphism and a suitable limiting behavior. The extraction supplies the group-theoretic data but no growth quantity, normalization, or convergence space.

Sources

• Local UnsolvedMath registry, permanent ID $30001031$, source code catalogue entry, "Asymptotic Growth Under Group Automorphisms." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, pp. 2337--2406, DOI https://doi.org/10.4171/owr/2008/42.
• Supplied literature triage, literature_checked_at: 2026-08-21: labeled open; says the growth question is incomplete, cannot be matched to a resolved theorem, and no authoritative resolution was found. This is supplied dated curation, not independently verified, and does not certify that every possible precise interpretation is open.