Boundedness of Volume-Preserving Diffeomorphism Groups of Spheres

Definitions and mathematical statement

For each integer $n\ge3$, let $\mu$ be the intended smooth volume form on $S^n$. Consider the group of volume-preserving diffeomorphisms isotopic to the identity, conventionally written \[ G_n=\operatorname{Diff}_0(S^n,\mu). \] Volume preservation means $f^*\mu=\mu$. The source does not say whether the isotopy must stay volume preserving or only be an ambient diffeomorphism isotopy, nor does it state the differentiability/topology convention. The standard smooth identity-component interpretation must be checked.

Is $G_n$ bounded or unbounded for $n\ge3$? The meaning of boundedness is not defined in the supplied statement. In a common group-theoretic interpretation, a conjugation-invariant norm is a finite-valued function $\nu:G_n\to[0,\infty)$ satisfying \[ \nu(g)=0\Longleftrightarrow g=1,\quad \nu(g^{-1})=\nu(g),\quad \nu(gh)\le\nu(g)+\nu(h),\quad \nu(hgh^{-1})=\nu(g). \] Under that interpretation boundedness would mean \[ \forall\nu\text{ of this kind}:\quad\sup_{g\in G_n}\nu(g)<\infty, \] and unboundedness would mean existence of one such $\nu$ with infinite supremum. The bound may depend on $\nu$ and $n$; no single constant uniform over all norms or dimensions is required.

This norm-based interpretation is explanatory, not a supplied definition. If the report instead fixes a particular metric, restricts to continuous norms, or uses another boundedness property, the question changes. The source gives no such choice. The requested task should determine the behavior for each $n\ge3$, allowing any dimension dependence, rather than merely exhibit one dimension with a stated behavior.

Short English statement

The question asks whether the identity-isotopic volume-preserving diffeomorphism group of each sphere of dimension at least three is bounded. A usual interpretation uses all conjugation-invariant norms, but the source must specify boundedness and the isotopy convention.

Sources

• Local UnsolvedMath registry, permanent ID 30000541, code catalogue entry, "Boundedness of Volume-Preserving Diffeomorphism Groups of Spheres." Proposed year 2006; proposer unspecified.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2007), Oberwolfach Reports, pp. 1991--2058, DOI 10.4171/owr/2006/33; https://doi.org/10.4171/owr/2006/33. The displayed report year 2007 differs from the DOI and proposed year 2006.
• Statement status corrected_verified; the supplied note says editorial and LaTeX cleanup preserved meaning and added no source-dependent content.
• Literature triage checked 2026-08-21: open. It describes the boundedness question for volume-preserving diffeomorphisms of $S^n$, $n\ge3$, as unresolved in this generality, without specifying a norm or metric. Created 2026-08-25; updated 2026-09-16. The intended property and current status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.