Hofer Norms on Fundamental Groups of Symplectic Manifolds

Definitions and mathematical statement

Let $(M,\omega)$ be a symplectic manifold, and let $B$ be the ball mentioned in the supplied source record. Choose a base point $x_0$ in the component under consideration and write $G=\pi_1(M,x_0)$, with identity element $e$. The source denotes an induced Hofer norm by \[ N_B(\gamma)=\lVert\gamma\rVert_B,\qquad \gamma\in G, \] and describes it as conjugation invariant. The stated description as a conjugation-invariant norm entails \[ \begin{gathered} N_B(e)=0,\qquad N_B(\gamma)>0\quad(\gamma\ne e),\\ N_B(\gamma^{-1})=N_B(\gamma),\qquad N_B(\gamma\delta)\le N_B(\gamma)+N_B(\delta),\\ N_B(\eta\gamma\eta^{-1})=N_B(\gamma) \qquad(\gamma,\delta,\eta\in G). \end{gathered} These are an unpacking of the source terminology, not a verification that its unspecified construction has these properties. The supplied record does not define the construction of $N_B$, the type or size of $B$, or the permitted Hamiltonian motions and support conditions. Consequently, the following formulation is conditional on specifying a construction that defines a finite norm on all of $G$. For the settings intended by the source, determine when this norm is bounded, meaning \[ \exists K=K(M,\omega,B)<\infty\quad \forall\gamma\in G,\qquad N_B(\gamma)\le K. \] Also determine whether it is equivalent to word length. To express one possible precise meaning of the latter question, if $G$ has a finite symmetric generating set $S$, define \[ \ell_S(\gamma)=\min\{k\in\mathbb Z_{\ge0}:\gamma=s_1\cdots s_k,\ s_i\in S\}, \] where the empty product is $e$. Multiplicative equivalence would mean \[ \exists c,C>0\quad\forall\gamma\in G,\qquad c\ell_S(\gamma)\le N_B(\gamma)\le C\ell_S(\gamma). \] The source does not specify a generating set, assume finite generation, or state whether equivalence instead permits additive constants. Thus this displayed comparison is a possible interpretation, not an asserted resolution of the intended meaning.

The supplied dated assessment labels the problem partially solved: it reports boundedness on the two-torus and unboundedness on an annulus when the fixed ball is sufficiently large, while describing general boundedness and word-length equivalence as unresolved. The relevant manifold, ball-size, and support hypotheses for these reported results are absent from the record; these reports must therefore retain those unresolved qualifications.

Short English statement

The question asks when the Hofer norm associated with a fixed ball on a symplectic manifold's fundamental group is bounded, and how it compares with word length. The construction of the norm and the comparison convention need clarification. Supplied dated curation reports partial results for the two-torus and annuli; this draft does not independently verify those results or the current status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002137$, titled Hofer Norms on Fundamental Groups of Symplectic Manifolds; exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is specified; the proposed year is recorded as 2012.
• Supplied report citation: Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2013), Oberwolfach Reports, pp. 2139--2203, DOI 10.4171/owr/2012/35. https://doi.org/10.4171/owr/2012/35. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's year 2013 and the record's 2012 identifiers are retained as supplied.
• Supplied literature source: Khanevsky, Oberwolfach lecture report. https://oa.tib.eu/renate/bitstreams/2d68ade4-b554-459d-9f08-bb2793165aa0/download.
• Supplied literature source: Khanevsky, Hofer norm and disk translations. https://arxiv.org/abs/1111.1923.
• Supplied literature source: Oberwolfach Report 35. https://doi.org/10.4171/owr/2012/35.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem partially solved, reports boundedness on the two-torus and unboundedness on an annulus when the fixed ball is sufficiently large, and describes general boundedness and word-length equivalence as unresolved. The background explicitly calls this dated literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not independent verification performed for this draft.