Complex-Hyperbolic Lattices in Hamiltonian Diffeomorphism Groups

Definitions and mathematical statement

Hamiltonian embedding question. If $G$ is a lattice in $\mathrm{PU}(n,1)$, $n\ge2$, does there exist a symplectic manifold $(M,\omega)$ and an injective homomorphism\[\rho:G\hookrightarrow\operatorname{Ham}(M,\omega),\] possibly with interesting dynamical properties? A lattice is a discrete subgroup of finite covolume in the ambient Lie group, the projective unitary isometry group of complex hyperbolic $n$-space. The source does not impose torsion-freeness or cocompactness. A monomorphism requires a faithful action: no nonidentity element may map to the identity diffeomorphism. The manifold and its dimension may depend on $G$ unless an omitted uniformity condition says otherwise.

A symplectic manifold has a closed nondegenerate two-form $\omega$. On a closed manifold, $\operatorname{Ham}(M,\omega)$ consists of time-one maps of Hamiltonian isotopies generated by smooth time-dependent functions $H_t$, with $\iota_{X_{H_t}}\omega=-dH_t$ as a fixed sign convention. It is not silently replaced by the full symplectomorphism group. On a noncompact manifold, compactly supported Hamiltonian maps and time-one maps of more general complete Hamiltonian flows give different target groups. The source's phrase some symplectic manifold does not decide compactness, connectedness, boundary, support, or completeness conventions. Those are substantial existence restrictions, not cosmetic choices; unrestricted noncompact linear Hamiltonian settings must not automatically be assigned the same status as closed or compactly supported targets.

The clause possibly with interesting dynamical properties specifies no mathematical property, so the core question is the faithful embedding, with an optional refinement whose conditions remain to be supplied. No ergodicity, fixed-point behavior, entropy, or other dynamical requirement is invented. Nor is a continuous action of the entire Lie group required by a question about its discrete lattice.

The supplied triage checked 2026-08-21 labels the problem open and says no general embedding with the requested dynamical properties is known. That assessment treats dynamics as a requested condition whereas the statement makes it optional. Its meaning also depends on the omitted compactness and support conventions. The dated assessment is retained as supplied provenance rather than independently applied to every literal choice of target group.

Short English statement

Ask whether each complex-hyperbolic lattice in the stated dimensions acts faithfully by Hamiltonian diffeomorphisms on some symplectic manifold. Closed versus noncompact targets and support conventions are missing. The optional phrase about interesting dynamics is not a defined additional requirement.

Sources

• Local UnsolvedMath registry, permanent ID 30000545, code catalogue entry; Complex-Hyperbolic Lattices in Hamiltonian Diffeomorphism Groups, Oberwolfach Reports Open Problems.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/33, pp. 1991--2058. https://doi.org/10.4171/owr/2006/33. Proposed year and year tag: 2006; no proposer named.
• Supplied literature label OWR Geometric Group Theory report: https://doi.org/10.4171/owr/2006/33. No separate authoritative embedding theorem or precise dynamical condition is supplied.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field claims editorial and notation cleanup only. Verify dated claims independently.