Boundedness of the contactomorphism group for closed contact manifolds

Determine whether, for every closed contact manifold (M,\xi), the contactomorphism group Cont(M,\xi) is bounded with respect to conjugation-invariant norms, i.e., whether Cont(M,\xi) admits no unbounded conjugation-invariant norm. This parallels the known boundedness of the identity component of the diffeomorphism group of a closed manifold.

Background

The paper constructs a stably unbounded conjugation-invariant pseudo-norm on the universal cover \widetilde{Cont}(M,\xi) under periodic Reeb flow, raising natural questions about the base group Cont(M,\xi). In contrast, it is known that the identity component of the diffeomorphism group of a closed manifold is bounded (BIP, Tsuboi), i.e., admits no unbounded conjugation-invariant norms.

Whether an analogous boundedness property holds for contactomorphism groups remains unresolved in general and has implications for large-scale geometry and dynamics in contact topology.