Existence of measurable embeddings between distinct mapping class groups

Determine whether measurable embeddings exist between mapping class groups associated with distinct connected orientable finite-type surfaces, beyond the established necessary condition that the source surface’s topological complexity is at most that of the target surface.

Background

The main theorem proves rigidity for self-measurable embeddings of a mapping class group associated with a surface of complexity at least two: every such embedding is a measure equivalence. The paper also proves monotonicity of topological complexity for measurable embeddings between mapping class groups of potentially different surfaces.

Despite this necessary monotonicity condition, the general existence problem for measurable embeddings between different mapping class groups remains unresolved. The paper explicitly identifies this as an unknown question rather than resolving it.

References

It is unknown in general whether measurable embeddings between two different mapping class groups exist, although we do know the following monotone invariance.

— Self measurable embeddings of mapping class groups  (2610.12054 - Gurieva, 8 Oct 2026) in Introduction, paragraph beginning “It is unknown in general whether measurable embeddings between two different mapping class groups exist”