Coarse embeddings

Definitions and mathematical statement

A map $f:(X,d_X)\to(Y,d_Y)$ is a coarse embedding if there exist nondecreasing finite-valued control functions $\rho_-,\rho_+:[0,\infty)\to[0,\infty)$ with $\rho_-(t)\to\infty$ as $t\to\infty$ and \[ \rho_-(d_X(x,x'))\le d_Y(f(x),f(x')) \le\rho_+(d_X(x,x'))\qquad(x,x'\in X). \] No continuity or small-scale injectivity is required by this large-scale definition.

The question is to find further numerical assignments $I$ on metric spaces, or on a clearly specified class of them, that are nondecreasing under coarse embeddings: \[ X\text{ coarsely embeds into }Y\quad\Longrightarrow\quad I(X)\le I(Y). \] The target may be an ordered numerical range, such as $[0,\infty]$, provided its meaning is specified. Such an assignment must in particular take equal values when coarse embeddings exist in both directions.

The word "more" refers to an unstated baseline of known examples; the source does not prescribe a list to extend, a completeness criterion, a minimum distinguishing power, or restrictions on the metric spaces. This is a request to develop useful monotone invariants, not a conjecture that a particular proposed formula is monotone or a demand to classify every possible invariant.

Short English statement

Find useful numerical measures of large-scale complexity that cannot decrease when a space is coarsely embedded into another. The source leaves the space class and desired new distinguishing power open.

The supplied summary mentions asymptotic dimension, coarse growth/entropy, asymptotic-dimension growth, expander-type ideas, and Pansu's work as related context. It gives no formulas or hypotheses showing that every listed quantity is monotone for all metric spaces; growth and entropy can require additional structure and normalization. The OPEN-TRIAGE label describes an ongoing research direction rather than a single unresolved theorem.

Sources

• Local UnsolvedMath registry, permanent ID 6800012, source code catalogue entry, "Coarse embeddings," proposed by Dominique Hulin and Pierre Pansu, Paris-Sud (Paris-Saclay), 2018. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Morgan and Pansu, A List of Open Problems in Differential Geometry (2018), Question environment 12, https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex; source-TeX extraction, accessed 2026-07-29.
• The supplied summary's list of related invariant themes and Pansu's work has no complete primary references or exact monotonicity statements.
• Source-specific provenance: the item is labelled a question, current status NEEDS_REVIEW; the public source TeX is on a coauthor's publication page, with redistribution terms NEEDS_REVIEW, separately from the dataset licence.