Boundaries of Groups and Kleinian Groups — Problem 51

Definitions and mathematical statement

A map $f:X\to Y$ is a quasi-isometric embedding if there are constants $L\geq1$, $A\geq0$ such that \[ L^{-1}d_X(x,x')-A\leq d_Y(f(x),f(x'))\leq Ld_X(x,x')+A \qquad(x,x'\in X). \] No coarse-surjectivity condition is imposed. The source asks whether all such embeddings between higher-rank symmetric spaces are "either isometries or algebraic in this way".

A common intended setting would use Riemannian symmetric spaces of noncompact type and rank at least two, but the source does not specify irreducibility, rank-one or Euclidean factors, relative ranks, or metric normalizations. More importantly, "in this way" refers to a preceding algebraic construction not included in the record. No class of algebraic maps is defined.

Rigidity conclusions also usually distinguish equality from finite distance: being at bounded distance from a model $F$ means $\sup_{x\in X}d_Y(f(x),F(x))<\infty$. Quasi-isometric embeddings are unchanged by bounded perturbations, so literal pointwise identification with an isometry or a rigid algebraic map is much stronger than coarse identification. The record does not choose this equivalence or allow rescaling of factors. Consequently the precise conclusion and its hypotheses cannot be recovered from the fragment; no theorem classifying all embeddings is asserted.

Short English statement

Are quasi-isometric embeddings of higher-rank symmetric spaces forced to follow specified algebraic models, perhaps up to bounded distance? The source omits the model construction, metric conventions, and intended equivalence.

Sources

• Local UnsolvedMath registry, permanent ID 6200051, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 51"; proposed by David Fisher, 2005. Registry status partially solved, classification PARTIAL-PROGRESS.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 51, PDF page 15, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. The construction referenced by "in this way" is absent; no PDF was accessed here.
• The supplied summary names Eskin--Fisher--Whyte and Kleiner--Leeb for related rigidity results and distinguishes quasi-isometries from embeddings. It provides no precise titles, theorems, or applicable hypotheses, and calls the embedding question partial/open. These statements are retained as supplied contextual assessment.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the document is publicly author-hosted.