Boundaries of Groups and Kleinian Groups — Problem 70

Definitions and mathematical statement

Let $(X,d)$ be a proper metric space, meaning closed bounded subsets are compact, and choose $o\in X$. The normalized distance functions are \[ h_x(z)=d(z,x)-d(o,x). \] A usual horofunction compactification is obtained by taking the closure of these functions in the topology of uniform convergence on compact subsets; limit functions not representing points of $X$ form the horofunction boundary. The source proposes an incidence geometry at infinity built from horofunctions, halfspaces, and limiting objects called stars.

The definitions of those halfspaces and stars, their limit topology, and their incidence relation are not supplied. Write $\mathcal I(X)$ for this intended incidence structure, without inventing its missing operations. An incidence-preserving map must take incident objects to incident objects; whether it also reflects incidence or preserves object types is not specified.

Karlsson's problem asks for situations in which homomorphisms induce incidence-preserving maps between these structures at infinity, and asks the same for quasi-isometries. If the former means a group homomorphism $\varphi:G\to H$ between groups acting on proper metric spaces $X,Y$, the action hypotheses and the map relating $X$ to $Y$ need to be stated before there is an induced boundary map. A homomorphism alone is not such data in the extracted record.

A quasi-isometry $f:X\to Y$ has constants $\lambda\ge1$, $c\ge0$ with \[ \lambda^{-1}d_X(x,x')-c\le d_Y(fx,fx') \le\lambda d_X(x,x')+c, \] and image at uniformly bounded distance from every point of $Y$. The question is to determine hypotheses giving a well-defined induced map $\mathcal I(X)\to\mathcal I(Y)$ preserving the intended incidences, not to assume every quasi-isometry automatically has one.

The source identifies the CAT(0) case as closely connected to Tits geometry and suggests that the Teichmüller-space case should relate to the curve complex. The precise correspondences, metrics, and domains are not supplied, and the latter is explicitly an expected relation rather than a theorem stated here.

Short English statement

When do homomorphisms or quasi-isometries induce maps preserving the horofunction/halfspace/star geometry at infinity? The incidence construction and the needed action or map hypotheses are part of the missing source context.

The supplied summary calls this an open program, mentions Gromov-product/horo-space ideas and work of Caprace and Hume, and reports no full resolution. It gives no exact papers or correspondence theorem. The OPEN-TRIAGE label and 2005 source are retained with current status explicitly marked for review.

Sources

• Local UnsolvedMath registry, permanent ID 6200070, source code catalogue entry, "Boundaries of Groups and Kleinian Groups --- Problem 70," proposed by Anders Karlsson, 2005. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 70, PDF page 19, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf; normalized pdf-text extraction, accessed 2026-07-29.
• The research summary mentions Caprace, Hume, and Gromov-product/horo-space theory without complete primary references or exact results. Source-specific provenance marks current status and statement-reuse terms NEEDS_REVIEW for the public author-hosted PDF, separately from the dataset licence.