Fixed Points, Expanding Combings, and Lattices in Metric Geometry

Definitions and mathematical statement

This record combines several questions. Some terms, hypotheses, and referenced examples are missing, so the formulations below retain explicit qualifications rather than assert a fully specified common conjecture.

1. Fixed points in CAT(0) spaces. A global fixed point for an action of a group $G$ on a space $X$ is an element of \[ X^G:=\{x\in X: gx=x\text{ for every }g\in G\}. \] Under the interpretation that the action is by isometries and that "locally elliptic" means that every individual group element fixes some point, the question is: for every finitely generated group $G$, every nonempty finite-dimensional CAT(0) space $X$, and every such action, does \[ \left(\forall g\in G\;\exists x_g\in X:\;gx_g=x_g\right) \quad\Longrightarrow\quad X^G\ne\varnothing \] hold? The source singles out Euclidean buildings of arbitrary finite dimension as a particular case. Here CAT(0) means that the space is geodesic and satisfies the Euclidean triangle comparison inequality. The source does not specify completeness, the notion of dimension, or the permitted types of building actions. The elementwise interpretation of local ellipticity is provisional: requiring every finitely generated subgroup to have a common fixed point would already include $G$ itself and would change the question substantially.

2. Fixed points in Helly complexes. Does the analogous implication hold for actions of finitely generated groups on nonempty finite-dimensional Helly complexes? In the usual graph-based setting, the Helly property requires every finite family of pairwise intersecting combinatorial balls to have nonempty total intersection, and one may associate the flag complex of the graph. The record does not specify its convention for a Helly complex, the action category, or whether a fixed point must be a vertex or may lie in a geometric realization. Consequently, this question cannot be given a unique precise formulation from the supplied record alone; the meaning of local ellipticity also requires confirmation.

3. Central extensions and combings. A central extension is an exact sequence of groups \[ 1\longrightarrow A\xrightarrow{\iota}E\xrightarrow{\pi}Q\longrightarrow 1, \qquad \iota(A)\subseteq Z(E), \] where $Z(E)$ is the center of $E$. A hyperbolic group here means a finitely generated group whose Cayley graph, for a finite generating set, has uniformly thin geodesic triangles. The question asks whether central extensions $E$ of hyperbolic groups $Q$ admit coherent expanding combings. The record does not specify restrictions on the kernel $A$, finite generation of $E$, the metric or coarse structure used on $E$, or the definitions of coherence and expansion. Thus the exact class of extensions and the precise combing predicate remain unresolved. A combing generally selects paths from a basepoint to group elements; the necessary uniform control, coherence, and expansion conditions must be recovered from the source before this part can be fully formalized.

4. Automatic groups and expansion. Construct an automatic group that admits no expanding combing. An automatic group is understood in the usual sense of a finitely generated group with a regular language of representatives satisfying a uniform synchronous fellow-traveller condition for representatives of adjacent group elements. The question concerns absence of any expanding combing, rather than merely failure of expansion for one chosen automatic structure. The required definition of an expanding combing is not supplied.

   Separately, determine whether the automatic structures referred to as "the listed automatic structures" are expanding. No list of groups, generating sets, representative languages, or automatic structures appears in this record, so the objects quantified over in this separate question are missing.

5. Lattice generation. Are all irreducible lattices in products of two trees finitely generated? Are all non-uniform lattices in two-dimensional right-angled hyperbolic buildings finitely generated? A group is finitely generated if it has a finite subset whose generated subgroup is the whole group. In the usual locally compact group setting, a lattice is a discrete subgroup $\Gamma\leq H$ for which $H/\Gamma$ has finite invariant measure; it is non-uniform if this quotient is noncompact. These definitions require a specified ambient group $H$. The record does not specify the tree or building automorphism groups, their topology, local finiteness assumptions, the meaning of irreducibility, or any further restrictions on the buildings. The universal questions are retained, but their precise domains cannot be recovered from the supplied text alone.

The supplied registry labels this compound record open. Its dated literature assessment says that the questions remain open in their stated generality and that low-dimensional fixed-point theorems and examples of proper actions do not settle the arbitrary-dimensional building, combing, or lattice-generation questions. These are supplied curation claims, not independently verified mathematical conclusions.

Short English statement

The record asks about global fixed points for locally elliptic actions, coherent expanding combings of central extensions, automatic groups and automatic structures with expansion properties, and finite generation of two classes of lattices. Several definitions, hypotheses, and the referenced list of automatic structures are absent. The supplied open status is dated curation; this statement-only draft requires human mathematical and source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004627$, exact source problem code catalogue entry, titled Fixed Points, Expanding Combings, and Lattices in Metric Geometry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2021$ and no named proposer.
• Supplied source citation: Nonpositively Curved Complexes (2022), Oberwolfach Reports, pp. 383--417, DOI 10.4171/owr/2021/8. Supplied source URL and literature-source link, labelled "Oberwolfach report": https://doi.org/10.4171/owr/2021/8. The background attributes the collection to workshop open-problem discussions documented in Oberwolfach Reports. The supplied title year is retained alongside the proposed year and DOI containing $2021$; their relationship has not been verified.
• Supplied literature-source link, labelled "Haettel, triangle-free Artin groups": https://doi.org/10.4171/GGD/648. No fuller bibliographic information or explanation of its relevance to individual component questions is supplied.
• Literature assessment checked at 2026-08-22, exactly as supplied. This is supplied dated curation, not independently verified. It labels the compound questions open in their stated generality and says that low-dimensional fixed-point theorems and examples of proper actions do not settle the arbitrary-dimensional building, combing, or lattice-generation questions. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the target. That review claim is source metadata; it does not resolve the missing formulations identified in this draft. Neither linked reference has been independently consulted for this draft.