research/AMR-010-0202.md

metadata

id: AMR-010-0202
classification: OPEN-TRIAGE
wording_corrected: 'no'

AMR-010-0202 — Gromov's question: finite-dimensional K(G,1) ⇒ proper isometric action on a complete CAT(0) space?

Problem (corrected statement if needed)

The dataset transcription was checked against the source PDF and is verbatim correct; no correction needed.

Question 2.2 of Bestvina, "Questions in Geometric Group Theory" (updated July 2004):

(Gromov) If $G$ admits a finite dimensional $K(G,1)$, does $G$ act properly discontinuously by isometries on a complete CAT(0) space?

Key features of the statement that matter for the literature triage:

• The CAT(0) space is only required to be complete — it need not be proper (i.e., closed balls may be non-compact; infinite-dimensional Hilbert spaces are allowed).
• The action is only required to be properly discontinuous — no cocompactness, and isometries need not be semisimple (parabolics are allowed).
• The hypothesis (finite-dimensional $K(G,1)$) forces $G$ to be torsion-free, finitely presented, and of type FP; it is far weaker than being a "CAT(0) group" in the standard sense (proper and cocompact action on a CAT(0) space).

Status / Literature

Open as of August 2026, to the best of my verification. No solution (positive or negative) appears in the literature I could verify. Supporting evidence and surrounding results (all citations verified against the source PDF, Crossref, the arXiv API, or publisher pages):

1. Source. Bestvina's problem list, §2 "CAT(0) groups", Q 2.2 (PDF p. 7). The 2004 update gives no solution or partial-result annotation, unlike many other items on the list.

2. The question is open even at a much coarser level. Button, "Groups acting purely loxodromically on products of hyperbolic graphs" (arXiv:2009.10575, 2020; verified via arXiv API), states in the introduction: "It is an open question whether every countable group acts properly on some CAT(0) space, whereas every countable group $G$ does act properly on some hyperbolic space." Q 2.2 is the special case of this for groups with a finite-dimensional $K(G,1)$; since even the all-countable-groups version is open, so is Q 2.2.

3. Dimension-gap results do NOT answer Q 2.2 (important nuance). There is a body of work showing that the CAT(0) dimension of a group can exceed its geometric dimension:

   • Bridson, "Length functions, curvature and the dimension of discrete groups", Math. Res. Lett. 8 (2001), 557–567, DOI 10.4310/MRL.2001.v8.n4.a14 (verified via Crossref reference records).
   • Crisp, "On the CAT(0) dimension of 2-dimensional Bestvina–Brady groups", Algebr. Geom. Topol. 2 (2002), 921–936, DOI 10.2140/agt.2002.2.921 (verified via Crossref): Bestvina–Brady groups $\Gamma_K$ of geometric dimension 2 that do not act properly on any 2-dimensional CAT(0) space, but act properly cocompactly on 3-dimensional ones.
   • Brady–Crisp, "Two-Dimensional Artin Groups with CAT(0) Dimension Three", Geom. Dedicata 94 (2002), 185–214 (verified via Crossref).
   • Tomiyoshi, "Parabolic isometries of CAT(0) spaces and CAT(0) dimensions", Algebr. Geom. Topol. 4 (2004) (verified via the MSP page): groups of geometric dimension 2 that do not act properly on any proper CAT(0) space of dimension 2 by semisimple isometries — but which do act properly on proper CAT(−1) spaces (of higher dimension) once parabolics are allowed (his Theorem 1.1, 1.2, Corollary 5.1).

   All of these concern proper CAT(0) spaces of a bounded dimension, often with semisimplicity imposed. Q 2.2 allows arbitrary complete (possibly non-proper, infinite-dimensional) CAT(0) spaces and arbitrary isometries, so none of these examples obstructs Q 2.2 — indeed Tomiyoshi's groups satisfy its conclusion.

4. Even the hyperbolic special case is open. Gromov's question whether every word-hyperbolic group acts properly and cocompactly on a CAT(0) (or CAT(−1)) space — the "Jugendtraum" — is a famous open problem; see Nica, "Two applications of strong hyperbolicity", Kyoto J. Math. 59 (2019), which calls it "still wildly open" (verified via Project Euclid). Since every hyperbolic group has a finite K(G,1) (Rips complex, mod finite subgroups — for torsion-free hyperbolic groups a finite K(G,1) exists), a positive answer to Q 2.2 in the hyperbolic case would already be a major advance; conversely a negative answer to Q 2.2 would most plausibly come from (or at least illuminate) this case.

5. The class of groups acting properly on complete CAT(0) spaces is very broad, which makes a negative answer hard to engineer:

   • It contains all CAT(0) groups, all a-(T)-menable (Haagerup) groups (Hilbert spaces are CAT(0)), and is closed under passing to subgroups and direct products.
   • It even contains infinite finitely generated torsion groups: Schneeberger, "Proper actions of Grigorchuk groups on a CAT(0) cube complex", Geom. Dedicata (2024), DOI 10.1007/s10711-024-00948-6 (verified via Springer).
   • No algebraic or analytic consequence of "acts properly on some complete CAT(0) space" is known that some group with a finite K(G,1) could fail. Property (T) is not an obstruction (cocompact lattices in $\mathrm{Sp}(n,1)$, $n\ge 2$, have (T) yet act properly cocompactly on quaternionic hyperbolic space, which is CAT(−1)). Note the contrast with CAT(0) cube complexes: a proper cubical action implies the Haagerup property, so infinite property-(T) groups admit no proper cubical actions — but cube complexes are a much smaller class of CAT(0) spaces.
   • Recent tool-building: Petyt, "Hyperbolic models for CAT(0) spaces" (arXiv:2207.14127; published in Adv. Math. 2024; verified via arXiv/Warwick repository) shows any group acting properly on a CAT(0) space inherits a well-behaved action on an associated hyperbolic space — but since every countable group acts properly on some hyperbolic space (see item 2), this yields no obstruction.

6. Background monograph for all CAT(0) terminology: Bridson–Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999, DOI 10.1007/978-3-662-12494-9 (verified via Crossref).

Work done

• Verified the dataset wording character-for-character against the original Bestvina PDF (Q 2.2, p. 7): exact match, attribution "(Gromov)" included. wording_corrected: no.
• Established that the frequently-cited "dimension gap" literature (Bridson 2001; Brady–Crisp 2002; Crisp 2002; Tomiyoshi 2004) answers only stronger variants (proper spaces, bounded dimension, semisimple isometries) and does not decide Q 2.2 as stated; on the contrary, Tomiyoshi's exotic examples do act properly on complete CAT(−1) spaces, so they confirm rather than refute the conjectural implication in those cases.
• Reasoned through both directions:
  • Positive direction (attempted): the naive strategy — equip the universal cover of a finite-dimensional $K(G,1)$ with a $G$-invariant CAT(0) metric — fails in general: there are closed aspherical manifolds admitting no non-positively curved metric, and finite aspherical complexes whose universal covers carry no NPC metric (this is precisely the content of the dimension-gap papers above). Allowing non-proper/infinite-dimensional CAT(0) spaces removes the dimensional obstruction in principle, but no general construction is known — indeed none is known even for arbitrary countable groups (Button's remark, item 2).
  • Negative direction (attempted): any counterexample $G$ must be a group with finite K(G,1) that does not embed in any group acting properly on a complete CAT(0) space (the class is subgroup-closed). All standard candidates are excluded: it cannot be a subgroup of a CAT(0) group, a Haagerup group, a cubulated group, or a lattice in a rank-1 group. No known invariant (bounded cohomology, property (T), Dehn function, torsion) separates "finite K(G,1)" groups from this class. Note that Dehn-function obstructions (e.g., the Baumslag–Gersten group's enormous Dehn function) only obstruct cocompact actions on proper CAT(0) spaces (which force quadratic Dehn function via quasi-isometry to the space); a merely proper action carries no such isoperimetric constraint, since orbits are distorted.
• Searched for post-2020 developments (arXiv API full-text queries on the question's exact phrasing and variants; web searches for 2021–2026 preprints). Found no claimed solution or partial resolution of Q 2.2 itself.

Result

Open — rigorous triage. The problem is unsolved in both directions as of August 2026:

• No group with a finite-dimensional $K(G,1)$ is known that provably fails to act properly discontinuously by isometries on a complete CAT(0) space.
• No theorem establishes such an action for all (or even for all hyperbolic) groups with finite-dimensional $K(G,1)$.
• The strongest surrounding facts: (i) the more general question for arbitrary countable groups is explicitly open (Button 2020); (ii) the cocompact/proper-space/semisimple strengthenings are known to fail even in geometric dimension 2 (Bridson, Brady–Crisp, Crisp, Tomiyoshi 2001–2004), but their counterexamples still satisfy the conclusion of Q 2.2; (iii) the hyperbolic special case (Gromov's Jugendtraum) remains open.

I could not solve or make substantive new mathematical progress on the problem itself; the difficulty is that the hypothesis gives a finite-dimensional, possibly non-positively-curved classifying space while the conclusion allows arbitrary complete CAT(0) spaces, and the two sides are connected by no known construction or invariant.

What remains

• The full question: construct, for every $G$ with finite-dimensional $K(G,1)$, a proper isometric action on a complete CAT(0) space — or produce a counterexample.
• Natural attackable sub-problems:
  1. Hyperbolic case: does every word-hyperbolic group act properly (not necessarily cocompactly) on some complete CAT(0) space? This is weaker than the open Jugendtraum and might be more accessible; note (T) hyperbolic groups would need non-cubical CAT(0) targets.
  2. Baumslag–Gersten-type examples: groups with finite $K(G,1)$ and non-elementary-recursive Dehn functions are natural stress tests; no obstruction to proper CAT(0) actions is known for them, and no action is known either.
  3. Find any invariant of discrete groups that is forced by proper actions on arbitrary complete CAT(0) spaces but is not already forced by proper actions on hyperbolic spaces (Petyt's work suggests such invariants may be scarce), or prove none exists — which would point to a positive answer.
• Also open and strictly harder: the same question with "complete" strengthened to "proper", or with cocompactness added (false in general, by Tomiyoshi's Corollary 5.1 — those strengthenings are known to fail, unlike Q 2.2 itself).