research/AMR-010-0105.md

metadata

id: AMR-010-0105
classification: OPEN-TRIAGE
wording_corrected: 'no'

AMR-010-0105 — Equivariant negatively curved metric on the Rips complex of a hyperbolic group (Davis)

Problem (corrected statement if needed)

The dataset transcription matches the source verbatim; no correction was needed. The source is Mladen Bestvina's curated list Questions in Geometric Group Theory (major revision August 2000, updated July 2004), Question 1.5, hosted at the University of Utah (questions-updated.pdf):

Q 1.5. (Davis) If $G$ is word-hyperbolic, does the Rips complex $P_d(G)$ have an equivariant negatively curved metric for $d$ sufficiently large?

Bestvina's list appends a remark (reproduced here in full, since the "Background" of the dataset omits it):

A potential counterexample is the mapping torus of a hyperbolic automorphism of a free group, or perhaps the quotient of a uniform lattice in $Sp(n,1)$ by a "random" element. For a related example see [...]

(the trailing reference was truncated in the source extraction; from context it is presumably Gromov's Asymptotic invariants of infinite groups [Gro93]).

Conventions: for $G$ finitely generated with word metric from a finite generating set, the Rips complex $P_d(G)$ is the flag simplicial complex whose simplices are finite subsets of $G$ of diameter $\le d$. "Equivariant negatively curved metric" means a $G$-invariant CAT($-\kappa$) metric (some $\kappa>0$), typically piecewise-hyperbolic, with the left action by isometries; for large $d$ the action is then automatically proper and cocompact. Recall Rips's theorem: for $d \ge 4\delta+2$ (with $\delta$ the hyperbolicity constant) $P_d(G)$ is contractible, and Meintrup–Schick showed it is a finite model for the universal proper $G$-space $\underline{E}G$ (D. Meintrup & T. Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002), 1–7; its existence and citation data were confirmed via the Crossref-deposited reference list of [Lang 2013], DOI 10.1142/S1793525313500118).

Status / Literature

The question is open, and it is one concrete incarnation of a famous open problem, Gromov's "curvature conjecture" / Jungentraum ([Gro93, §7.B, p. 193], M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory Vol. 2, LMS Lecture Note Ser. 182, Cambridge Univ. Press, 1993): does every word-hyperbolic group act properly and cocompactly by isometries on a CAT($-1$) space? (Even the CAT(0) version is open.) Verified evidence that it remains open:

• P.-E. Caprace, Y. de Cornulier, N. Monod, R. Tessera, Amenable hyperbolic groups, J. Eur. Math. Soc. 17 (2015), 2903–2947, DOI 10.4171/JEMS/575 (verified via Crossref). They state: "for general hyperbolic locally compact groups (even discrete ones), it is an outstanding problem to determine if they can act properly cocompactly on any CAT(−1) (or even CAT(0)) space [Gro93, §7.B]."
• A 2026 preprint, The variety of group actions on all algebraic real hyperbolic spaces, arXiv:2603.03863, states in its introduction: "Gromov's Jungentraum ([Gro93, p. 193]) is to show that every (finitely generated) hyperbolic group admits [a] geometric action ... on a CAT(−1) space. While this is wide open, it is expected to fail, but no counterexamples are known." (Recent preprint; used only as evidence of current status, not as a source of theorems.)
• J. McCammond, Constructing non-positively curved spaces and groups, in Geometric Methods in Group Theory, Contemp. Math. 372, AMS, 2005 (author PDF; DOI not independently verified — my guessed DOI 10.1090/conm/372/06885 in fact resolves to a different paper, so I cite only the author copy). The survey explicitly discusses Davis's strategy: "As defined above, the Rips complex is a simplicial complex with no natural metric. One approach to the curvature conjecture would be to try and add a metric to ..." — confirming Q 1.5 is viewed as an approach to Gromov's conjecture, not a settled statement.

Verified partial results in the vicinity:

• S. Brown, A gluing theorem for negatively curved complexes, J. London Math. Soc. 93(3) (2016), 741–762, DOI 10.1112/jlms/jdw021 (verified via arXiv API). Consequence: hyperbolic limit groups, and hyperbolic groups whose JSJ components are fundamental groups of negatively curved 2-complexes (e.g., finite graphs of free groups with cyclic edge groups), are CAT(−1). This is progress on Gromov's conjecture for large classes, but the CAT(−1) spaces produced are glued 2-complexes, not the Rips complex itself.
• N. Brady & J. Crisp, CAT(0) and CAT(−1) dimensions of torsion free hyperbolic groups, Comment. Math. Helv. 82(1) (2007), 61–85, DOI 10.4171/cmh/85 (verified via Crossref). They exhibit a free-by-cyclic group with CAT(0) dimension 2 but CAT(−1) dimension 3, and an infinite family of 2-dimensional hyperbolic groups (including a free-by-cyclic group with rank-6 free kernel) that do not act properly discontinuously by isometries on any proper CAT(0) space of dimension 2. This is directly relevant to Bestvina's proposed counterexample class (hyperbolic free-by-cyclic groups) and shows that CAT(−1) realizations, when they exist, may require more dimensions than the group's geometric/cohomological dimension — a warning sign for the Rips-complex version.
• M. F. Hagen & D. T. Wise, Cubulating hyperbolic free-by-cyclic groups: the general case, Geom. Funct. Anal., DOI 10.1007/s00039-015-0314-y (verified via arXiv API): every word-hyperbolic free-by-cyclic group $F \rtimes_\Phi \mathbb{Z}$ acts freely and cocompactly on a CAT(0) cube complex. So the candidate counterexamples are CAT(0), but this says nothing about CAT(−1): hyperbolic CAT(0) cube complexes can still fail to support any CAT(−1) structure (cf. Brady–Crisp).
• U. Lang, Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5(3) (2013), 297–331, DOI 10.1142/S1793525313500118 (verified via Crossref): every word-hyperbolic group acts properly and cocompactly by isometries on its injective hull $E(\Gamma)$, a finite-dimensional polyhedral complex enjoying a weak (non-coarse) form of non-positive curvature — evidence "one level down" from CAT(0)/CAT(−1), and Lang explicitly relates it to this long-standing question.
• Basic topological facts: $P_d(G)$ is contractible for $d \ge 4\delta+2$ (Rips; see Bridson–Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999, DOI 10.1007/978-3-662-12494-9, III.Γ.3), and a finite $\underline{E}G$ (Meintrup–Schick, above).

Work done

• Read worklist/AMR-010-0105.md; identified the source list and fetched Bestvina's PDF. The transcription is exact (Question 1.5, attributed to Davis); recovered the remark about candidate counterexamples.
• Web-searched the status of Davis's question and of Gromov's curvature conjecture; the consistent picture across sources from 2004 to 2026 is: open, no counterexample known, no solution claimed.
• Verified every cited item against Crossref or the arXiv API: Caprace–Cornulier–Monod–Tessera (10.4171/JEMS/575), Brown (arXiv:1510.02716 / 10.1112/jlms/jdw021), Brady–Crisp (10.4171/cmh/85), Hagen–Wise (arXiv:1406.3292 / 10.1007/s00039-015-0314-y), Lang (10.1142/S1793525313500118), Bridson–Haefliger (10.1007/978-3-662-12494-9). Two citation attempts were rejected by verification and corrected: a guessed DOI for McCammond's survey resolved to a Baumslag paper, and McCammond's survey appears not to be on arXiv under its title.
• No computation was used; the analysis below is by hand.

Result

No solution exists in the literature, and I could not solve it (a solution would resolve Gromov's conjecture). The rigorous synthesis:

1. Logical position. A positive answer to Q 1.5 for all $G$ implies Gromov's conjecture, since $P_d(G)$ with an invariant CAT($-\kappa$) metric is a proper cocompact $G$-model (finiteness of $P_d(G)/G$ is automatic, and properness follows from Meintrup–Schick). Conversely Q 1.5 is in principle strictly stronger than Gromov's conjecture: metrics do not transfer across equivariant homotopy equivalences, so a group could be CAT(−1) on some space while its Rips complex supports no invariant CAT(−1) metric. Davis's question is thus a canonical-model strengthening of the curvature conjecture.

2. Why the naive approach fails (the precise obstruction). Any $G$-invariant piecewise-hyperbolic metric on $P_d(G)$ must satisfy Gromov's link condition: every closed geodesic in the link of every simplex must have length $\ge 2\pi$. The link of a $k$-simplex $\sigma$ in $P_d(G)$ is itself a Rips-type complex — the complex of diameter-$\le d$ subsets of $G$ whose union with $\sigma$ still has diameter $\le d$ — built from the "corona" $B(x,d)\setminus N_k(\sigma)$ in the Cayley graph. As the local combinatorics of the Cayley graph grows complex (think of the thin quadrilaterals forced by long relators in quotients, or by the train-track dynamics of a free-group automorphism), these links acquire short essential loops that no uniform choice of simplex scale kills: shrinking simplices worsens angles in higher links, enlarging them breaks the homotopy type needed for contractibility. There is no known uniform combinatorial invariant of hyperbolicity that controls link girth in all dimensions simultaneously; hyperbolicity is a coarse condition, while the link condition is local and dimension-dependent. This gap is exactly why the CAT(0) analogue (equivariant CAT(0) metric on $P_d(G)$) is equally open, and why alternative canonical models (Lang's injective hull) only reach weaker curvature properties.

3. State of the candidate counterexamples. Bestvina's proposed counterexample class — hyperbolic mapping tori $F_n \rtimes_\Phi \mathbb{Z}$ — is now known to be CAT(0) (Hagen–Wise), but Brady–Crisp show that even 2-dimensional hyperbolic (free-by-cyclic) groups can force CAT(−1) dimension 3 while being CAT(0) in dimension 2. Since for a torsion-free group of cohomological dimension $n$ the Rips complex is a model of dimension potentially much larger than $n$, a dimension-counting obstruction to Q 1.5 is not currently derivable, but the Brady–Crisp phenomenon shows the "expected" dimension is genuinely wrong in this class. The second candidate class (random quotients of uniform $Sp(n,1)$ lattices) retains property (T) from the ambient lattice; property (T) obstructs proper actions on CAT(0) cube complexes but is fully compatible with CAT(−1) actions (the $Sp(n,1)$ lattices themselves are CAT(−1) and (T)), so no known mechanism makes these counterexamples either.

4. Known positive territory. All groups with "negatively curved 2-dimensional JSJ structure" — hyperbolic limit groups, graphs of free groups with cyclic edges — are CAT(−1) (Brown), but via ad hoc glued 2-complexes, not via $P_d(G)$. Nothing in the literature puts a negatively curved metric on the Rips complex of even a single non-elementary infinite-ended example class as far as I could verify.

What remains

• The full question is open for every group not already covered by the 2-dimensional/gluing results; the first genuinely unknown cases are hyperbolic free-by-cyclic groups with fully irreducible atoroidal monodromy, and (even earlier in difficulty) whether any uniform $Sp(n,1)$ lattice's Rips complex carries an invariant CAT(−1) metric — the group is CAT(−1) on quaternionic hyperbolic space, but the Rips-complex metric is a separate matter.
• A natural weakening with current traction: does $P_d(G)$ admit an invariant CAT(0) metric? This too is open and equivalent in spirit to "every hyperbolic group is CAT(0)."
• A plausible attack on the negative side: find a hyperbolic group with a coarse obstruction to CAT(−1) actions (none is known; this is the bottleneck for Gromov's conjecture itself), or show the links in $P_d(G)$ of Brady–Crisp-type groups necessarily contain sub-$2\pi$ loops for all $d$ — which would refute Q 1.5 without refuting Gromov's conjecture.
• A plausible attack on the positive side: exploit the quasi-tree / finite-complexity structure of links for specific classes (e.g., free groups, where $P_d$ is built from diameter-$d$ subsets of a tree) — even the free-group case of Q 1.5 does not appear to be written down in the literature.