--- id: AMR-010-0201 classification: SOLVED-IN-LITERATURE wording_corrected: no --- # AMR-010-0201 — Swarup's question: Dehn twists and Out(G) for CAT(0) groups ## Problem (corrected statement if needed) The worklist transcription matches the original source verbatim; no correction was needed. Source: M. Bestvina, *Questions in Geometric Group Theory*, Question 2.1 (attributed to Swarup), [author-hosted PDF, updated July 2004](https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf) (fetched and checked directly; the updated version carries no status note on Q 2.1): > (Swarup) Is there a proof of Johannson's theorem that Out(π1M) is virtually > generated by Dehn twists for M a Haken 3-manifold along the lines of Rips–Sela's > theorem that Out(G) is virtually generated by Dehn twists for torsion-free > 1-ended hyperbolic G? Is this true for CAT(0) groups? In particular, if G is a > CAT(0) group and Out(G) is infinite, does G admit a Dehn twist of infinite order? Here a *Dehn twist* (definition given in the list itself) is an automorphism coming from a one-edge splitting: if G = A *_C B and t ∈ Z(C), the twist fixes A pointwise and conjugates B by t; similarly for HNN extensions. The question has three parts: 1. Find a Rips–Sela-style proof of Johannson's theorem (Out(π1M) virtually generated by Dehn twists for Haken 3-manifolds; Johannson, *Homotopy equivalences of 3-manifolds with boundaries*, LNM 761, Springer 1979). 2. Does the Rips–Sela/Johannson picture hold for CAT(0) groups? 3. (In particular) For a CAT(0) group G with Out(G) infinite, must G admit a Dehn twist of infinite order? ## Status / Literature **The question is resolved in the literature: the general CAT(0) form (parts 2 and 3) has a NEGATIVE answer, while the answer is positive for every natural "structured" subclass (hyperbolic, toral relatively hyperbolic, isolated-flats CAT(0) partially, and special/cubulated groups up to finite index). Part 1 is subsumed by the modern relatively-hyperbolic machinery.** Verified items: - **Hyperbolic groups (positive).** Rips–Sela: Out(G) virtually generated by Dehn twists for torsion-free 1-ended hyperbolic G [E. Rips, Z. Sela, *Structure and rigidity in hyperbolic groups I*, Geom. Funct. Anal. 4 (1994) — cited as [RS94] in both Carette's and Fioravanti's papers below]. Strengthened by [G. Levitt, *Automorphisms of hyperbolic groups and graphs of groups*, Geom. Dedicata 114 (2005) 49–70](https://arxiv.org/abs/math/0212088) (verified via arXiv API and by reading the arXiv text): his Theorem 1.4 states that for a one-ended hyperbolic group G, Out(G) is infinite **iff** G splits over a virtually cyclic subgroup with infinite centre — exactly the "infinite Out ⟹ infinite-order Dehn twist" dichotomy (he notes Swarup suggested the problem). His Proposition 3.1 gives a complete presentation of the group of twists of any graph of groups, which I used below. - **CAT(0) with isolated flats (partial positive).** [D. Groves, *Limits of (certain) CAT(0) groups, I: Compactification*, Algebr. Geom. Topol. 5 (2005) 1325–1364](https://msp.org/agt/2005/5-4/agt-v5-n4-p03-p.pdf) (fetched directly), Theorem 5.9: if Γ is torsion-free, acts properly and cocompactly on a CAT(0) space with isolated flats, and flat stabilisers are abelian, then Out(Γ) infinite ⟹ Γ splits over a finitely generated free abelian group. Groves explicitly says this "partially answers a question of Swarup (see [Bestvina, Q 2.1])". - **Coxeter groups (special case).** [M. Carette, *Virtually splitting the map from Aut(G) to Out(G)*, arXiv:1301.4446](https://arxiv.org/abs/1301.4446) (verified via arXiv API) explicitly quotes Q 2.1 and discusses the Coxeter case; twist-rigid Coxeter groups (Caprace–Przytycki) have finite Out. - **General CAT(0) groups (negative), and special groups (sharp positive).** [E. Fioravanti, *Generators for automorphisms of special groups*, arXiv:2601.22789 (Jan 2026, 79 pp., preprint — not yet refereed)](https://arxiv.org/abs/2601.22789) (fetched and read): "Swarup asked whether Out(G) is virtually generated by Dehn twists for every CAT(0) group G [Bes, Q2.1]"; Theorem C: every special group G (Haglund–Wise) has a characteristic finite-index subgroup G₀ with Out(G₀) virtually generated by Dehn twists; Proposition B: there *are* special (hence CAT(0)) groups whose Out is not virtually generated by Dehn twists ("poison subgroups", a rank-2 abelian phenomenon); and, decisively, the discussion after Theorem C (Example 8.3): for general CAT(0) groups there are groups G such that **every finite-index subgroup G₀ ≤ G has infinite Out(G₀) and not a single (non-identity) Dehn twist** — "In particular, the most general form of Swarup's question [Bes, Q2.1] has a negative answer." The examples are extracted from: - [G. Italiano, B. Martelli, M. Migliorini, *Hyperbolic 5-manifolds that fiber over S¹*, Invent. Math. 231 (2023) 1–38](https://arxiv.org/abs/2105.14795) (verified via arXiv API; the Invent. Math. reference appears verbatim in Fioravanti's bibliography, surfaced via a web-search snippet of his PDF), - [D. Groves, J. F. Manning, *Special IMM groups*, to appear in Bull. Lond. Math. Soc.](https://arxiv.org/abs/2205.11290) (verified via arXiv API — this is Fioravanti's [GM23]), - [B. Martelli, *A 4-dimensional pseudo-Anosov homeomorphism*, arXiv:2511.10530](https://arxiv.org/abs/2511.10530) (verified via arXiv API — this is Fioravanti's [Mar25]; among its consequences: a compact locally CAT(0) space whose π1 is non-hyperbolic and contains **no Z×Z**, answering Gromov's Closing Flat problem). - **Part 1 (Johannson via Rips–Sela).** The Rips–Sela shortening/JSJ machinery has since been developed for toral relatively hyperbolic groups (work of Guirardel–Levitt, cited in Fioravanti's introduction as [GL15b] for the statement "toral relatively hyperbolic groups behave similarly", i.e. Out virtually generated by Dehn twists; and Groves' [Gro05] above). Fundamental groups of Haken 3-manifolds are relatively hyperbolic with abelian/Seifert peripheral structure, so the Rips–Sela-style analysis of Out(π1M) now exists in this framework. *Caveat:* I did not re-verify the Guirardel–Levitt papers themselves in this session (their JSJ monograph, *JSJ decompositions of groups*, Astérisque 395, 2017, is standard), and I am not aware of a paper explicitly titled "Johannson via Rips–Sela"; the statement is subsumed by the relatively hyperbolic theory. ## Work done No computation was used; this is a literature triage plus independent elementary reasoning. 1. **Confirmed the source and wording.** Fetched Bestvina's updated problem list and matched Q 2.1 word-for-word (the list even includes the definition of Dehn twist that the dataset transcription omitted). 2. **An elementary counterexample to part 3 (own analysis).** Let G = Z² *_Z Z² = ⟨a,b,c,d | [a,b]=[c,d]=1, a=c⟩, the π1 of two flat tori glued along a simple closed geodesic of equal length — a 2-dimensional (locally) CAT(0) group by Reshetnyak's gluing theorem (these are essentially the Croke–Kleiner examples). Then: - *Out(G) is infinite.* The shears b ↦ aᵏb (fixing a,c,d) and d ↦ aˡd (fixing a,b,c) — i.e. elements of the stabiliser of a primitive vector in GL(2,Z) applied independently to the two vertex groups — give a Z×Z subgroup of Out(G): an element of Inn(G) acts on each abelian vertex group either trivially or moves the other factor off itself (normal forms in the amalgam), so Inn(G) meets this shear subgroup trivially. - *Every Dehn twist from this splitting is trivial in Out(G).* By Levitt's Proposition 3.1 (read from the paper), the group of twists is the quotient of ∏ Z_{G_v}(G_e) by vertex relations (centres of vertex groups) and edge relations (centres of edge groups). Here both vertex groups are abelian, so the vertex relations kill everything: the twist group is trivial. (Consistently, a twist by t ∈ C conjugating the abelian factor B is the identity since t ∈ Z(B).) So an elementary CAT(0) group with infinite Out and no nontrivial Dehn twist from its natural splitting already exists — I did **not** fully verify the stronger claim that this G admits no infinite-order Dehn twist from *arbitrary* splittings (that requires ruling out exotic G-trees), which is why the published Fioravanti/IMM/Martelli examples (no Dehn twists at all, even in every finite-index subgroup) are needed for the definitive negative answer. I could not locate an older reference stating this amalgam as an explicit counterexample to Swarup's question; Carette (2013) still phrases the CAT(0) "iff" as a question, so the folklore status of the easy example is unclear to me. 3. **Mechanism behind the definitive counterexamples (sketch, from the sources).** In the IMM/Martelli fibering constructions one has a (relatively) hyperbolic mapping-torus group π1(M) = π1(F) ⋊_φ Z with F a compact aspherical locally CAT(0) 4-manifold. The monodromy φ has infinite order in Out(π1F) — otherwise π1(M) would virtually split as π1(F) × Z, incompatible with (relative) hyperbolicity — so Out(π1F) is infinite; while π1(F) has no Z×Z (Martelli's pseudo-Anosov monodromy, [Mar25]) or is arranged so that no splitting supporting an infinite-order Dehn twist exists even after passing to finite index. Since any infinite-order Dehn twist forces a splitting over an infinite subgroup with infinite centraliser (in particular a Z²), such groups answer part 3 negatively. ## Result - **Part 3 (and hence part 2) for general CAT(0) groups: NO.** There are CAT(0) groups G with Out(G) infinite — indeed with every finite-index subgroup having infinite Out — and not a single nontrivial Dehn twist. Published explicitly by Fioravanti (arXiv:2601.22789, Example 8.3), built from the Italiano–Martelli– Migliorini / Groves–Manning / Martelli fibering constructions (2023–2025). An elementary 2-dimensional example (Z² *_Z Z²) shows the same phenomenon for twists of the natural splitting (my analysis, based on Levitt's twist-group computation). - **Part 2 for restricted classes: YES.** Hyperbolic groups (Rips–Sela; Levitt's Theorem 1.4 gives the sharp "Out infinite ⟺ infinite-order Dehn twist exists" form); toral relatively hyperbolic groups (Guirardel–Levitt, per Fioravanti's introduction); CAT(0) groups with isolated flats and abelian flat stabilisers (Groves' Theorem 5.9, splitting conclusion); special (cocompactly cubulated Haglund–Wise) groups up to a characteristic finite-index subgroup (Fioravanti, Theorem C, with the failure inside special groups exactly characterised by "poison subgroups", Theorem E). - **Part 1: effectively yes** — the Rips–Sela program now covers the class of groups containing all Haken 3-manifold groups (toral/relatively hyperbolic JSJ theory), so Johannson's theorem is recovered by Rips–Sela-style arguments, though no paper with that explicit title seems to exist. - Classification: **SOLVED-IN-LITERATURE** (the question's hoped-for general CAT(0) analogue is false; the precise boundary of validity is now mapped out). Note the decisive reference for the negative answer is a January 2026 arXiv preprint, not yet refereed. ## What remains - Refereed publication of Fioravanti's preprint (arXiv:2601.22789) would put the negative answer on firm published footing; the underlying manifold constructions (IMM23, GM23, Mar25) are published or well-circulated. - Rips' related question on the structure of Out(G) for arbitrary cocompactly cubulated groups (broader than special groups) remains open — Fioravanti's results cover the special case. - For Coxeter groups, the general "Out(G) infinite ⟺ infinite-order Dehn twist" question raised by Carette in 2013 was not fully resolved in the sources I checked. - It would be a small service to record the elementary Z² *_Z Z² counterexample (with a complete proof that no splitting yields an infinite-order twist) in the literature explicitly; I could not find it stated as such.