research/AMR-010-0201.md

metadata

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 (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 (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 (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 (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) (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 (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. (verified via arXiv API — this is Fioravanti's [GM23]),
  • B. Martelli, A 4-dimensional pseudo-Anosov homeomorphism, arXiv: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.