research/AMR-010-0201.md

---
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.