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arxiv: 2605.23829 · v1 · pith:G2WKRG3Xnew · submitted 2026-05-22 · 🧮 math.GR · math.GT

Outer automorphism groups of hyperbolic groups, bounded extensions, and hierarchical hyperbolicity

Ervin Hadziosmanovic , Giorgio Mangioni This is my paper

Pith reviewed 2026-05-25 02:27 UTC · model grok-4.3

classification 🧮 math.GR math.GT
keywords hyperbolic groupsouter automorphism groupshierarchically hyperbolic groupsJSJ decompositioncentral extensionsEuler classorbifold mapping class groupssurface amalgams
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The pith

The outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group under mild JSJ orientability conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for one-ended hyperbolic groups satisfying mild orientability conditions on their JSJ decomposition, the outer automorphism group is virtually hierarchically hyperbolic. It reaches this by proving a finite-index subgroup arises as a central extension of a product of orbifold mapping class groups whose Euler class is bounded. A reader would care because the bounded extension transfers the hierarchical hyperbolicity from the product to the extension, yielding geometric control over these automorphism groups. The claim is sharp, as the authors construct a surface amalgam whose fundamental group has outer automorphism group that fails to be hierarchically hyperbolic.

Core claim

We prove that the outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group (HHG), under mild orientability conditions on the associated JSJ decomposition. This is done by proving that a finite-index subgroup is a central extension of a product of orbifold mapping class groups, and the extension has bounded Euler class. Our theorem is sharp: we exhibit a surface amalgam whose fundamental group has full outer automorphism group which is not a HHG. To prove this, the main technical tool is the fact that a top-dimensional Abelian subgroup of a HHG is a standard flat.

What carries the argument

A central extension with bounded Euler class of a product of orbifold mapping class groups, which transfers hierarchical hyperbolicity to the extension group.

If this is right

  • Outer automorphism groups of many one-ended hyperbolic groups inherit the coarse geometric features of hierarchically hyperbolic groups.
  • The top-dimensional abelian subgroups of these outer automorphism groups are standard flats.
  • Hierarchical hyperbolicity techniques become available for studying the geometry and subgroups of these outer automorphism groups.
  • The result covers all one-ended hyperbolic groups whose JSJ decompositions satisfy the stated orientability conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bounded-extension technique may apply to outer automorphism groups of other classes of groups that decompose into mapping class group products.
  • One could check whether the same virtual hierarchical hyperbolicity holds when the JSJ orientability conditions are dropped.
  • The sharpness example suggests that surface amalgams mark a boundary case separating groups whose outer automorphisms remain hierarchically hyperbolic from those that do not.

Load-bearing premise

The Euler class of the central extension remains bounded so that hierarchical hyperbolicity passes from the product of mapping class groups to the extension.

What would settle it

An explicit one-ended hyperbolic group meeting the orientability conditions whose associated central extension has unbounded Euler class, or a direct computation showing its outer automorphism group fails to be hierarchically hyperbolic.

Figures

Figures reproduced from arXiv: 2605.23829 by Ervin Hadziosmanovic, Giorgio Mangioni.

Figure 1
Figure 1. Figure 1: Depiction of the complex K, obtained by gluing three tori with one bound￾ary to a common circle (here, the dashed line) along the boundaries. The cyclic per￾mutation of the three surfaces descends to an order-three element σ of Outpπ1pKqq. The decomposition is invariant under automorphisms of G, in a suitable sense, and this makes it possible to study automorphisms of G by their action on the vertex groups… view at source ↗
Figure 2
Figure 2. Figure 2: Consider the genus two closed surface Σ2, and choose a hyperbolic metric making the rotation ρ by an angle of π around the dashed axes an isometry (one can either explicitly construct such a metric, or invoke Nielsen’s realization theorem, see e.g. [FM12, Theorem 7.1]). The quotient Σ2{xρy is a sphere with six cone points, each of weight 2, and indeed the total angle around each cone point is π [PITH_FULL… view at source ↗
Figure 3
Figure 3. Figure 3: The above is [GL17, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The various balls appearing in the proof of Claim 3.17. Every point in the ball of radius R around x1 is D-close to a point in F 1 ¨ x0 inside the ball of radius R ` 2D around f1 ¨ x0. Similarly, every point in the ball of radius R ` 2D around f2 ¨ x0 is D-close to a point in O2 inside the ball of radius R ` 4D around x2. For every U P U, let RU be the collection of hierarchy rays in the factor FU that app… view at source ↗
read the original abstract

We prove that the outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group (HHG), under mild orientability conditions on the associated JSJ decomposition. This is done by proving that a finite-index subgroup is a central extension of a product of orbifold mapping class groups, and the extension has bounded Euler class. Our theorem is sharp: we exhibit a surface amalgam whose fundamental group has full outer automorphism group which is not a HHG. To prove this, the main technical tool is the fact that a top-dimensional Abelian subgroup of a HHG is a standard flat.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves that for a one-ended hyperbolic group G, under mild orientability conditions on its JSJ decomposition, Out(G) is virtually a hierarchically hyperbolic group. The argument realizes a finite-index subgroup of Out(G) as a central extension (with bounded Euler class) of a product of orbifold mapping class groups, which are already known to be HHGs; the boundedness transfers the HHG structure. The main technical tool invoked is that top-dimensional abelian subgroups of an HHG are standard flats. Sharpness is shown by a surface amalgam example where the full Out(G) fails to be an HHG.

Significance. If the central claims hold, the result would link outer automorphism groups of hyperbolic groups to the HHG framework, enabling transfer of geometric and algebraic properties such as acylindrical hyperbolicity and quasi-isometric rigidity. The bounded-extension technique and the auxiliary fact on abelian subgroups are reusable tools. The orientability hypotheses and the explicit counterexample make the statement precise and falsifiable.

minor comments (3)
  1. [Abstract] The abstract states that the central extension has bounded Euler class but does not indicate where this boundedness is established (e.g., which lemma or proposition).
  2. [Introduction] Clarify whether the fact that top-dimensional abelian subgroups of HHGs are standard flats is proved in the paper or cited from prior work; if cited, add the precise reference in the introduction.
  3. In the sharpness counterexample, explicitly verify that the surface amalgam violates one of the stated orientability conditions on the JSJ decomposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance, and recommendation of minor revision. As the major comments section contains no specific points, we have no items to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external HHG facts

full rationale

The paper's core reduction realizes a finite-index subgroup of Out(G) as a central extension (bounded Euler class) of a product of orbifold mapping class groups already known to be HHGs; the boundedness transfers the HHG structure. The auxiliary fact that top-dimensional abelian subgroups of HHGs are standard flats is invoked as an external technical tool, not derived within the paper or via self-citation chain. No step reduces by construction to the target result, no fitted inputs are relabeled as predictions, and no uniqueness theorem is smuggled from the authors' prior work. The argument is self-contained against external benchmarks on HHGs and mapping class groups.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, background axioms, or newly postulated entities; the result is framed as a proof that builds on standard notions such as JSJ decompositions and HHGs.

pith-pipeline@v0.9.0 · 5627 in / 1224 out tokens · 35437 ms · 2026-05-25T02:27:47.034394+00:00 · methodology

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Works this paper leans on

73 extracted references · 73 canonical work pages · 1 internal anchor

  1. [1]

    Levitt, Gilbert , TITLE =. Geom. Dedicata , FJOURNAL =. 2005 , PAGES =. doi:10.1007/s10711-004-1492-1 , URL =

    work page doi:10.1007/s10711-004-1492-1 2005

  2. [2]

    Alperin, J. L. , TITLE =. Pacific J. Math. , FJOURNAL =. 1962 , PAGES =

    work page 1962

  3. [3]

    , TITLE =

    Pettet, Martin R. , TITLE =. Proc. Edinburgh Math. Soc. (2) , FJOURNAL =. 1995 , NUMBER =. doi:10.1017/S0013091500019271 , URL =

    work page doi:10.1017/s0013091500019271 1995

  4. [4]

    and Guirardel, V

    Dahmani, F. and Guirardel, V. and Osin, D. , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =. doi:10.1090/memo/1156 , URL =

    work page doi:10.1090/memo/1156 2017

  5. [5]

    2008 , PAGES =

    Bogopolski, Oleg , TITLE =. 2008 , PAGES =. doi:10.4171/041 , URL =

    work page doi:10.4171/041 2008

  6. [6]

    2003 , PAGES =

    Serre, Jean-Pierre , TITLE =. 2003 , PAGES =

    work page 2003

  7. [7]

    Metric spaces of non-positive curvatu re

    Bridson, Martin R. and Haefliger, Andr\'e , TITLE =. 1999 , PAGES =. doi:10.1007/978-3-662-12494-9 , URL =

    work page doi:10.1007/978-3-662-12494-9 1999

  8. [8]

    Masur and Yair N

    Masur, Howard A. and Minsky, Yair N. , title =. Invent. Math. , issn =. 1999 , language =. doi:10.1007/s002220050343 , keywords =

    work page doi:10.1007/s002220050343 1999

  9. [9]

    Masur, H. A. and Minsky, Y. N. , title =. Geom. Funct. Anal. , issn =. 2000 , language =. doi:10.1007/PL00001643 , keywords =

    work page doi:10.1007/pl00001643 2000

  10. [10]

    Caprace, Pierre-Emmanuel and Cornulier, Yves and Monod, Nicolas and Tessera, Romain , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2015 , NUMBER =. doi:10.4171/JEMS/575 , URL =

    work page doi:10.4171/jems/575 2015

  11. [11]

    Kapovich, Ilya and Lukyanenko, Anton , TITLE =. Conform. Geom. Dyn. , FJOURNAL =. 2012 , PAGES =. doi:10.1090/S1088-4173-2012-00246-9 , URL =

    work page doi:10.1090/s1088-4173-2012-00246-9 2012

  12. [12]

    and Myasnikov, A

    Kharlampovich, O. and Myasnikov, A. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1998 , NUMBER =. doi:10.1090/S0002-9947-98-01773-5 , URL =

    work page doi:10.1090/s0002-9947-98-01773-5 1998

  13. [13]

    Diagram rigidity for geometric amalgamations of free groups , JOURNAL =

    Lafont, Jean-Fran. Diagram rigidity for geometric amalgamations of free groups , JOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jpaa.2006.07.004 , URL =

    work page doi:10.1016/j.jpaa.2006.07.004 2007

  14. [14]

    , TITLE =

    Gromov, M. , TITLE =. Essays in group theory , SERIES =. 1987 , ISBN =. doi:10.1007/978-1-4613-9586-7\_3 , URL =

    work page doi:10.1007/978-1-4613-9586-7 1987

  15. [15]

    Turkish J

    Korkmaz, Mustafa , TITLE =. Turkish J. Math. , FJOURNAL =. 2002 , NUMBER =

    work page 2002

  16. [16]

    1971 , PAGES =

    Stallings, John , TITLE =. 1971 , PAGES =

    work page 1971

  17. [17]

    and Sisto, Alessandro , TITLE =

    Durham, Matthew Gentry and Hagen, Mark F. and Sisto, Alessandro , TITLE =. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/gt.2017.21.3659 , URL =

    work page doi:10.2140/gt.2017.21.3659 2017

  18. [18]

    2009 , PAGES =

    Calegari, Danny , TITLE =. 2009 , PAGES =. doi:10.1142/e018 , URL =

    work page doi:10.1142/e018 2009

  19. [19]

    Epstein, D. B. A. , TITLE =. Acta Math. , FJOURNAL =. 1966 , PAGES =. doi:10.1007/BF02392203 , URL =

    work page doi:10.1007/bf02392203 1966

  20. [20]

    4th Edition, 2025 , note =

    Bruno Martelli , title =. 4th Edition, 2025 , note =

    work page 2025

  21. [21]

    Ast\'erisque , FJOURNAL =

    Guirardel, Vincent and Levitt, Gilbert , TITLE =. Ast\'erisque , FJOURNAL =. 2017 , PAGES =

    work page 2017

  22. [22]

    Correction to the article

    Durham, Matthew and Hagen, Mark and Sisto, Alessandro , year =. Correction to the article. Geometry & Topology , doi =

    work page

  23. [23]

    and Osin, Denis , TITLE =

    Abbott, Carolyn and Balasubramanya, Sahana H. and Osin, Denis , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/agt.2019.19.1747 , URL =

    work page doi:10.2140/agt.2019.19.1747 2019

  24. [24]

    and Ng, Thomas and Spriano, Davide and Gupta, Radhika and Petyt, Harry , title =

    Abbott, Carolyn R. and Ng, Thomas and Spriano, Davide and Gupta, Radhika and Petyt, Harry , title =. Math. Z. , issn =. 2024 , language =. doi:10.1007/s00209-023-03411-6 , keywords =

    work page doi:10.1007/s00209-023-03411-6 2024

  25. [25]

    2012 , PAGES =

    Farb, Benson and Margalit, Dan , TITLE =. 2012 , PAGES =

    work page 2012

  26. [26]

    Gersten, S. M. , title =. Int. J. Algebra Comput. , issn =. 1992 , language =. doi:10.1142/S0218196792000190 , keywords =

    work page doi:10.1142/s0218196792000190 1992

  27. [27]

    Groups Geom

    Edletzberger, Alexandra , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2024 , NUMBER =. doi:10.4171/ggd/779 , URL =

    work page doi:10.4171/ggd/779 2024

  28. [28]

    and Mangioni, G

    Jones, O. and Mangioni, G. and Sartori, G. , year=. 2506.17157 , archivePrefix=

    work page arXiv

  29. [29]

    Groups Geom

    Margolis, Alexander , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2020 , NUMBER =. doi:10.4171/ggd/584 , URL =

    work page doi:10.4171/ggd/584 2020

  30. [30]

    Dani, Pallavi and Thomas, Anne , TITLE =. J. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/topo.12033 , URL =

    work page doi:10.1112/topo.12033 2017

  31. [31]

    Baik, Hyungryul and Jang, Wonyong , TITLE =. J. Group Theory , FJOURNAL =. 2025 , NUMBER =. doi:10.1515/jgth-2024-0126 , URL =

    work page doi:10.1515/jgth-2024-0126 2025

  32. [32]

    2025 , eprint=

    Short hierarchically hyperbolic groups I: uncountably many coarse median structures , author=. 2025 , eprint=

    work page 2025

  33. [33]

    The isomorphism problem for all hyperbolic groups , JOURNAL =

    Dahmani, Fran. The isomorphism problem for all hyperbolic groups , JOURNAL =. 2011 , NUMBER =. doi:10.1007/s00039-011-0120-0 , URL =

    work page doi:10.1007/s00039-011-0120-0 2011

  34. [34]

    , TITLE =

    Thurston, William P. , TITLE =. 1997 , PAGES =

    work page 1997

  35. [35]

    Groups Geom

    Berlai, Federico and Robbio, Bruno , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2020 , NUMBER =. doi:10.4171/ggd/576 , URL =

    work page doi:10.4171/ggd/576 2020

  36. [36]

    Bestvina, Mladen and Feighn, Mark , TITLE =. Invent. Math. , FJOURNAL =. 1995 , NUMBER =. doi:10.1007/BF01884300 , URL =

    work page doi:10.1007/bf01884300 1995

  37. [37]

    Pénélope Azuelos and Mark Hagen , title =

    work page

  38. [38]

    Abbott, Carolyn and Hagen, Mark and Petyt, Harry and Zalloum, Abdul , TITLE =. Anal. Geom. Metr. Spaces , FJOURNAL =. 2025 , NUMBER =. doi:10.1515/agms-2025-0027 , URL =

    work page doi:10.1515/agms-2025-0027 2025

  39. [39]

    Abbott, Carolyn and Behrstock, Jason and Durham, Matthew Gentry , TITLE =. Trans. Amer. Math. Soc. Ser. B , FJOURNAL =. 2021 , PAGES =. doi:10.1090/btran/50 , URL =

    work page doi:10.1090/btran/50 2021

  40. [40]

    2026 , eprint=

    Random quotients preserve acylindrical and hierarchical hyperbolicity , author=. 2026 , eprint=

    work page 2026

  41. [41]

    and Sisto, Alessandro , TITLE =

    Behrstock, Jason and Hagen, Mark F. and Sisto, Alessandro , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/plms.12026 , URL =

    work page doi:10.1112/plms.12026 2017

  42. [42]

    Haettel, Thomas and Hoda, Nima and Petyt, Harry , TITLE =. Geom. Topol. , FJOURNAL =. 2023 , NUMBER =. doi:10.2140/gt.2023.27.1587 , URL =

    work page doi:10.2140/gt.2023.27.1587 2023

  43. [43]

    2024 , eprint=

    Mapping class groups are quasicubical , author=. 2024 , eprint=

    work page 2024

  44. [44]

    Internat

    Clay, Matt , TITLE =. Internat. J. Algebra Comput. , FJOURNAL =. 2014 , NUMBER =. doi:10.1142/S0218196714500350 , URL =

    work page doi:10.1142/s0218196714500350 2014

  45. [45]

    2026 , eprint=

    A combination theorem for hierarchically quasiconvex subgroups, and application to geometric subgroups of mapping class groups , author=. 2026 , eprint=

    work page 2026

  46. [46]

    and Sisto, Alessandro , TITLE =

    Behrstock, Jason and Hagen, Mark F. and Sisto, Alessandro , TITLE =. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/gt.2017.21.1731 , URL =

    work page doi:10.2140/gt.2017.21.1731 2017

  47. [47]

    Pacific J

    Behrstock, Jason and Hagen, Mark and Sisto, Alessandro , TITLE =. Pacific J. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.2140/pjm.2019.299.257 , URL =

    work page doi:10.2140/pjm.2019.299.257 2019

  48. [48]

    and Sisto, Alessandro , TITLE =

    Behrstock, Jason and Hagen, Mark F. and Sisto, Alessandro , TITLE =. Duke Math. J. , FJOURNAL =. 2021 , NUMBER =. doi:10.1215/00127094-2020-0056 , URL =

    work page doi:10.1215/00127094-2020-0056 2021

  49. [49]

    Groups Geom

    Petyt, Harry and Spriano, Davide , Title =. Groups Geom. Dyn. , ISSN =. 2023 , Language =. doi:10.4171/GGD/706 , Keywords =

    work page doi:10.4171/ggd/706 2023

  50. [50]

    2026 , eprint=

    Bounded cohomology, quotient extensions, and hierarchical hyperbolicity , author=. 2026 , eprint=

    work page 2026

  51. [51]

    2002 , PAGES =

    Hatcher, Allen , TITLE =. 2002 , PAGES =

    work page 2002

  52. [52]

    , TITLE =

    Brown, Kenneth S. , TITLE =. 1994 , PAGES =

    work page 1994

  53. [53]

    and Harvey, W

    Maclachlan, C. and Harvey, W. J. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1975 , NUMBER =. doi:10.1112/plms/s3-30.4.496 , URL =

    work page doi:10.1112/plms/s3-30.4.496 1975

  54. [54]

    2025 , eprint=

    An extension theorem for quasimorphisms , author=. 2025 , eprint=

    work page 2025

  55. [55]

    Manning, Jason Fox , TITLE =. Algebr. Geom. Topol. , FJOURNAL =. 2008 , NUMBER =. doi:10.2140/agt.2008.8.1371 , URL =

    work page doi:10.2140/agt.2008.8.1371 2008

  56. [56]

    and Sela, Z

    Rips, E. and Sela, Z. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 1994 , NUMBER =. doi:10.1007/BF01896245 , URL =

    work page doi:10.1007/bf01896245 1994

  57. [57]

    , TITLE =

    Sela, Z. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/s000390050019 , URL =

    work page doi:10.1007/s000390050019 1997

  58. [58]

    , TITLE =

    Bowditch, Brian H. , TITLE =. Acta Math. , FJOURNAL =. 1998 , NUMBER =. doi:10.1007/BF02392898 , URL =

    work page doi:10.1007/bf02392898 1998

  59. [59]

    Swarup, G. A. , TITLE =. Electron. Res. Announc. Amer. Math. Soc. , FJOURNAL =. 1996 , NUMBER =. doi:10.1090/S1079-6762-96-00013-3 , URL =

    work page doi:10.1090/s1079-6762-96-00013-3 1996

  60. [60]

    Outer automorphisms of hyperbolic groups and small actions on

    Paulin, Fr\'. Outer automorphisms of hyperbolic groups and small actions on. Arboreal group theory (. 1991 , ISBN =. doi:10.1007/978-1-4612-3142-4\_12 , URL =

    work page doi:10.1007/978-1-4612-3142-4 1991

  61. [61]

    Ordered groups and infinite permutation groups , SERIES =

    Macpherson, Dugald , TITLE =. Ordered groups and infinite permutation groups , SERIES =. 1996 , ISBN =

    work page 1996

  62. [62]

    2025 , eprint=

    Proper actions on finite products of hyperbolic spaces , author=. 2025 , eprint=

    work page 2025

  63. [63]

    Heuer, Nicolaus , TITLE =. Math. Scand. , FJOURNAL =. 2020 , NUMBER =. doi:10.7146/math.scand.a-114969 , URL =

    work page doi:10.7146/math.scand.a-114969 2020

  64. [64]

    Israel J

    Fujiwara, Koji , TITLE =. Israel J. Math. , FJOURNAL =. 2002 , PAGES =. doi:10.1007/BF02785862 , URL =

    work page doi:10.1007/bf02785862 2002

  65. [65]

    Zieschang, Heiner , TITLE =. Math. Ann. , FJOURNAL =. 1973 , PAGES =. doi:10.1007/BF01431525 , URL =

    work page doi:10.1007/bf01431525 1973

  66. [66]

    1992 , PAGES =

    Katok, Svetlana , TITLE =. 1992 , PAGES =

    work page 1992

  67. [67]

    Elysia Wang , year=. The. 2604.18815 , archivePrefix=

    work page internal anchor Pith review Pith/arXiv arXiv

  68. [68]

    Embedding the braid group in mapping class groups , JOURNAL =

    Szepietowski, B. Embedding the braid group in mapping class groups , JOURNAL =. 2010 , NUMBER =. doi:10.5565/PUBLMAT\_54210\_04 , URL =

    work page doi:10.5565/publmat 2010

  69. [69]

    Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces , JOURNAL =

    Lima Gon. Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces , JOURNAL =. 2018 , NUMBER =. doi:10.5802/cml.45 , URL =

    work page doi:10.5802/cml.45 2018

  70. [70]

    , TITLE =

    Ruane, Kim E. , TITLE =. Geom. Dedicata , FJOURNAL =. 2001 , NUMBER =. doi:10.1023/A:1010301824765 , URL =

    work page doi:10.1023/a:1010301824765 2001

  71. [71]

    and Chillingworth, D

    Birman, Joan S. and Chillingworth, D. R. J. , TITLE =. Proc. Cambridge Philos. Soc. , FJOURNAL =. 1972 , PAGES =. doi:10.1017/s0305004100050714 , URL =

    work page doi:10.1017/s0305004100050714 1972

  72. [72]

    Macbeath, A. M. , TITLE =. Canadian J. Math. , FJOURNAL =. 1967 , PAGES =. doi:10.4153/CJM-1967-108-5 , URL =

    work page doi:10.4153/cjm-1967-108-5 1967

  73. [73]

    2017 , PAGES =

    Frigerio, Roberto , TITLE =. 2017 , PAGES =. doi:10.1090/surv/227 , URL =

    work page doi:10.1090/surv/227 2017