pith. sign in

arxiv: 2606.31774 · v1 · pith:JAR2JLF5new · submitted 2026-06-30 · 🧮 math.GR · math.GT

Thurston norm, polytopes and splitting complexity

Andrei Jaikin-Zapirain , Monika Kudlinska , Pablo S\'anchez-Peralta This is my paper

Pith reviewed 2026-07-01 02:42 UTC · model grok-4.3

classification 🧮 math.GR math.GT
keywords Thurston normL2-Betti numbersStrong Atiyah Conjecturefree-by-cyclic groupsBieri-Neumann-Strebel invariantpolytopesplitting complexity3-manifold groups
0
0 comments X

The pith

For finitely generated torsion-free groups satisfying the Strong Atiyah Conjecture with vanishing first L²-Betti number, the first L²-Betti numbers of kernels of integral characters extend to a seminorm on real cohomology that is induced by

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

The paper establishes that under these hypotheses on a group G, the function sending each surjective integral character to the first L²-Betti number of its kernel extends continuously to a seminorm on the real first cohomology group. This seminorm, termed the Thurston norm, is realized by a polytope in the dual real homology group. The construction generalizes to higher L²-Betti numbers of kernels and confirms an existing conjecture in that direction. When G is free-by-cyclic or the fundamental group of an admissible 3-manifold, the norm admits a combinatorial description in terms of splitting complexity of characters, which in turn yields an algorithm for computing the Bieri-Neumann-Strebel invariant.

Core claim

If G is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first L²-Betti number, then the map that assigns to each surjective integral character the first L²-Betti number of the kernel extends to a seminorm on the first cohomology group of G with real coefficients; this seminorm is induced by a polytope in the first homology group with real coefficients. The result also holds for higher L²-Betti numbers.

What carries the argument

The Thurston norm on H¹(G, ℝ) obtained by extending the assignment χ ↦ b₁⁽²⁾(ker χ) for surjective integral characters χ, realized geometrically by a polytope in H₁(G, ℝ).

If this is right

  • The Thurston norm admits a combinatorial interpretation in terms of splitting complexity for free-by-cyclic groups and admissible 3-manifold groups.
  • There exists an algorithm to compute the Bieri-Neumann-Strebel invariant of any free-by-cyclic group.
  • The construction confirms the conjecture of Friedl, Lück and Tillmann on higher L²-Betti numbers.
  • The same combinatorial description confirms the conjecture of Gardam and Kielak relating the Thurston norm to splitting complexity.

Where Pith is reading between the lines

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

  • The polytope construction may supply new computable invariants for deciding isomorphism questions among free-by-cyclic groups.
  • The same extension technique could be tested on other classes of groups known to satisfy the Strong Atiyah Conjecture, such as certain one-relator groups.
  • If the polytope is rational, its vertices would give finitely many extremal characters whose kernels realize the minimal L²-Betti numbers.

Load-bearing premise

The group must be finitely generated and torsion-free, must satisfy the Strong Atiyah Conjecture, and must have vanishing first L²-Betti number.

What would settle it

A concrete finitely generated torsion-free group that satisfies the Strong Atiyah Conjecture, has vanishing first L²-Betti number, yet for which the function sending integral characters to kernel L²-Betti numbers fails to extend continuously to a seminorm on real cohomology.

read the original abstract

We show that if $G$ is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first $L^{2}$-Betti number, then the map that assigns to each surjective integral character the first $L^2$-Betti number of the kernel extends to a seminorm on the first cohomology group of $G$ with real coefficients. We call this seminorm the Thurston norm. Moreover, we show that this norm is induced by a polytope in the first homology group with real coefficients. We also generalize this result to higher $L^{2}$-Betti numbers of the kernels, thereby confirming a conjecture of Friedl, L\"uck and Tillmann. In the case where $G$ is either a free-by-cyclic group or the fundamental group of an admissible $3$-manifold, we show that the Thurston norm of $G$ admits a combinatorial interpretation that relates it to the splitting complexity of the character. This confirms a conjecture of Gardam and Kielak. As an application, we show that there exists an algorithm to compute the Bieri--Neumann--Strebel invariant of free-by-cyclic groups, and discuss connections to the isomorphism problem in free-by-cyclic groups.

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 / 2 minor

Summary. The paper shows that if G is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with b_1^{(2)}(G)=0, then the map sending each surjective integral character χ to b_1^{(2)}(ker χ) extends to a seminorm (the Thurston norm) on H^1(G;ℝ) and is induced by a polytope in H_1(G;ℝ). The result is generalized to higher L²-Betti numbers of kernels, confirming a conjecture of Friedl-Lück-Tillmann. For free-by-cyclic groups and fundamental groups of admissible 3-manifolds, a combinatorial interpretation in terms of splitting complexity is given, confirming a conjecture of Gardam-Kielak; this yields an algorithm for the BNS invariant of free-by-cyclic groups and connections to the isomorphism problem.

Significance. If the claims hold, the work supplies a group-theoretic extension of the Thurston norm via L²-Betti numbers under standard hypotheses, realizes it by a polytope, and confirms two existing conjectures. The combinatorial interpretation for free-by-cyclic groups and the resulting algorithm for the BNS invariant constitute concrete, falsifiable advances with potential implications for the isomorphism problem in that class. The hypotheses (including the Strong Atiyah Conjecture) are stated explicitly and the construction is direct from independently defined L² invariants.

minor comments (2)
  1. [§1] §1: the statement of the main theorem could explicitly record that the polytope is defined over ℤ (or ℚ) before passing to real coefficients, to clarify the integral structure used in the applications.
  2. [§4] §4 (combinatorial interpretation): the relation between the Thurston norm and splitting complexity is stated for admissible 3-manifolds; a brief remark on whether the same formula holds verbatim for free-by-cyclic groups would help the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report, which recommends acceptance of the manuscript. We appreciate the recognition of the results confirming the conjectures of Friedl-Lück-Tillmann and Gardam-Kielak, as well as the algorithmic application to the BNS invariant.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central result defines the Thurston norm explicitly as the (proven) extension of the independently defined map χ ↦ b₁⁽²⁾(ker χ) from surjective integral characters to a seminorm on H¹(G; ℝ), under the stated hypotheses on G (finitely generated, torsion-free, Strong Atiyah Conjecture, b₁⁽²⁾(G)=0). The polytope realization and higher Betti generalizations are likewise derived from these external inputs rather than by re-labeling or fitting. No self-definitional equations, fitted-input predictions, or load-bearing self-citations appear; the derivation remains self-contained against the external L²-Betti theory.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the Strong Atiyah Conjecture (domain assumption) and the vanishing of the first L2-Betti number (domain assumption); both are stated as hypotheses rather than derived.

axioms (3)
  • domain assumption Strong Atiyah Conjecture
    Invoked to ensure that L2-Betti numbers behave well enough to define a seminorm (abstract).
  • domain assumption vanishing first L2-Betti number
    Required for the extension map to be a seminorm (abstract, first sentence).
  • domain assumption G finitely generated and torsion-free
    Basic structural hypothesis on the group (abstract).

pith-pipeline@v0.9.1-grok · 5763 in / 1607 out tokens · 54410 ms · 2026-07-01T02:42:04.328772+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

298 extracted references · 196 canonical work pages · 2 internal anchors

  1. [1]

    Boileau, Michel and Friedl, Stefan , TITLE =. Osaka J. Math. , FJOURNAL =. 2019 , NUMBER =

    2019

  2. [2]

    Baumslag, Gilbert , TITLE =. Bull. Austral. Math. Soc. , FJOURNAL =. 1971 , PAGES =. doi:10.1017/S0004972700046906 , URL =

    work page doi:10.1017/s0004972700046906 1971

  3. [3]

    Bieri, Robert and Strebel, Ralph , TITLE =. Comment. Math. Helv. , FJOURNAL =. 1978 , NUMBER =. doi:10.1007/BF02566077 , URL =

    work page doi:10.1007/bf02566077 1978

  4. [4]

    , TITLE =

    Berberian, Sterling K. , TITLE =. 1972 , PAGES =

    1972

  5. [5]

    Rocky Mountain J

    Berberian, Sterling Khazaz , TITLE =. Rocky Mountain J. Math. , FJOURNAL =. 1982 , NUMBER =. doi:10.1216/RMJ-1982-12-1-149 , URL =

    work page doi:10.1216/rmj-1982-12-1-149 1982

  6. [6]

    Bestvina, Mladen and Feighn, Mark and Handel, Michael , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2000 , NUMBER =. doi:10.2307/121043 , URL =

    work page doi:10.2307/121043 2000

  7. [7]

    Bestvina, Mladen and Fujiwara, Koji and Wigglesworth, Derrick , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2023 , NUMBER =. doi:10.1093/imrn/rnac012 , URL =

    work page doi:10.1093/imrn/rnac012 2023

  8. [8]

    Bestvina, Mladen and Handel, Michael , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1992 , NUMBER =. doi:10.2307/2946562 , URL =

    work page doi:10.2307/2946562 1992

  9. [9]

    and Strebel, Ralph , TITLE =

    Bieri, Robert and Neumann, Walter D. and Strebel, Ralph , TITLE =. Invent. Math. , FJOURNAL =. 1987 , NUMBER =

    1987

  10. [10]

    Bogopolski, Oleg and Martino, Armando and Ventura, Eric , TITLE =. Comm. Algebra , FJOURNAL =. 2007 , NUMBER =. doi:10.1080/00927870601169333 , URL =

    work page doi:10.1080/00927870601169333 2007

  11. [11]

    What's next?---the mathematical legacy of

    Boileau, Michel and Friedl, Stefan , TITLE =. What's next?---the mathematical legacy of. 2020 , MRCLASS =. doi:10.2307/j.ctvthhdvv.5 , URL =

    work page doi:10.2307/j.ctvthhdvv.5 2020

  12. [12]

    and Groves, Daniel , TITLE =

    Bridson, Martin R. and Groves, Daniel , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =. doi:10.1090/S0065-9266-09-00578-X , URL =

    work page doi:10.1090/s0065-9266-09-00578-x 2010

  13. [13]

    and Haefliger, Andr\'

    Bridson, Martin R. and Haefliger, Andr\'. Metric spaces of non-positive curvature , SERIES =. 1999 , PAGES =

    1999

  14. [14]

    and McReynolds, David B

    Bridson, Martin R. and McReynolds, David B. and Reid, Alan W. and Spitler, Ryan , Title =. Ann. Math. (2) , ISSN =. 2020 , Language =. doi:10.4007/annals.2020.192.3.1 , Keywords =

    work page doi:10.4007/annals.2020.192.3.1 2020

  15. [15]

    and Conder, Marston and Reid, Alan , TITLE =

    Bridson, Martin R. and Conder, Marston and Reid, Alan , TITLE =. Israel J. Math. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s11856-016-1341-6 , URL =

    work page doi:10.1007/s11856-016-1341-6 2016

  16. [16]

    and Reid, Alan W

    Bridson, Martin R. and Reid, Alan W. , TITLE =. What's next?---the mathematical legacy of. 2020 , MRCLASS =. doi:10.2307/j.ctvthhdvv.6 , URL =

    work page doi:10.2307/j.ctvthhdvv.6 2020

  17. [17]

    and Reid, Alan W

    Bridson, Martin R. and Reid, Alan W. and Wilton, Henry , TITLE =. Bull. Lond. Math. Soc. , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/blms.12076 , URL =

    work page doi:10.1112/blms.12076 2017

  18. [18]

    Brinkmann, Peter , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2000 , NUMBER =. doi:10.1007/PL00001647 , URL =

    work page doi:10.1007/pl00001647 2000

  19. [19]

    , TITLE =

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

    1994

  20. [20]

    Button, J. O. , TITLE =. Israel J. Math. , FJOURNAL =. 2008 , PAGES =. doi:10.1007/s11856-008-1043-9 , URL =

    work page doi:10.1007/s11856-008-1043-9 2008

  21. [21]

    2013 , eprint=

    Free by cyclic groups are large , author=. 2013 , eprint=

    2013

  22. [22]

    and Levitt, Gilbert , TITLE =

    Cashen, Christopher H. and Levitt, Gilbert , TITLE =. J. Group Theory , FJOURNAL =. 2016 , NUMBER =

    2016

  23. [23]

    Cohn, P. M. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 1971 , PAGES =. doi:10.1112/plms/s3-23.2.193 , URL =

    work page doi:10.1112/plms/s3-23.2.193 1971

  24. [24]

    Cohn, P. M. , TITLE =. 2006 , PAGES =. doi:10.1017/CBO9780511542794 , URL =

    work page doi:10.1017/cbo9780511542794 2006

  25. [25]

    Collins, D. J. and Turner, E. C. , TITLE =. Michigan Math. J. , FJOURNAL =. 1994 , NUMBER =. doi:10.1307/mmj/1029005072 , URL =

    work page doi:10.1307/mmj/1029005072 1994

  26. [26]

    Collins, D. J. and Turner, E. C. , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 1996 , NUMBER =. doi:10.1017/S0305004100074466 , URL =

    work page doi:10.1017/s0305004100074466 1996

  27. [27]

    2022 , eprint=

    Absolute profinite rigidity of some closed fibered hyperbolic 3-manifolds , author=. 2022 , eprint=

    2022

  28. [28]

    Culler, Marc and Vogtmann, Karen , TITLE =. Invent. Math. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF01388734 , URL =

    work page doi:10.1007/bf01388734 1986

  29. [29]

    , TITLE =

    Dicks, Warren and Sontag, Eduardo D. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1978 , NUMBER =. doi:10.1016/0022-4049(78)90011-7 , URL =

    work page doi:10.1016/0022-4049(78)90011-7 1978

  30. [30]

    Dicks, Warren and Ventura, Enric , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 1993 , NUMBER =. doi:10.1016/0022-4049(93)90012-I , URL =

    work page doi:10.1016/0022-4049(93)90012-i 1993

  31. [31]

    Dieudonn\'. Les d\'. Bull. Soc. Math. France , FJOURNAL =. 1943 , PAGES =

    1943

  32. [32]

    and Formanek, Edward W

    Dixon, John D. and Formanek, Edward W. and Poland, John C. and Ribes, Luis , Title =. J. Pure Appl. Algebra , ISSN =. 1982 , Language =. doi:10.1016/0022-4049(82)90098-6 , Keywords =

    work page doi:10.1016/0022-4049(82)90098-6 1982

  33. [33]

    , TITLE =

    Dowdall, Spencer and Taylor, Samuel J. , TITLE =. Geom. Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.2140/gt.2018.22.517 , URL =

    work page doi:10.2140/gt.2018.22.517 2018

  34. [34]

    and Friedl, Stefan and Jackson, Nicholas , TITLE =

    Dunfield, Nathan M. and Friedl, Stefan and Jackson, Nicholas , TITLE =. Exp. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1080/10586458.2012.669268 , URL =

    work page doi:10.1080/10586458.2012.669268 2012

  35. [35]

    and Friedl, Stefan and Jackson, Nicholas , TITLE =

    Dunfield, Nathan M. and Friedl, Stefan and Jackson, Nicholas , TITLE =. 2010 , NOTE =

    2010

  36. [36]

    Eckmann, Beno , TITLE =. Arch. Math. (Basel) , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/s000130050565 , URL =

    work page doi:10.1007/s000130050565 2001

  37. [37]

    Elek, G\'abor and Szab\'o, Endre , TITLE =. J. Algebra , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/j.jalgebra.2004.06.023 , URL =

    work page doi:10.1016/j.jalgebra.2004.06.023 2004

  38. [38]

    Feighn, Mark and Handel, Michael , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1999 , NUMBER =. doi:10.2307/121081 , URL =

    work page doi:10.2307/121081 1999

  39. [39]

    and S\'anchez-Peralta, Pablo , TITLE =

    Fisher, Sam P. and S\'anchez-Peralta, Pablo , TITLE =. J. Comb. Algebra , FJOURNAL =. 2026 , NUMBER =. doi:10.4171/jca/89 , URL =

    work page doi:10.4171/jca/89 2026

  40. [40]

    Illinois J

    Francaviglia, Stefano and Martino, Armando , TITLE =. Illinois J. Math. , FJOURNAL =. 2015 , NUMBER =

    2015

  41. [41]

    Francaviglia, Stefano and Martino, Armando , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2021 , NUMBER =. doi:10.1090/tran/8333 , URL =

    work page doi:10.1090/tran/8333 2021

  42. [42]

    Topology , FJOURNAL =

    Friedl, Stefan and Kim, Taehee , TITLE =. Topology , FJOURNAL =. 2006 , NUMBER =

    2006

  43. [43]

    Universal

    Friedl, Stefan and L\". Universal. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2017 , NUMBER =. doi:10.1112/plms.12035 , URL =

    work page doi:10.1112/plms.12035 2017

  44. [44]

    Friedl, Stefan and L\". Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2019 , NUMBER =. doi:10.1112/plms.12202 , URL =

    work page doi:10.1112/plms.12202 2019

  45. [45]

    Friedl, Stefan and Schreve, Kevin and Tillmann, Stephan , TITLE =. Geom. Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/gt.2017.21.3759 , URL =

    work page doi:10.2140/gt.2017.21.3759 2017

  46. [46]

    Friedl, Stefan and Tillmann, Stephan , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2020 , NUMBER =

    2020

  47. [47]

    Friedl, Stefan and Vidussi, Stefano , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2008 , NUMBER =

    2008

  48. [48]

    Friedl, Stefan and Vidussi, Stefano , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2011 , NUMBER =

    2011

  49. [49]

    The mathematics of knots , SERIES =

    Friedl, Stefan and Vidussi, Stefano , TITLE =. The mathematics of knots , SERIES =. 2011 , MRCLASS =

    2011

  50. [50]

    Friedl, Stefan and Vidussi, Stefano , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2013 , NUMBER =

    2013

  51. [51]

    Funar, Louis , Title =. Geom. Topol. , ISSN =. 2013 , Language =. doi:10.2140/gt.2013.17.2289 , Keywords =

    work page doi:10.2140/gt.2013.17.2289 2013

  52. [52]

    Gaboriau, Damien , title =. Publ. Math. Inst. Hautes \'. 2002 , pages =. doi:10.1007/s102400200002 , url =

    work page doi:10.1007/s102400200002 2002

  53. [53]

    Wong, Peng Choon and Gan, Hui Woan , TITLE =. Bull. Malaysian Math. Soc. (2) , FJOURNAL =. 1999 , NUMBER =

    1999

  54. [54]

    2021 , eprint=

    JSJ decompositions and polytopes for two-generator one-relator groups , author=. 2021 , eprint=

    2021

  55. [55]

    Genevois, Anthony and Horbez, Camille , TITLE =. J. Topol. , FJOURNAL =. 2021 , NUMBER =

    2021

  56. [56]

    and Stallings, John R

    Gersten, Stephen M. and Stallings, John R. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 1991 , NUMBER =. doi:10.2307/2048318 , URL =

    work page doi:10.2307/2048318 1991

  57. [57]

    Goodearl, K. R. , TITLE =. 1991 , PAGES =

    1991

  58. [58]

    Free division rings of fractions of crossed products of groups with

    Gr. Free division rings of fractions of crossed products of groups with. Forum Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1515/forum-2019-0264 , URL =

    work page doi:10.1515/forum-2019-0264 2020

  59. [59]

    Grunewald, Fritz and Jaikin-Zapirain, Andrei and Pinto, Aline G. S. and Zalesskii, Pavel A. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2014 , NUMBER =. doi:10.1016/j.jpaa.2013.10.003 , URL =

    work page doi:10.1016/j.jpaa.2013.10.003 2014

  60. [60]

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

    2017

  61. [61]

    and Sapir, Mark and Wise, Daniel T

    Geoghegan, Ross and Mihalik, Michael L. and Sapir, Mark and Wise, Daniel T. , TITLE =. Bull. London Math. Soc. , FJOURNAL =. 2001 , NUMBER =. doi:10.1017/S0024609301007986 , URL =

    work page doi:10.1017/s0024609301007986 2001

  62. [62]

    and Wise, Daniel T

    Hagen, Mark F. and Wise, Daniel T. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00039-015-0314-y , URL =

    work page doi:10.1007/s00039-015-0314-y 2015

  63. [63]

    Special cube complexes , JOURNAL =

    Haglund, Fr\'. Special cube complexes , JOURNAL =. 2008 , NUMBER =. doi:10.1007/s00039-007-0629-4 , URL =

    work page doi:10.1007/s00039-007-0629-4 2008

  64. [64]

    , TITLE =

    Hamilton, Emily and Wilton, Henry and Zalesskii, Pavel A. , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2013 , NUMBER =. doi:10.1112/jlms/jds040 , URL =

    work page doi:10.1112/jlms/jds040 2013

  65. [65]

    Healy, Brendan Burns , Title =. Topol. Proc. , ISSN =. 2021 , Language =

    2021

  66. [66]

    Henneke, Fabian and Kielak, Dawid , Title =. Fundam. Math. , ISSN =. 2021 , Language =. doi:10.4064/fm808-4-2021 , Keywords =

    work page doi:10.4064/fm808-4-2021 2021

  67. [67]

    2014 , eprint=

    Some 3-manifold groups with the same finite quotients , author=. 2014 , eprint=

    2014

  68. [68]

    Hughes, Ian , title =. Comm. Pure Appl. Math. , fjournal =. 1970 , pages =. doi:10.1002/cpa.3160230205 , url =

    work page doi:10.1002/cpa.3160230205 1970

  69. [69]

    2022 , eprint=

    Profinite rigidity of fibring , author=. 2022 , eprint=

    2022

  70. [70]

    1983 , PAGES =

    Jiang, Boju , TITLE =. 1983 , PAGES =

    1983

  71. [71]

    Jaikin-Zapirain, Andrei , TITLE =. Groups. 2019 , ISBN =

    2019

  72. [72]

    Jaikin-Zapirain, Andrei , Title =. Geom. Topol. , ISSN =. 2020 , Language =. doi:10.2140/gt.2020.24.409 , Keywords =

    work page doi:10.2140/gt.2020.24.409 2020

  73. [73]

    Jaikin-Zapirain, Andrei , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00039-019-00487-3 , URL =

    work page doi:10.1007/s00039-019-00487-3 2019

  74. [74]

    Selecta Math

    Jaikin-Zapirain, Andrei , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00029-021-00691-w , URL =

    work page doi:10.1007/s00029-021-00691-w 2021

  75. [75]

    Jaikin-Zapirain, Andrei and L\'opez-\'Alvarez, Diego , TITLE =. Math. Ann. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s00208-019-01926-0 , URL =

    work page doi:10.1007/s00208-019-01926-0 2020

  76. [76]

    Jaikin-Zapirain, Andrei and Linton, Marco , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2025 , NUMBER =. doi:10.4007/annals.2025.201.3.4 , URL =

    work page doi:10.4007/annals.2025.201.3.4 2025

  77. [77]

    Jankiewicz, Kasia and Schreve, Kevin , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2025 , NUMBER =. doi:10.1016/j.jpaa.2024.107775 , URL =

    work page doi:10.1016/j.jpaa.2024.107775 2025

  78. [78]

    2019 , PAGES =

    Kammeyer, Holger , TITLE =. 2019 , PAGES =

    2019

  79. [79]

    1972 , PAGES =

    Kaplansky, Irving , TITLE =. 1972 , PAGES =

    1972

  80. [80]

    Kapovich, Ilya and Weidmann, Richard , TITLE =. Math. Z. , FJOURNAL =. 1999 , NUMBER =. doi:10.1007/PL00004753 , URL =

    work page doi:10.1007/pl00004753 1999

Showing first 80 references.