pith. sign in

arxiv: 2411.02516 · v3 · submitted 2024-11-04 · 🧮 math.GR · math.GT

Thurston norm for coherent right-angled Artin groups via L²-invariants

Monika Kudlinska This is my paper

Pith reviewed 2026-05-23 17:22 UTC · model grok-4.3

classification 🧮 math.GR math.GT
keywords right-angled Artin groupsThurston normL2-invariantssplitting complexityL2-polytopecoherent groupsgroup cohomology
0
0 comments X

The pith

For one-ended coherent right-angled Artin groups, splitting complexity along epimorphisms to Z equals the L2-Euler characteristic of the kernel.

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

The paper defines splitting complexity of a group along a nontrivial integral character. For one-ended coherent right-angled Artin groups this quantity along any epimorphism to the integers equals the L2-Euler characteristic of the kernel. The equality is obtained from Friedl-Lück's L2-polytope and is used to produce a Thurston-type semi-norm on real first cohomology that records the minimal splitting complexity of integral characters. A reader would care because the construction supplies an explicit L2-computable norm that generalizes a classical invariant from three-manifold topology to this family of groups.

Core claim

If G is a one-ended coherent right-angled Artin group, then the splitting complexity along an epimorphism φ: G → Z equals the L2-Euler characteristic of the kernel of φ. This allows defining a Thurston-type semi-norm on H1(G; R) that measures the splitting complexity of integral characters, with the main tool being Friedl-Lück's L2-polytope.

What carries the argument

The equality of splitting complexity with the L2-Euler characteristic of the kernel, obtained by evaluating the L2-polytope of Friedl-Lück on the character.

If this is right

  • The resulting semi-norm is well-defined on the real cohomology and can be evaluated using only L2 data.
  • Minimal splitting complexity is achieved precisely by the kernels whose L2-Euler characteristics attain the norm.
  • The construction produces a semi-norm that vanishes exactly on the characters for which the associated kernels have vanishing L2-Euler characteristic.
  • Integral points in the unit ball of the norm correspond to characters realizing the minimal possible splitting complexity.

Where Pith is reading between the lines

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

  • The same L2-polytope technique might produce analogous norms for other coherent groups whose kernels admit computable L2-invariants.
  • Results about the unit ball of the classical Thurston norm on manifold groups could be tested for transfer to these Artin groups via the new equality.
  • The method supplies a concrete way to decide whether a given character realizes minimal splitting complexity by direct L2 computation.

Load-bearing premise

The groups under consideration are one-ended coherent right-angled Artin groups, so that the L2-polytope applies directly without further restrictions.

What would settle it

An explicit one-ended coherent right-angled Artin group together with an epimorphism to Z for which the splitting complexity differs from the L2-Euler characteristic of the kernel.

read the original abstract

We define a new notion of splitting complexity for a group $G$ along a non-trivial integral character $\phi \in H^1(G; \mathbb{Z})$. If $G$ is a one-ended coherent right-angled Artin group, we show that the splitting complexity along an epimorphism $\phi \colon G \to \mathbb{Z}$ equals the $L^2$-Euler characteristic of the kernel of $\phi$. This allows us to define a Thurston-type semi-norm $\| \cdot \|_T \colon H^1(G ; \mathbb{R}) \to \mathbb{R}$ that measures the splitting complexity of integral characters. Our main tool is Friedl--L\"{u}ck's $L^2$-polytope.

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 defines a new notion of splitting complexity for a group G along a non-trivial integral character φ ∈ H¹(G; ℤ). For one-ended coherent right-angled Artin groups, it shows that the splitting complexity along an epimorphism φ: G → ℤ equals the L²-Euler characteristic of the kernel of φ. This is used to define a Thurston-type semi-norm ‖⋅‖_T : H¹(G; ℝ) → ℝ measuring splitting complexity of integral characters. The main tool is Friedl–Lück's L²-polytope.

Significance. If the equality holds under the stated hypotheses, the result links a new group-theoretic invariant (splitting complexity) to an existing L²-invariant for coherent one-ended RAAGs, allowing a Thurston-type norm to be defined via L²-Euler characteristics. The explicit restriction to one-ended coherent RAAGs and reliance on the external Friedl–Lück tool are appropriately scoped; the approach could aid computations in geometric group theory where L²-methods apply.

minor comments (3)
  1. The abstract states the main equality cleanly but the manuscript should include an explicit statement of the main theorem (with hypotheses) in the introduction or §1 to make the central claim immediately locatable.
  2. Notation for L²-invariants is not fully standardized in the provided abstract (mix of L^2 and L²); ensure consistent use throughout, e.g., in the definition of the polytope and the Euler characteristic.
  3. The definition of splitting complexity is introduced but its precise formula (likely in terms of minimal number of splittings or generators) should be displayed as a numbered equation for reference when equating it to χ^{(2)}(ker φ).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the significance statement, and for recommending minor revision. The report lists no major comments under the MAJOR COMMENTS section.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external L²-polytope result

full rationale

The paper defines splitting complexity for groups G along characters φ, then for one-ended coherent right-angled Artin groups equates it to χ^{(2)}(ker φ) via the external Friedl-Lück L²-polytope. This is an application of an independent theorem rather than any self-definitional reduction, fitted prediction, or self-citation chain. The hypotheses are explicitly scoped and the central equality does not reduce to inputs by construction within the paper. No load-bearing self-citations or ansatz smuggling are present in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard properties of L2-invariants and the coherence/one-endedness assumptions for right-angled Artin groups; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Friedl-Lück L2-polytope computes the relevant L2-Euler characteristic for the groups in question
    Invoked as the main tool for the equality.
  • domain assumption Coherent right-angled Artin groups admit the stated splittings along integral characters
    Required for the equality to be stated.

pith-pipeline@v0.9.0 · 5656 in / 1352 out tokens · 37239 ms · 2026-05-23T17:22:11.444538+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

29 extracted references · 29 canonical work pages

  1. [1]

    When does a right-angled A rtin group split over Z ? Internat

    Matt Clay. When does a right-angled A rtin group split over Z ? Internat. J. Algebra Comput. , 24(6):815--825, 2014

    work page 2014

  2. [2]

    Skew field constructions , volume No

    Paul Moritz Cohn. Skew field constructions , volume No. 27 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge-New York-Melbourne, 1977

    work page 1977

  3. [3]

    Limit groups over coherent right-angled A rtin groups

    Montserrat Casals-Ruiz, Andrew Duncan, and Ilya Kazachkov. Limit groups over coherent right-angled A rtin groups. Publ. Mat. , 67(1):199--257, 2023

    work page 2023

  4. [4]

    Rank gradients of infinite cyclic covers of 3-manifolds

    Jason DeBlois, Stefan Friedl, and Stefano Vidussi. Rank gradients of infinite cyclic covers of 3-manifolds. Michigan Math. J. , 63(1):65--81, 2014

    work page 2014

  5. [5]

    M. W. Davis and I. J. Leary. The l^2 -cohomology of A rtin groups. J. London Math. Soc. (2) , 68(2):493--510, 2003

    work page 2003

  6. [6]

    Graph groups, coherence, and three-manifolds

    Carl Droms. Graph groups, coherence, and three-manifolds. J. Algebra , 106(2):484--489, 1987

    work page 1987

  7. [7]

    Fisher, Sam Hughes, and Ian J

    Sam P. Fisher, Sam Hughes, and Ian J. Leary. Homological growth of A rtin kernels in positive characteristic. Math. Ann. , 389(1):819--843, 2024

    work page 2024

  8. [8]

    Alexander and T hurston norms, and the B ieri- N eumann- S trebel invariants for free-by-cyclic groups

    Florian Funke and Dawid Kielak. Alexander and T hurston norms, and the B ieri- N eumann- S trebel invariants for free-by-cyclic groups. Geom. Topol. , 22(5):2647--2696, 2018

    work page 2018

  9. [9]

    Universal L^2 -torsion, polytopes and applications to 3-manifolds

    Stefan Friedl and Wolfgang L\" u ck. Universal L^2 -torsion, polytopes and applications to 3-manifolds. Proc. Lond. Math. Soc. (3) , 114(6):1114--1151, 2017

    work page 2017

  10. [10]

    L^2 - E uler characteristics and the T hurston norm

    Stefan Friedl and Wolfgang L\"uck. L^2 - E uler characteristics and the T hurston norm. Proc. Lond. Math. Soc. (3) , 118(4):857--900, 2019

    work page 2019

  11. [11]

    Groups and polytopes

    Stefan Friedl, Wolfgang L\"uck, and Stephan Tillmann. Groups and polytopes. In Breadth in contemporary topology , volume 102 of Proc. Sympos. Pure Math. , pages 57--77. Amer. Math. Soc., Providence, RI, 2019

    work page 2019

  12. [12]

    Limit Groups over Coherent Right-Angled Artin Groups Are Cyclic Subgroup Separable

    Jonathan Fruchter. Limit Groups over Coherent Right-Angled Artin Groups Are Cyclic Subgroup Separable . Michigan Mathematical Journal , 73(5):909 -- 923, 2023

    work page 2023

  13. [13]

    Two-generator one-relator groups and marked polytopes

    Stefan Friedl and Stephan Tillmann. Two-generator one-relator groups and marked polytopes. Ann. Inst. Fourier (Grenoble) , 70(2):831--879, 2020

    work page 2020

  14. [14]

    The L^2 -Torsion Polytope of Groups and the Integral Polytope Group

    Florian Funke. The L^2 -Torsion Polytope of Groups and the Integral Polytope Group . PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, January 2018

    work page 2018

  15. [15]

    Computing fibring of free-by-cyclic groups

    Giles Gardam and Dawid Kielak. Computing fibring of free-by-cyclic groups. In preparation

    work page

  16. [16]

    The agrarian polytope of two-generator one-relator groups

    Fabian Henneke and Dawid Kielak. The agrarian polytope of two-generator one-relator groups. J. Lond. Math. Soc. (2) , 102(2):722--748, 2020

    work page 2020

  17. [17]

    The strong A tiyah and L \"uck approximation conjectures for one-relator groups

    Andrei Jaikin-Zapirain and Diego L\'opez-\'Alvarez. The strong A tiyah and L \"uck approximation conjectures for one-relator groups. Math. Ann. , 376(3-4):1741--1793, 2020

    work page 2020

  18. [18]

    The B ieri- N eumann- S trebel invariants via N ewton polytopes

    Dawid Kielak. The B ieri- N eumann- S trebel invariants via N ewton polytopes. Invent. Math. , 219(3):1009--1068, 2020

    work page 2020

  19. [19]

    Kochloukova and Jone Lopez de Gamiz Zearra

    Dessislava H. Kochloukova and Jone Lopez de Gamiz Zearra . On the bieri-neumann-strebel-renz invariants and limit groups over droms raags. Journal of Algebra , 606:170--194, 2022

    work page 2022

  20. [20]

    Geometry and combinatorics via right-angled A rtin groups

    Thomas Koberda. Geometry and combinatorics via right-angled A rtin groups. In In the tradition of T hurston II . G eometry and groups , pages 475--518. Springer, Cham, [2022] 2022

    work page 2022

  21. [21]

    Agrarian and ^2 - B etti numbers of locally indicable groups, with a twist

    Dawid Kielak and Bin Sun. Agrarian and ^2 - B etti numbers of locally indicable groups, with a twist. Math. Ann. , 390(3):3567--3619, 2024

    work page 2024

  22. [22]

    Peter A. Linnell. Division rings and group von N eumann algebras. Forum Math. , 5(6):561--576, 1993

    work page 1993

  23. [23]

    The strong A tiyah conjecture for right-angled A rtin and C oxeter groups

    Peter Linnell, Boris Okun, and Thomas Schick. The strong A tiyah conjecture for right-angled A rtin and C oxeter groups. Geom. Dedicata , 158:261--266, 2012

    work page 2012

  24. [24]

    L^2 -invariants: theory and applications to geometry and K -theory , volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete

    Wolfgang L \" u ck. L^2 -invariants: theory and applications to geometry and K -theory , volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] . Springer-Verlag, Berlin, 2002

    work page 2002

  25. [25]

    The B ieri- N eumann- S trebel invariants for graph groups

    John Meier and Leonard VanWyk. The B ieri- N eumann- S trebel invariants for graph groups. Proc. London Math. Soc. (3) , 71(2):263--280, 1995

    work page 1995

  26. [26]

    Oberwolfach Rep

    Geometric structures in group theory (hybrid meeting). Oberwolfach Rep. , 17(2-3):877--918, 2020. Abstracts from the conference held June 21--27, 2020, Organized by Martin Bridson, Cornelia Dru t u, Linus Kramer, Bertrand R\' e my and Petra Schwer

    work page 2020

  27. [27]

    Torsion invariants of complexes of groups

    Boris Okun and Kevin Schreve. Torsion invariants of complexes of groups. Duke Math. J. , 173(2):391--418, 2024

    work page 2024

  28. [28]

    Jean-Pierre Serre. Trees . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation

    work page 2003

  29. [29]

    Thurston

    William P. Thurston. A norm for the homology of 3 -manifolds. Mem. Amer. Math. Soc. , 59(339):i--vi and 99--130, 1986

    work page 1986