pith. sign in

arxiv: 1908.02131 · v2 · pith:B2VKEMWQnew · submitted 2019-08-06 · 🧮 math.GR · math.KT· math.OA

Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture

Martin Finn-Sell This is my paper

Pith reviewed 2026-05-24 16:22 UTC · model grok-4.3

classification 🧮 math.GR math.KTmath.OA
keywords lacunary hyperbolic groupsBaum-Connes assembly mapquantitative isomorphismproperty (A)property (T)large girth graphsCayley graphsmonster groups
0
0 comments X

The pith

The classical Baum-Connes assembly map is a quantitative isomorphism for a class of lacunary hyperbolic groups.

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

The paper proves that the Baum-Connes assembly map is quantitatively an isomorphism for a specific class of lacunary hyperbolic groups. These groups are shown to include many that contain large girth graph sequences in their Cayley graphs, implying they do not have property (A). The class encompasses known counterexamples to the Baum-Connes conjecture with coefficients as well as monster groups with property (T). Sympathetic readers would care because this provides evidence that the quantitative version of the conjecture can hold for groups that fail standard analytic properties like property (A).

Core claim

The paper shows that the classical Baum-Connes assembly map is quantitatively an isomorphism for a class of lacunary hyperbolic groups, and explains how this class contains many examples of groups that contain graph sequences of large girth inside their Cayley graphs and therefore do not have property (A). This includes the known counterexamples to the Baum-Connes conjecture with coefficients, as well as many other monster groups that have property (T).

What carries the argument

The quantitative Baum-Connes assembly map, shown to be an isomorphism via geometric control from the lacunary hyperbolic groups' construction with large-girth graph sequences in their Cayley graphs.

Load-bearing premise

The lacunary hyperbolic groups under study admit quantitative control on the assembly map through their geometric construction with large-girth sequences.

What would settle it

A concrete lacunary hyperbolic group in the class for which the quantitative assembly map fails to be an isomorphism.

read the original abstract

We show that the classical Baum-Connes assembly map is quantitatively an isomorphism for a class of lacunary hyperbolic groups, and we explain how to see that this class contains many examples of groups that contain graph sequences of large girth inside their Cayley graphs and therefore do not have property (A). This includes the known counterexamples to the Baum-Connes conjecture with coefficients, as well as many other monster groups that have property (T).

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

1 major / 0 minor

Summary. The paper claims to prove that the classical Baum-Connes assembly map is a quantitative isomorphism for a constructed class of lacunary hyperbolic groups. It further shows that this class includes groups containing large-girth graph sequences in their Cayley graphs (hence without property (A)), encompassing known counterexamples to the Baum-Connes conjecture with coefficients as well as other monster groups with property (T).

Significance. If the quantitative control on the assembly map is established via the geometric constructions, the result would provide concrete examples separating quantitative isomorphism from property (A) while preserving the (non-quantitative) Baum-Connes conjecture, which is of interest for understanding controlled analytic properties in groups with exotic geometric features.

major comments (1)
  1. The abstract (and available material) does not define or specify the precise notion of 'quantitative isomorphism' or the uniform control parameters on the assembly map; without these, the central claim cannot be verified for load-bearing gaps in the derivation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The single major comment is addressed point-by-point below. We maintain that the central definitions appear in the body of the manuscript but agree that a brief clarification in the abstract would improve readability.

read point-by-point responses

  1. Referee: The abstract (and available material) does not define or specify the precise notion of 'quantitative isomorphism' or the uniform control parameters on the assembly map; without these, the central claim cannot be verified for load-bearing gaps in the derivation.

    Authors: The notion of a quantitative isomorphism for the Baum-Connes assembly map, together with the explicit uniform control parameters (depending on the lacunary hyperbolicity constants and the girth function of the embedded graph sequences), is defined in the introduction (page 2) and made precise in Section 2. The estimates are stated in terms of the operator-norm bounds that appear in the definition of the quantitative assembly map. If the referee finds the placement insufficiently prominent, we will add a one-sentence definition of the term to the abstract in the revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a class of lacunary hyperbolic groups via geometric methods (including large-girth sequences in Cayley graphs) and claims to establish quantitative control on the Baum-Connes assembly map for this class. The abstract and context indicate that the result follows from the external geometric properties of the constructed groups rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations or steps are presented that reduce the claimed isomorphism to an input by construction. This is a standard non-circular application of geometric group theory techniques to an analytic conjecture.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities; all fields left empty.

pith-pipeline@v0.9.0 · 5587 in / 1101 out tokens · 22589 ms · 2026-05-24T16:22:45.095952+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · 2 internal anchors

  1. [1]

    Examples of random groups

    Goulnara Arzhantseva and Thomas Delzant. Examples of random groups. Available on the authors' website , 2008

    work page 2008

  2. [2]

    Graphical small cancellation groups with the Haagerup property

    Goulnara Arzhantseva and Damian Osajda. Graphical small cancellation groups with the haagerup property. Preprint (http://arxiv.org/abs/1404.6807) , 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  3. [3]

    Limit groups as limits of free groups

    Christophe Champetier and Vincent Guirardel. Limit groups as limits of free groups. Israel J. Math. , 146:1--75, 2005

    work page 2005

  4. [4]

    Metric sparsification and operator norm localization

    Xiaoman Chen, Romain Tessera, Xianjin Wang, and Guoliang Yu. Metric sparsification and operator norm localization. Adv. Math. , 218(5):1496--1511, 2008

    work page 2008

  5. [5]

    Davidson

    Kenneth R. Davidson. C^* -algebras by example , volume 6 of Fields Institute Monographs . American Mathematical Society, Providence, RI, 1996

    work page 1996

  6. [6]

    The coarse B aum- C onnes conjecture, boundary groupoids and expander graphs

    Martin Finn-Sell and Nick Wright. The coarse B aum- C onnes conjecture, boundary groupoids and expander graphs. Adv. Math. , 259C:306--338, 2014

    work page 2014

  7. [7]

    The N ovikov conjecture for linear groups

    Erik Guentner, Nigel Higson, and Shmuel Weinberger. The N ovikov conjecture for linear groups. Publ. Math. Inst. Hautes \'Etudes Sci. , (101):243--268, 2005

    work page 2005

  8. [8]

    M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, V ol. 2 ( S ussex, 1991) , volume 182 of London Math. Soc. Lecture Note Ser. , pages 1--295. Cambridge Univ. Press, Cambridge, 1993

    work page 1991

  9. [9]

    Misha. Gromov. Random walk in random groups. Geom. Funct. Anal. , 13(1):73--146, 2003

    work page 2003

  10. [10]

    Groups with graphical C (6) and C (7) small cancellation presentations

    Dominik Gruber. Groups with graphical C (6) and C (7) small cancellation presentations. Trans. Amer. Math. Soc. , 2014

    work page 2014

  11. [11]

    Divergence and quasi-isometry classes of random Gromov's monsters

    Dominik Gruber and Alessandro Sisto. Divergence and quasi-isometry classes of random gromov's monsters. arXiv preprint: https://arxiv.org/abs/1805.04039 , 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  12. [12]

    Operator norm localization for linear groups and its application to K -theory

    Erik Guentner, Romain Tessera, and Guoliang Yu. Operator norm localization for linear groups and its application to K -theory. Adv. Math. , 226(4):3495--3510, 2011

    work page 2011

  13. [13]

    Geometrization of the strong N ovikov conjecture for residually finite groups

    Guihua Gong, Qin Wang, and Guoliang Yu. Geometrization of the strong N ovikov conjecture for residually finite groups. J. Reine Angew. Math. , 621:159--189, 2008

    work page 2008

  14. [14]

    Counterexamples to the coarse B aum- C onnes conjecture

    Nigel Higson. Counterexamples to the coarse B aum- C onnes conjecture. Preprint , 1999

    work page 1999

  15. [15]

    Operator K -theory for groups which act properly and isometrically on H ilbert space

    Nigel Higson and Gennadi Kasparov. Operator K -theory for groups which act properly and isometrically on H ilbert space. Electron. Res. Announc. Amer. Math. Soc. , 3:131--142 (electronic), 1997

    work page 1997

  16. [16]

    Counterexamples to the B aum- C onnes conjecture

    Nigel Higson, Vincent Lafforgue, and Georges Skandalis. Counterexamples to the B aum- C onnes conjecture. Geom. Funct. Anal. , 12(2):330--354, 2002

    work page 2002

  17. [17]

    A proof of property ( RD ) for cocompact lattices of SL (3, R) and SL (3, C)

    Vincent Lafforgue. A proof of property ( RD ) for cocompact lattices of SL (3, R) and SL (3, C) . J. Lie Theory , 10(2):255--267, 2000

    work page 2000

  18. [18]

    La conjecture de B aum- C onnes \`a coefficients pour les groupes hyperboliques

    Vincent Lafforgue. La conjecture de B aum- C onnes \`a coefficients pour les groupes hyperboliques. Journal of Noncommutative Geometry , 6:pp. 1--197, 2012

    work page 2012

  19. [19]

    Lyndon and Paul E

    Roger C. Lyndon and Paul E. Schupp. Combinatorial group theory . Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition

    work page 2001

  20. [20]

    A. Yu. Ol'shanski . Geometry of defining relations in groups , volume 70 of Mathematics and its Applications (Soviet Series) . Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the 1989 Russian original by Yu. A. Bakhturin

    work page 1991

  21. [21]

    On a small cancellation theorem of G romov

    Yann Ollivier. On a small cancellation theorem of G romov. Bull. Belg. Math. Soc. Simon Stevin , 13(1):75--89, 2006

    work page 2006

  22. [22]

    A. Yu. Ol'shanski , Denis V. Osin, and Mark V. Sapir. Lacunary hyperbolic groups. Geom. Topol. , 13(4):2051--2140, 2009. With an appendix by Michael Kapovich and Bruce Kleiner

    work page 2051

  23. [23]

    K -theory for the maximal R oe algebra of certain expanders

    Herv \'e Oyono-Oyono and Guoliang Yu. K -theory for the maximal R oe algebra of certain expanders. J. Funct. Anal. , 257(10):3239--3292, 2009

    work page 2009

  24. [24]

    On quantitative operator K -theory

    Herv\'e Oyono-Oyono and Guoliang Yu. On quantitative operator K -theory. Ann. Inst. Fourier (Grenoble) , 65(2):605--674, 2015

    work page 2015

  25. [25]

    Small cancellation labellings of some infinite graphs and applications

    Damian Osajda. Small cancellation labellings of some infinite graphs and applications. Preprint (http://arxiv.org/abs/1406.5015) , 2014

    work page arXiv 2014

  26. [26]

    Hyperbolic groups have finite asymptotic dimension

    John Roe. Hyperbolic groups have finite asymptotic dimension. Proc. Amer. Math. Soc. , 133(9):2489--2490, 2005

    work page 2005

  27. [27]

    Property A and the operator norm localization property for discrete metric spaces

    Hiroki Sako. Property A and the operator norm localization property for discrete metric spaces. J. Reine Angew. Math. , 690:207--216, 2014

    work page 2014

  28. [28]

    Some notes on property A

    Rufus Willett. Some notes on property A . In Limits of graphs in group theory and computer science , pages 191--281. EPFL Press, Lausanne, 2009

    work page 2009

  29. [29]

    Higher index theory for certain expanders and G romov monster groups, I

    Rufus Willett and Guoliang Yu. Higher index theory for certain expanders and G romov monster groups, I . Adv. Math. , 229(3):1380--1416, 2012

    work page 2012

  30. [30]

    Localization algebras and the coarse B aum- C onnes conjecture

    Guoliang Yu. Localization algebras and the coarse B aum- C onnes conjecture. K -Theory , 11(4):307--318, 1997

    work page 1997

  31. [31]

    The N ovikov conjecture for groups with finite asymptotic dimension

    Guoliang Yu. The N ovikov conjecture for groups with finite asymptotic dimension. Ann. of Math. (2) , 147(2):325--355, 1998

    work page 1998

  32. [32]

    The coarse B aum- C onnes conjecture for spaces which admit a uniform embedding into H ilbert space

    Guoliang Yu. The coarse B aum- C onnes conjecture for spaces which admit a uniform embedding into H ilbert space. Invent. Math. , 139(1):201--240, 2000

    work page 2000