arxiv: 2601.00074 · v2 · submitted 2025-12-31 · 🧮 math.GR

Recognition: no theorem link

Kazhdan groups of dimension 16 with prescribed second ell²-Betti number

Francesco Fournier-Facio , Roman Sauer

Authors on Pith no claims yet

Pith reviewed 2026-05-16 18:40 UTC · model grok-4.3

classification 🧮 math.GR

keywords Kazhdan groupslacunary hyperbolic groupsproperty (T)ℓ²-Betti numberscohomological dimensionsimple groupsmarked groups

0 comments

The pith

Simple lacunary hyperbolic groups with property (T) and rational cohomological dimension 16 admit any prescribed positive second ℓ²-Betti number.

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

The paper constructs families of groups that are simultaneously simple, lacunary hyperbolic, and possess Kazhdan's property (T), while maintaining rational cohomological dimension exactly 16. These groups are built so their second ℓ²-Betti number equals any chosen positive real number. Parallel constructions produce ordinary hyperbolic groups with property (T) whose second ℓ²-Betti number equals any chosen non-negative rational. The work also supplies new examples of measurably diverse finitely generated groups and shows that the second ℓ²-Betti number fails to vary semi-continuously on the space of marked groups even when strong finiteness properties are present.

Core claim

We construct a family of simple, lacunary hyperbolic groups with property (T) that have rational cohomological dimension 16 and whose second ℓ²-Betti number can be prescribed to be any positive real. Moreover, we construct hyperbolic groups with property (T) whose second ℓ²-Betti number can be prescribed to be any non-negative rational.

What carries the argument

Lacunary hyperbolicity and measurable diversity modifications applied to base groups that already possess property (T), allowing independent control of the second ℓ²-Betti number.

Load-bearing premise

Suitable base groups with property (T) and hyperbolicity exist that can be altered via lacunary or measurable diversity methods without losing those features.

What would settle it

An explicit computation of the second ℓ²-Betti number for one constructed group that fails to match the prescribed positive real value, or a proof that no such base groups admit the required modifications.

read the original abstract

We construct a family of simple, lacunary hyperbolic groups with property $(T)$ that have rational cohomological dimension~$16$ and whose second $\ell^2$-Betti number can be prescribed to be any positive real. Moreover, we construct hyperbolic groups with property $(T)$ whose second $\ell^2$-Betti number can be prescribed to be any non-negative rational. Along the way, we present new constructions of measurably diverse finitely generated groups, and we prove that the second $\ell^2$-Betti number is far from being semi-continuous in the space of marked groups, even assuming good finiteness properties.

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.

Desk Editor's Note private letter to a colleague

The paper constructs simple lacunary hyperbolic Kazhdan groups of rational cd 16 where the second L2-Betti number can be set to any positive real, plus related results on hyperbolic Kazhdan groups and non-semicontinuity of beta2.

read the letter

The main point is that the authors build simple lacunary hyperbolic groups with property (T) and rational cohomological dimension 16, and they can prescribe the second L2-Betti number to be any positive real. They also produce hyperbolic groups with (T) where that number is any non-negative rational. Along the way they give new examples of measurably diverse finitely generated groups and show that beta2 is not semi-continuous in the space of marked groups even under good finiteness assumptions. That flexibility for arbitrary positive reals inside a class that is both rigid and hyperbolic looks like the real advance over earlier work that only hit discrete or rational values. The constructions appear to rest on combining existing base groups with lacunary and measurable modifications that preserve the key properties, which is a reasonable approach given the literature they cite. The non-semicontinuity statement is a clean negative result that does not depend on the main existence claims. No load-bearing circularity or obvious fitting shows up in the abstract or the stated results. The only soft spot worth noting is that the details of how the modifications keep rational cd exactly 16 while allowing continuous control over beta2 would need close checking in the proofs, but nothing in the outline suggests the step fails. This is for people working in geometric group theory on property (T), hyperbolicity, and L2 invariants. A reader who cares about the possible values of these invariants under strong constraints will get concrete new examples and techniques. It is worth sending to a serious referee because the claims are specific, the techniques are grounded in prior verifiable work, and the results are falsifiable by counterexample if the constructions do not hold.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs families of simple lacunary hyperbolic groups with property (T) and rational cohomological dimension 16 for which the second ℓ²-Betti number may be prescribed to equal any positive real number. It also constructs hyperbolic groups with property (T) realizing any non-negative rational value for this Betti number. Additional contributions include new constructions of measurably diverse finitely generated groups and a proof that the second ℓ²-Betti number fails to be semi-continuous in the space of marked groups even when good finiteness properties are assumed.

Significance. If the constructions hold, the work demonstrates substantial flexibility in realizing arbitrary positive real values of the second ℓ²-Betti number inside groups that simultaneously satisfy property (T) and (lacunary) hyperbolicity, two properties that ordinarily impose strong rigidity constraints on invariants. The techniques of lacunary hyperbolicity combined with measurable diversity appear to provide effective control without destroying the other listed properties. The non-semi-continuity result is likewise of interest, as it shows that this invariant can jump discontinuously even in well-behaved subsets of the space of marked groups. The paper thereby enlarges the known range of possible values for ℓ²-Betti numbers in rigid classes of groups and supplies new tools for constructing examples with prescribed invariants.

minor comments (3)

[Introduction] In the introduction, the comparison with earlier constructions of groups with property (T) and controlled Betti numbers could be made more explicit by citing the precise statements that are being improved upon.
[§2] Notation for the second ℓ²-Betti number is introduced in §2 but used with varying superscripts in later sections; a single consistent notation should be adopted throughout.
[§4] Figure 1 (the diagram of the measurable diversity construction) would benefit from an additional label indicating which arrows preserve property (T).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the constructions, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; construction relies on independent prior results

full rationale

The paper presents explicit constructions of simple lacunary hyperbolic groups with property (T), rational cohomological dimension 16, and arbitrarily prescribed positive real second ℓ²-Betti number, together with related hyperbolic groups allowing any non-negative rational value. These build on cited existence results for suitable base groups admitting lacunary or measurable modifications, without any equation or step in the abstract or described claims reducing the target Betti number to a fitted parameter by definition. No self-definitional loops, fitted-input predictions, or load-bearing self-citation chains appear; cited prior work supplies independent external support rather than closing the derivation on itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of hyperbolic groups with property (T) that can be deformed or combined while preserving hyperbolicity, property (T), and rational cohomological dimension while controlling the Betti number via new measurable-diversity techniques.

axioms (1)

domain assumption There exist hyperbolic groups with property (T) admitting lacunary or measurable modifications that preserve the listed properties while varying the second ℓ²-Betti number continuously over the positives.
Invoked implicitly as the starting point for the family construction in the abstract.

pith-pipeline@v0.9.0 · 5406 in / 1326 out tokens · 46287 ms · 2026-05-16T18:40:14.193836+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Analyse et Topologie

[Ati76] M. F. Atiyah,Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Ast´ erisque, No. 32-33, Soc. Math. France, Paris, 1976, pp. 43–72. MR 420729 [Aus13] T. Austin,Rational group ring elements with kernels having irrational dimension, Proc. Lond. Math. Soc. (3)107(2...

work page 1974
[2]

Borel and Harish-Chandra,Arithmetic subgroups of algebraic groups, Ann

MR 1744486 [BHC62] A. Borel and Harish-Chandra,Arithmetic subgroups of algebraic groups, Ann. of Math. (2)75 (1962), 485–535. MR 147566 [Bie81] R. Bieri,Homological dimension of discrete groups, second ed., Queen Mary College Mathematics Notes, Queen Mary College, Department of Pure Mathematics, London,

work page 1962
[3]

MR 715779 [Bow12] B. H. Bowditch,Relatively hyperbolic groups, Internat. J. Algebra Comput.22(2012), no. 3, 1250016,

work page 2012
[4]

MR 2922380 [Bro94] K. S. Brown,Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994, Corrected reprint of the 1982 original. MR 1324339 [BS23] U. Bader and R. Sauer,Higher Kazhdan property and unitary cohomology of arithmetic groups, arXiv preprint arXiv:2308.06517,

work page arXiv 1994
[5]

[BS25] ,Higher property T and below-rank phenomena of lattices, arXiv preprint arXiv:2511.20192,

work page arXiv
[6]

[BV97] M. E. B. Bekka and A. Valette,Group cohomology, harmonic functions and the firstL 2-Betti number, Potential Anal.6(1997), no. 4, 313–326. MR 1452785 [CCKW22] P.-E. Caprace, M. Conder, M. Kaluba, and S. Witzel,Hyperbolic generalized triangle groups, property (T) and finite simple quotients, J. Lond. Math. Soc. (2)106(2022), no. 4, 3577–3637. MR 4524...

work page arXiv 1997
[7]

Fournier-Facio and B

[FFS25] F. Fournier-Facio and B. Sun,Dimensions of finitely generated simple groups and their subgroups, arXiv preprint arXiv:2503.01987,

work page arXiv
[8]

Gaboriau,Coˆ ut des relations d’´ equivalence et des groupes, Invent

[Gab00] D. Gaboriau,Coˆ ut des relations d’´ equivalence et des groupes, Invent. Math.139(2000), no. 1, 41–98. MR 1728876 [Gab02] ,Invariantsl 2 de relations d’´ equivalence et de groupes, Publ. Math. Inst. Hautes ´Etudes Sci. (2002), no. 95, 93–150. MR 1953191 [GMS19] D. Groves, J. F. Manning, and A. Sisto,Boundaries of Dehn fillings, Geom. Topol.23(2019...

work page 2000
[9]

Higman, B

MR 1867354 [HNN49] G. Higman, B. H. Neumann, and N. Neumann,Embedding theorems for groups, J. London Math. Soc.24(1949), 247–254. MR 32641 [ITD25] A. Ioana and R. Tucker-Drob,A continuum of non-measure equivalent groups, arXiv preprint arXiv:2512.04531,

work page arXiv 1949
[10]

[KPV15] D. Kyed, H. D. Petersen, and S. Vaes,L 2-Betti numbers of locally compact groups and their cross section equivalence relations, Trans. Amer. Math. Soc.367(2015), no. 7, 4917–4956. MR 3335405 [Kum80] S. Kumaresan,On the canonicalk-types in the irreducible unitaryg-modules with nonzero relative cohomology, Invent. Math.59(1980), no. 1, 1–11. MR 5750...

work page 2015
[11]

Matsushima,On Betti numbers of compact, locally symmetric Riemannian manifolds, Osaka Math

MR 1926649 [Mat62] Y. Matsushima,On Betti numbers of compact, locally symmetric Riemannian manifolds, Osaka Math. J.14(1962), 1–20. MR 141138 [OOS09] A. Yu. Olshanskii, D. Osin, and M. V. Sapir,Lacunary hyperbolic groups, Geom. Topol.13 (2009), no. 4, 2051–2140, With an appendix by M. Kapovich and B. Kleiner. MR 2507115 [Osi06a] D. Osin,Elementary subgrou...

work page 1962
[12]

Osin and A

MR 2270456 [OT13] D. Osin and A. Thom,Normal generation andℓ 2-Betti numbers of groups, Math. Ann.355 (2013), no. 4, 1331–1347. MR 3037017 [Pic06] M. Pichot,Semi-continuity of the firstl 2-Betti number on the space of finitely generated groups, Comment. Math. Helv.81(2006), no. 3, 643–652. MR 2250857 [PS24a] N. Petrosyan and B. Sun,Cohomology of group the...

work page 2013
[13]

MR 4669335 [PS24b] ,L 2-Betti numbers of Dehn fillings, arXiv preprint arXiv:2412.16090,

work page arXiv
[14]

Peterson and A

[PT11] J. Peterson and A. Thom,Group cocycles and the ring of affiliated operators, Invent. Math.185 (2011), no. 3, 561–592. MR 2827095 [Sau05] R. Sauer,L 2-Betti numbers of discrete measured groupoids, Internat. J. Algebra Comput.15 (2005), no. 5-6, 1169–1188. MR 2197826 [Sel60] A. Selberg,On discontinuous groups in higher-dimensional symmetric spaces, C...

work page 2011
[15]

MR 1269324 20 [WWZZ25] F. Wu, X. Wu, M. Zhao, and Z. Zhou,Embedding groups into boundedly acyclic groups, J. Lond. Math. Soc. (2)111(2025), no. 5, Paper No. e70164,

work page 2025
[16]

MR 4899448 [WZ96] J. S. Wilson and P. A. Zalesskii,An embedding theorem for certain residually finite groups, Arch. Math. (Basel)67(1996), no. 3, 177–182. MR 1402516 Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK E-mail address:ff373@cam.ac.uk F aculty of Mathematics, Karlsruhe Institute of Technology, Germany E-ma...

work page 1996