Higher Kazhdan projections and delocalised $\ell^ 2$-Betti numbers

Hang Wang; Sanaz Pooya

arxiv: 2405.03837 · v2 · pith:4SKNVK5Inew · submitted 2024-05-06 · 🧮 math.OA · math.GR· math.KT

Higher Kazhdan projections and delocalised ell^ 2-Betti numbers

Sanaz Pooya , Hang Wang This is my paper

Pith reviewed 2026-05-24 01:28 UTC · model grok-4.3

classification 🧮 math.OA math.GRmath.KT

keywords higher Kazhdan projectionsdelocalised ℓ²-Betti numbersLott's invariantsK-theory of group C*-algebrasfree product groupsCartesian product groupsnon-vanishing resultsoperator algebras

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The pith

Higher Kazhdan projections yield explicit K-classes that produce the first non-vanishing delocalised ℓ²-Betti numbers for infinite groups.

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

The paper supplies explicit descriptions of the K-classes of higher Kazhdan projections in positive degrees for selected free product groups and Cartesian product groups. These descriptions are applied to compute Lott's delocalised ℓ²-Betti numbers, producing the first non-vanishing values recorded for any infinite group. A reader would care because these numbers refine ordinary ℓ²-Betti numbers by incorporating delocalised information from the group C*-algebra, potentially distinguishing more groups topologically than classical invariants allow.

Core claim

For specific free product groups and Cartesian product groups, the K-classes of higher Kazhdan projections in degrees greater than zero admit explicit descriptions in the K-theory of the group C*-algebras. Employing this description yields new calculations of Lott's delocalised ℓ²-Betti numbers, including the first non-vanishing results for infinite groups.

What carries the argument

Higher Kazhdan projections, whose K-classes in the K-theory of group C*-algebras are computed explicitly to evaluate delocalised ℓ²-Betti numbers.

If this is right

Delocalised ℓ²-Betti numbers take non-zero values for some infinite groups.
Explicit K-class formulas are now available for higher Kazhdan projections in the studied families of groups.
New concrete values of Lott's invariants follow directly from the K-theory descriptions.
The method distinguishes infinite groups that standard ℓ²-Betti numbers leave indistinguishable.

Where Pith is reading between the lines

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

The same explicit-description technique could be tested on other groups built from free products or direct products.
Non-vanishing results may indicate that delocalised invariants detect finer group-theoretic features than their localised counterparts.
Connections between these K-theoretic calculations and other delocalised invariants in operator algebras remain open for exploration.

Load-bearing premise

The K-classes of the higher Kazhdan projections admit explicit descriptions in the K-theory of the group C*-algebras for the chosen free product and Cartesian product groups.

What would settle it

An independent computation of the delocalised ℓ²-Betti numbers for one of the specific free product or Cartesian product groups that shows vanishing in all degrees would contradict the non-vanishing claim.

read the original abstract

We provide an explicit description of the K-classes of higher Kazhdan projections in degrees greater than 0 for specific free product groups and Cartesian product groups. Employing this description, we obtain new calculations of Lott's delocalised $\ell^2$-Betti numbers. Notably, we establish the first non-vanishing results for infinite 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.

Desk Editor's Note private letter to a colleague

This paper gives explicit K-class descriptions for higher Kazhdan projections in some product groups and uses them for the first non-vanishing delocalised ℓ²-Betti numbers on infinite groups.

read the letter

The main point is that Pooya and Wang work out explicit K-classes for higher Kazhdan projections in degrees above zero for certain free product groups and Cartesian product groups, then plug those into Lott's delocalised ℓ²-Betti numbers to get non-zero values where only vanishing results existed before for infinite groups. That is the concrete advance claimed in the abstract. The paper does well by choosing groups where the K-theory of the reduced group C*-algebras is manageable enough to pin down the classes directly, turning an abstract invariant into actual numbers rather than just existence statements. The calculations appear to follow from the K-theory data without obvious fitting or circularity. The soft spots are that the groups are quite specific, so the method may not travel far beyond these constructions, and the non-vanishing examples are tied to those choices. Without the full derivations it is hard to judge how delicate the K-class computations are, but nothing in the abstract suggests a load-bearing gap. This is for specialists working on operator-algebraic invariants and K-theory of discrete groups. A reader already following Lott's invariants or Kazhdan projections would get direct value from the numbers and the explicit descriptions. The work shows clear engagement with the prior literature on these invariants. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript provides explicit descriptions of the K-classes of higher Kazhdan projections (in degrees >0) for certain free-product and Cartesian-product groups. These descriptions are then used to compute new values of Lott's delocalised ℓ²-Betti numbers, yielding the first non-vanishing examples for infinite groups.

Significance. If the explicit K-theory descriptions are correct, the work supplies the first concrete non-vanishing results for delocalised ℓ²-Betti numbers on infinite groups, a notable advance in the area. The direct link from K-classes in group C*-algebras to the Betti-number values is a strength of the approach.

minor comments (2)

The abstract refers to 'specific free product groups and Cartesian product groups' without naming them; a brief indication of the families (e.g., free products of finite groups or products with ℤ) would improve readability.
Notation for the higher Kazhdan projections and the associated K-theory classes could be introduced with a short table or diagram in the preliminaries to aid readers unfamiliar with the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by direct computation: the paper states it gives explicit K-theory descriptions of higher Kazhdan projections for chosen free-product and Cartesian-product groups, then feeds those descriptions into the definition of Lott's delocalised ℓ²-Betti numbers to obtain new values, including the first non-vanishing examples for infinite groups. No equation or step is shown to be equivalent to its own input by construction, no parameter is fitted on a subset and then relabeled a prediction, and no load-bearing uniqueness claim rests on a self-citation chain. The argument is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities introduced.

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