Higher Kazhdan Projections

• Higher Kazhdan projections are higher-dimensional analogs of the classical Kazhdan projection, defined via spectral gaps in combinatorial Laplacians associated with group cohomology.
• They are constructed as projections in matrix algebras over group C*-algebras or Roe algebras, with traces that recover ℓ₂-Betti numbers and yield nontrivial K-theory classes.
• The theory connects spectral methods, Baum–Connes assembly maps, and coarse geometric frameworks, providing explicit formulas and invariants for computational and structural analysis.

Searching arXiv for the cited papers to ground the article in current records. I’m going to look up the relevant arXiv entries on higher Kazhdan projections and related work. Higher Kazhdan projections are higher-dimensional analogs of the classical Kazhdan projection, realized as projections in matrix algebras over group $C^*$-algebras and Roe algebras. They are constructed from combinatorial Laplacians associated to cohomology with coefficients in unitary representations, and in favorable cases they define nontrivial $K$-theory classes whose traces recover $\ell_2$-Betti numbers. Their introduction provides a direct link between spectral gaps in group cohomology, harmonic cochains, assembly maps of Baum–Connes type, and coarse-geometric phenomena such as box spaces and uniform Roe algebras (Li et al., 2020).

1. Classical origin and higher-degree definition

The classical Kazhdan projection arises from degree-$0$ cohomology. For a finitely generated group $G$ with finite symmetric generating set $S$, one forms the degree-$0$ difference matrix

$d_0=[\,1-s\,]_{s\in S},$

and the corresponding Laplacian

$A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$

For any unitary representation $\pi$, the kernel of $K$0 is the space of invariant vectors. A theorem of Akemann–Walter identifies Kazhdan’s Property (T) with the existence of an idempotent $K$1 whose image in every unitary representation is the orthogonal projection onto $K$2; equivalently, $K$3 is the spectral projection of $K$4 at $K$5, given by

$K$6

under a uniform spectral gap condition at $K$7 (Li et al., 2020).

Higher Kazhdan projections replace degree $K$8 by arbitrary degree $K$9. Fix an Eilenberg–Mac Lane model $\ell_2$0 with finite $\ell_2$1-skeleton, and let $\ell_2$2 denote the number of $\ell_2$3-simplices of $\ell_2$4. For a unitary representation $\ell_2$5,

$\ell_2$6

and the coboundary operators are represented by matrices

$\ell_2$7

The $\ell_2$8-th combinatorial Laplacian is then

$\ell_2$9

For $0$0, the kernel of $0$1 is canonically identified with the harmonic $0$2-cochains, equivalently with the reduced cohomology $0$3. A projection

$0$4

is called a higher Kazhdan projection in degree $0$5 if $0$6 is the orthogonal projection onto $0$7 for every $0$8 (Li et al., 2020).

This construction generalizes the degree-$0$9 Kazhdan idempotent without reducing higher-degree cohomology to invariant vectors. In addition, one may define partial projections onto cocycles and cycles separately, by applying the same construction to the summands $G$0 and $G$1 (Li et al., 2020).

2. Spectral-gap mechanism and operator-algebraic realization

The existence theorem is entirely spectral. If the Laplacian $G$2 has a uniform gap at $G$3 in $G$4,

$G$5

then continuous functional calculus yields the spectral projection

$G$6

which exists in $G$7 and is the unique higher Kazhdan projection in degree $G$8. The same argument gives the partial projections associated to the two summands of $G$9 (Li et al., 2020).

The same spectral-gap/functional-calculus paradigm also appears in Roe algebras. For a discrete bounded-geometry metric space $S$0, the Roe algebra $S$1 is obtained by completing finite-propagation locally compact operators on $S$2. If $S$3 acts properly and cocompactly on $S$4, one has an equivariant Roe algebra $S$5, and an appropriate degree-$S$6 Laplacian with a gap again produces a Kazhdan projection. The higher theory extends this operator-algebraic viewpoint from invariant vectors to higher harmonic cochains (Li et al., 2020).

A useful way to situate the construction is to compare it with Banach-algebraic Kazhdan projections. For Banach families of representations, de la Salle and Liao characterize the existence of a Kazhdan projection by a local contraction condition on generator displacement, and analyze central versus non-central projections, including non-central examples from hyperbolic groups. They also formulate a program toward “higher” cohomological projections, but explicitly note that no general theory of higher-cohomological projections is yet in the literature (Salle, 2016). This suggests that the Hilbert-space theory of higher Kazhdan projections is presently the most developed operator-algebraic realization of such higher cohomological idempotents.

3. $S$7-theory, $S$8-Betti numbers, and Baum–Connes obstructions

When $S$9 is the left-regular representation, the relevant algebra is $0$0. If $0$1 has a spectral gap in $0$2, then the higher Kazhdan projection

$0$3

pairs with the canonical trace $0$4 to compute the $0$5-Betti number: $0$6 Consequently,

$0$7

Thus higher Kazhdan projections furnish concrete $0$8-classes whose trace detects higher $0$9-cohomology (Li et al., 2020).

This trace formula has an immediate Baum–Connes consequence. Let

$d_0=[\,1-s\,]_{s\in S},$0

be the maximal assembly map, and let

$d_0=[\,1-s\,]_{s\in S},$1

be the reduction map. Lück’s theorem implies that if $d_0=[\,1-s\,]_{s\in S},$2 is surjective, then the composition $d_0=[\,1-s\,]_{s\in S},$3 takes values in the subring $d_0=[\,1-s\,]_{s\in S},$4 generated by inverses of orders of finite subgroups. Therefore, for $d_0=[\,1-s\,]_{s\in S},$5 of type $d_0=[\,1-s\,]_{s\in S},$6, if $d_0=[\,1-s\,]_{s\in S},$7 has a spectral gap in $d_0=[\,1-s\,]_{s\in S},$8 and $d_0=[\,1-s\,]_{s\in S},$9 is onto, then

$A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$0

and in particular $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$1 when $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$2 is torsion-free. Equivalently, if $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$3, then $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$4 is not surjective (Li et al., 2020).

The $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$5-theoretic role of higher Kazhdan projections is therefore twofold. First, they encode harmonic cohomology classes as projections in operator algebras. Second, their traces constrain assembly maps by converting surjectivity questions into arithmetic restrictions on $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$6-Betti numbers. The paper also remarks that if $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$7 were $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$8-amenable, then the map $A_0=d_0d_0^*=\sum_{s\in S}(1-s)(1-s^{-1}).$9 would be an isomorphism; hence a nonzero higher Kazhdan class in $\pi$0 that vanishes after reduction would obstruct $\pi$1-amenability (Li et al., 2020).

4. Coarse-geometric version and box spaces

The higher theory has a coarse counterpart for box spaces. Let $\pi$2 be exact and residually finite, with decreasing finite-index normal subgroups $\pi$3, and form the box space

$\pi$4

with the disjoint union metric making levels diverge. For a fixed finite $\pi$5, the cochain-Laplacian $\pi$6 has a gap in $\pi$7 if and only if the corresponding finite-quotient Laplacians on the $\pi$8 have a uniform gap in the uniform Roe algebra $\pi$9. The resulting spectral projector is denoted

$K$00

If $K$01 for infinitely many $K$02, then the class

$K$03

is nonzero; its image under the box-trace map records $K$04 infinitely often (Li et al., 2020).

Under surjectivity of the coarse Baum–Connes assembly map

$K$05

the higher projection forces a strong stabilization phenomenon: $K$06 for all but finitely many $K$07. Equivalently,

$K$08

not only converges, but actually stabilizes exactly after finitely many indices. This strengthens Lück’s approximation theorem and yields a strategy for constructing counterexamples to the coarse Baum–Connes conjecture by seeking groups for which the normalized Betti numbers converge slowly but non-stably (Li et al., 2020).

The Roe-algebra perspective also connects higher Kazhdan projections to ghost operators. In $K$09, the kernel of the lifting map used in the nonvanishing result is exactly the ideal of ghosts. This suggests new candidates for noncompact ghost projections in situations where $K$10 but the higher projection remains nontrivial “at infinity” (Li et al., 2020).

5. Explicit calculations and formulas for $K$11-classes

Subsequent work makes the higher Kazhdan classes concrete in several families of groups. For non-amenable finitely generated virtually free groups acting properly and cocompactly on a tree $K$12, Pooya, Ren, and Wang show that the combinatorial Euler class of Emerson–Meyer is the preimage of the higher Kazhdan class under the Baum–Connes assembly map, and that when only $K$13 is nonzero,

$K$14

where

$K$15

is the averaging projection of the finite stabilizer. In the virtually free case, only $K$16 occurs, and

$K$17

(Pooya et al., 27 Jul 2025).

Pooya and Wang compute explicit higher Kazhdan classes for free products and Cartesian products. For

$K$18

the first higher Kazhdan class is

$K$19

For

$K$20

with $K$21 copies of $K$22 and $K$23 finite,

$K$24

These formulas are obtained from explicit resolutions in the free-product case and from a Künneth-type decomposition of cochains and Laplacians in the product case (Pooya et al., 2024).

Ren extends the free-product formula to the amalgamated product

$K$25

Writing

$K$26

the unique nonzero higher Kazhdan class satisfies

$K$27

When $K$28, this reduces to the free-product formula (Ren, 8 Jul 2025).

Group	Nonzero higher projection	$K$29-class
$K$30	$K$31	$K$32
$K$33	$K$34	$K$35
$K$36	$K$37	$K$38

These formulas show that, in several low-dimensional situations, higher Kazhdan projections can be represented in $K$39-theory by finite alternating sums of averaging idempotents attached to finite subgroups or stabilizers. A plausible implication is that this makes the otherwise spectral definition accessible to explicit computation in classes of virtually free groups.

6. Delocalized traces, examples, and open directions

Higher Kazhdan projections also define delocalized $K$40-Betti numbers. For a conjugacy class $K$41, the delocalized trace

$K$42

induces a map on $K$43, and one sets

$K$44

In the virtually free setting, pairing the alternating-sum formula with delocalized traces yields nonvanishing rationals whenever $K$45 fixes at least one vertex orbit but no edge orbit (Pooya et al., 27 Jul 2025).

The 2024 computations give the first non-vanishing examples for infinite groups. For $K$46,

$K$47

and for $K$48,

$K$49

Ren’s amalgamated-product formula produces, for

$K$50

the values

$K$51

with all others vanishing (Pooya et al., 2024, Ren, 8 Jul 2025).

The original theory already provided examples and structural questions. For free groups $K$52 with $K$53, the standard $K$54-cochain Laplacian has a gap, $K$55, and $K$56 is nonzero. Kähler hyperbolic groups have gaps in all degrees and nontrivial middle-degree $K$57-cohomology, so higher projections exist in all degrees. For lattices in $K$58, Garland-type vanishing and spectral-gap arguments yield partial Kazhdan projections in $K$59 (Li et al., 2020).

A recurrent misconception is that higher Kazhdan projections are already characterized by a known higher analog of Property (T). The available results do not establish this. Li, Nowak, and Pooya explicitly ask whether there is a purely spectral-gap-type characterization—a “Property $K$60”—equivalent to the existence of a higher Kazhdan projection $K$61 in $K$62 or $K$63 (Li et al., 2020). In parallel, the Banach-space framework of de la Salle and Liao suggests local and ultraproduct methods for higher cohomological idempotents, but it also records that no general theory of higher-cohomological projections is yet in the literature (Salle, 2016). The present state of the subject therefore combines a robust Hilbert-space theory, a growing body of explicit $K$64-theory computations, and a still-open problem of intrinsic higher rigidity characterizations.