l2 betti numbers, nonpositive immersions, and the … · l2 betti numbers, nonpositive immersions,...

L2 BETTI NUMBERS, NONPOSITIVE IMMERSIONS, AND THE

ENERGY CRITERION

MARK F. HAGEN

Abstract. After providing brief background on group von Neumann algebras and L2 Betti

numbers, we describe the energy criterion, due to Wise, for the vanishing of β(2)2 (X,G) for

a cocompact free G-complex X. We discuss some examples and applications of the energycriterion to the “nonpositive immersions property” and hence to local indicability.

Introduction

This expository paper has two parts. In the first section, we discuss the L2 homology ofa G-space X, following the discussion given by Luck in the survey [Luc09]. In the second

section, we interpret the definitions concretely in the situation in which X is the universalcover of a finite presentation complex of G. In the remainder of the paper, we focus on thevanishing of the 2nd L2 homology.

First, we relate this to Wise’s nonpositive immersions property of 2-complexes, and discussapplications of that property toward local indicability and coherence of G. We then describethe energy criterion, introduced in [Wis04]. This criterion on a 2-complex X guarantees that

b(2)2 (X) = 0, and is inherited by compact immersions Y → X. The nonpositive immersionsproperty follows from this.

Hopefully, this discussion, with its very simple examples, gives a small taste of the question

of vanishing of L(2)2 Betti numbers. A comprehensive discussion of specific groups from this

viewpoint can be found in Section 6 of [Luc09], while another intriguing discussion of thegeneric properties of L2 Betti numbers can be found in [Gro93].

1. Background on L2 Betti numbers

1.1. Group von Neumann algebras. Let G be a countable discrete group, and defineℓ2(G) in the usual way,

ℓ2(G) =⎧⎪⎪⎨⎪⎪⎩ϕ ∶ G→ C ∶ ∑

g∈G∣ϕ(g)∣2 <∞

⎫⎪⎪⎬⎪⎪⎭.

The scalar product

⟨ϕ,ψ⟩ = ∑g∈G

ϕ(g)ψ(g)

makes ℓ2(G) into a Hilbert space. A few things are worth noting. First, the complex groupring CG, which is the set of finitely-supported functions f ∶ G → C, can be endowed with

Date: November 24, 2011.

1

L2 BETTI NUMBERS, NONPOSITIVE IMMERSIONS, AND THE ENERGY CRITERION 2

the same scalar product, and ℓ2(G) is the completion of CG with respect to the resultingℓ2-norm. In particular, for each g ∈ G, the characteristic function Eg of the set {g} is inℓ2(G), and the set {Eg ∶ g ∈ G} forms an orthonormal basis for ℓ2(G).

Second, G acts isometrically on ℓ2(G). Indeed, for g ∈ G, define an operator ψg ∶ ℓ2(G) →ℓ2(G) by

ψg(ϕ)(h) = ϕ(hg−1),where ϕ ∈ ℓ2(G) and h ∈ G. Note that, for all g, k ∈ G, we have

ψgk(ϕ)(h) = ϕ(hk−1g−1) = ψg(ψk(ϕ))(h),

so this is really an action. Also,

∣∣ψg(ϕ)∣∣22 = ⟨ψg(ϕ), ψg(ϕ)⟩

= ∑h∈G

ϕ(hg−1)ϕ(hg−1)

= ∣∣ϕ∣∣22,

so the action is isometric.Let B(G) be the algebra of bounded (in operator norm) linear operators on ℓ2(G). Note

that ψg ∈ B(G) for all g ∈ G. We have an involution on B(G) coming from taking adjoints,and we define a trace by

tr(θ) = ∑g∈G⟨θ(Eg),Eg⟩

for θ ∈ B(G).The operator θ ∈ B(G) is equivariant if it commutes with the G-action on ℓ2(G), i.e. if for

all g ∈ G,ϕ ∈ ℓ2(G),θ(ψg(ϕ)) = ψg(θ(ϕ)).

The von Neumann algebraN (G) ofG is the algebra ofG-equivariant bounded linear operatorson ℓ2(G). In particular, each of the operators ψg ∈ N (G). Let us check that this is really avon Neumann algebra.

Proposition 1.1. N (G) is a unital ∗-subalgebra of B(G) and is closed in the weak-∗ topology.In other words, N (G) is a von Neumann algebra.

Proof. We first verify that N (G) is closed under taking adjoints. Let θ ∶ ℓ2(G) → ℓ2(G) bea G-equivariant bounded linear operator. Since θ is bounded, θ∗ is as well. Let g ∈ G. Thenfor all ϕ ∈ ℓ2(G),

θ(ψg(ϕ)) = ψg(θ(ϕ)).We need to show that, for all g, ϕ, the same equality holds with θ replaced by θ−1. We have

⟨θ∗(ψg(ϕ)) − ψg(θ∗(ϕ)), ψg(ϕ)⟩ = ⟨ψg(ϕ), θ(ψg(ϕ))⟩ − ⟨ψg(θ∗(ϕ)), ψg(ϕ)⟩= ⟨ψg(ϕ), ψg(θ(ϕ))⟩ − ⟨ψg(θ∗(ϕ)), ψg(ϕ)⟩= 0,

where the first equality follows from the fact that θ is g-equivariant and the second equalityholds because g preserves the scalar product.


We now verify that N (G)′′ = N (G), so that the proposition follows from von Neumann’sbicommutant theorem. Clearly N (G) ⊆ N (G)′′; this follows from the definition of a bicom-mutant. Let θ ∈ N (G)′′, i.e. let θ be a bounded linear operator on ℓ2(G) that commutes withevery element of N (G)′.

Suppose that there exists g ∈ G such that ψg(θ(ϕ)) ≠ θ(ψg(ϕ)) for some ϕ ∈ ℓ2(G). Butψg commutes with every G-equivariant bounded linear operator, by definition, so that ψg ∈N ′(G). Since θ ∈ N ′′(G), θ must therefore commute with ψg. �Example 1.2. In, for example, [Pic11], it is shown that N (Zn) is isomorphic to L∞(Tn, λ),where λ is the Lebesgue measure. If G is a finite group, then ℓ2(G) = CG, since every functionfrom G to C is finitely supported. Hence N (G) = CG.

We will consider N (G) both as a left N (G)-module and as a right ZG-module.

1.2. L2 Betti numbers of a G-space. Let A be topological space on which G acts. For themoment there are no restrictions on A or on the G action. Recall that a singular n-chain inA is a continuous map σ ∶ △n → A, where △n is an n-simplex, and Cn(A) is the free abeliangroup generated by the set of singular n-chains in A.

Since G acts (on the left) on A, the group Cn(A) is a left ZG-module. We thus define

C(2)n (A) = N (G)⊗ZG Cn(A)for each n ≥ 0. This has the makings of a chain complex; we need to define boundary maps.

To that end, first recall the definition of a the boundary map ∂′n ∶ Cn(A) → Cn−1(A). Ifσ ∶ [x0, . . . , xn]→ A is a singular simplex in A, then

∂′n(σ) =n

∑i=0(−1)iσ([x0, . . . , xi−1, xi, xi+1, . . . , xn]) ∈ Cn−1(A),

where [x0, . . . , xi−1, xi, xi+1, . . . , xn] is the (n − 1)-dimensional face obtained by omitting xi.This defines ∂′n on each singular chain, and does so G-equivariantly. Hence ∂′n is a well-definedZG-homomorphism. Now let

∂n = idN (G) ⊗ZG ∂′n ∶ C(2)n (A)→ C(2)n−1(A).

It is easy to see that ∂n ○ ∂n+1 = 0, since ∂′n ○ ∂′n+1 = 0. Hence ∂n is a boundary map, and thenth L2 homology of A is the nth homology group of the resulting N (G)-chain complex, i.e.

H(2)n (A,N (G)) =ker∂nim∂n+1

.

Note that H(2)n (A,N (G)), being a quotient of N (G)-modules, is again a N (G)-module.

1.2.1. The von Neumann dimension. To define L2 Betti numbers, we must define the vonNeumann dimension of a N (G)-module. This is quite technical, and we refer the readerto [Luc09] for a more comprehensive exposition. Using the trace onN (G), one first defines thevon Neumann dimension for finitely-generated projective N (G)-modules, and then extendsthis definition to arbitrary N (G)-modules.

We first recall the trace on N (G), which respects the fact that N (G) consists of G-equivariant operators. For each θ ∈ N (G), let

tr(θ) = ⟨θ(χ1G), χ1G⟩ = θ(χ1G)(1G).


For our purposes, a finitely-generated projective N (G)-module P is a module for whichthere exists a N (G)-module K and an isomorphism ι ∶ P ⊕ K → N (G)n for some n ∈ N.Let π ∶ N (G)n → N (G)n be the projection to P , so that π2 = π and imπ = ι(P ). For eachi ∈ {1, . . . , n}, we have a linear operator πi ∶ N (G)→ N (G) defined by restricting π to the ith

coordinate of N (G)n.Define the von Neumann dimension of P over N (G) by:

dimN (G)(P ) =n

∑i=1tr(πi) =

n

∑i=1πi(χ1G)(1G).

A priori, this depends on K, but it is easily verified that dimN (G)(P ) only depends on theisomorphism type of P as a N (G)-module.

In [Luc98], Luck showed that there is a unique function dimN (G) associating a real numberdimN (G)(M) to the N (G)-module M in such a way that, when M is finitely-generated andprojective, dimN (G) is equal to the above expression, and the following additional properties,which one would expect of any invariant called a “dimension”, are satisfied:

(1) If 0→M0 →M1 →M2 → 0 is an exact sequence of N (G)-modules, then

dimN (G)(M1) = dimN (G)(M0) + dimN (G)(M2).(2) If N is a submodule of M , then dimN (G)(N) ≤ dimN (G)(M).

Luck actually proved something stronger than (2), and needs two additional properties toensure uniqueness, but these facts are not needed in the remainder of our discussion.

1.2.2. Definition of L2 Betti numbers.

Definition 1.3 (L2 Betti numbers). Let G be a countable group and let A be a G-space.For n ≥ 0, then nth L2 Betti number of A is

b(2)n (A,G) = dimN (G) (H(2)n (A,N (G))) .

To define the L2 homology of G, let BG2 be the presentation complex coming from themultiplication table for G, and add higher-dimensional simplices in the usual way so that BGis a K(G,1) complex. The universal cover EG of BG is a contractible simplicial complex onwhich G acts freely by deck transformations. In fact, the graph EG1 is just the Cayley graphof G with respect to the generating set containing exactly one of g or g−1 for each g ∈ G. See,for example, [Hat02] for details of this construction.

Definition 1.4. Let EG be the universal cover of the K(G,1) space BG. Then

H(2)n (G) =H(2)n (EG,N (G))and

b(2)n (G) = b(2)n (EG,G).One can check that this definition is independent of the choice of K(G,1) space.

From the definition, we obtain the following proposition (compare Proposition 2.2.(1),(3)in [Wis04]). Many additional properties are provided by Theorem 2.7 of [Luc09], including L2

analogues of many standard facts about homology, such as a Kunneth formula and a versionof Poincare duality for finite-dimensional orientable G-manifolds. The next statement can


be verified using the definition of von Neumann dimension for finitely-generated projectivemodules. The two following it are more involved and are proved in [Luc02].

Proposition 1.5. Let A be a path-connected G-space. Then for all n ≥ 0, H(2)n (A,N (G)) istrivial if and only if b

(2)n (A,G) = 0.

Proof. If H(2)n (A,N (G)) = 0, then it is in particular finitely-generated and projective. The

projection used to define its von Neumann dimension is the zero map, whence b(2)n (A,G) = 0.

Conversely, suppose that b(2)n (A,G) = 0. As in Definition 1.10 of [Luc09], for any N (G)-

module M , there is an associated module

TM = ⋂f∈M∗

ker f

such that the quotient PM =M/TM is finitely-generated and projective, andM ≅ PM⊕TM .

Let P = P (H(2)n (A,N (G))) and let T = T (H(2)n (A,N (G))). Hence

dimN (G)(P ) = dimN (G)(T ) = 0.

Since P is projective, this implies that, if P ⊕K ≅ N (G)n for some n <∞ and some N (G)-module K, then the projection N (G)→ P is the zero map, whence P = 0.

Hence H(2)n (A,N (G)) = T , so that every G-equivariant linear map f ∶ H(2)n (A,N (G)) →

N (G) is trivial. Now T is the kernel of the map i ∶H(2)n (A,N (G)) =M → (M∗)∗ that sendsm ∈M to the map M∗ → N (G) given by f ↦ f(m). Therefore, for all m ∈M and all linearmaps f ∶M → N (G), we have f(m) = 0. This implies that M∗ = 0, whence M = 0. �

Proposition 1.6. b(2)0 (A,G) = 0 if and only if G is infinite. Otherwise, b

(2)0 (A,G) = ∣G∣

−1.

The preceding propositions are useful in the context of the following theorem, relating theL2 homology of a G-space to the Euler characteristic. The statement below is rather lessgeneral than the one in [Luc09] and more general than the one in [Wis04].

Theorem 1.7 (Euler characteristic). Let A be a path-connected, cocompact G-CW-complex.

Let A = G/A. Then

χ(A) =dimA

∑i=0

b(2)i (A,G),

where χ(A) is the Euler characteristic of A.

We will focus on L2 Betti numbers of a free, cocompact G-2-complex X, where G is

an infinite, finitely presented group. Proposition 1.6 tells us that b(2)0 (X,G) = 0. On the

other hand, the Euler characteristic of X = G/X is easily computed, so that the question of

computing all of the L2 Betti numbers of X is reduced to computing b(2)2 (X,G), say. For

many purposes, we are interested in whether or not b(2)2 (X,G) = 0.


2. L2 homology of G-2-complexes

In the remainder of this paper, X is a compact, connected 2-complex, and G = π1X actsby deck transformations on the universal cover X. For each n, when G is understood, weadopt the notation

H(2)n (X) =H(2)n (X,N (G))and

b(2)n (X) = b(2)n (X,G).

Let Sn denote the set of n-cells of X. As usual, Cn(X) is the free CG-module generated by

Sn, and this is contained in the CG-module C(2)n (X) of square-summable functions on Sn,

i.e. on formal sums

Z = ∑S∈Sn

ZSS

where ∑S ∣ZS ∣2 <∞. We define a boundary map dn ∶ C(2)n (X)→ C(2)n−1(X) by

dn(Z) = ∑S∈Sn

ZS∂′nS,

where ∂′n ∶ Cn(X)→ Cn(X) is the usual boundary used to define cellular homology. One cancheck that ∂nZ is an L2 (n − 1)-chain.C(2)n (X) has both an N (G)-module structure and a Hilbert space structure. To see this,

let x ∈ X be a base 0-cube, and let K be the union of all open cells of X whose closurescontain K. Let Sn(K) be the set of n-cells of K, so that any n-cell may be uniquely written

in the form gS, where S ∈ Sn(K) and g ∈ G. Hence any Z ∈ C(2)n (X) may be uniquely writtenin the form

Z = ∑S∈Sn(K)

ϕSS,

where each ϕS ∈ ℓ2(G). One verifies that this defines an isomorphism C(2)n (X)→⊕S∈Sn(K) ℓ

2(G).N (G) acts on each summand, and thus acts on C

(2)n .

For each n, let Zn = ker(dn) ⊆ C(2)n , and let Bn+1 = imdn+1 be the closure of the image of

dn+1 in the ℓ2-norm on C(2)n (X).

Lemma 2.1. For each n,

H(2)n (X) ≅Zn

Bn+1.

Proof. Let θ ⊗C ∈ N (G)⊗ZG Cn(X) be an n-chain, where

C = ∑S∈Sn(K)

ϕSS

with ϕS ∈ CG ⊆ ℓ2(G). Letα(θ ⊗C) = ∑

S∈Sn(K)θ(ϕS)S.


Since θ(ϕS) ∈ ℓ2(G) for all S, this is an element of C(2)2 (X). Moreover, if α(θ ⊗C) = 0, then

θ(ϕS) = 0 for all S, and thus θ ⊗C = 0. Hence

α ∶ N (G)⊗ZG Cn(X)→ C(2)n (X)

is a monomorphism. On the other hand, if C ∈ C(2)n (X), then since ℓ2(G) is the norm-

completion of CG, there is a sequence (Ci)i ⊂ Cn(X) converging to C. Now Ci = α(Id⊗Ci),so that imα = C(2)n (X). One now verifies that α ○ ∂n = dn ○ α, from which the conclusionfollows. �

Hence each element of H(2)n (X) is of the form [Z], where Z = ∑S ZSS ∈ C(2)n (X) is an L2

n-cycle, and we shall use this characterization to manipulate elements of the L2 homology.

2.1. Examples of H(2)∗ (G) and H

(2)∗ (X). L2 Betti numbers have been computed for many

groups and spaces. Here we focus on examples for which b(2)2 (X) or b

(2)2 (G) = 0. Dicks and

Linnell proved the following [DL07]; that b(2)2 (G) = 0 for one-relator groups was apparently

already established by Luck.

Theorem 2.2. Let G be a one-relator group. Then b(2)n (G) = 0 for n ≥ 2.

Very simple one-relator groups – finite cyclic groups – illustrate the fact that one mustuse a K(π,1) complex to define the L2 homology of a group, just as in the case of ordinaryhomology. Indeed, let X be the 2-complex associated to the presentation ⟨a ∣ an⟩ for n ≥ 2.Then χ(X) = 1, while b(2)0 (X) =

1n and b

(2)1 (X) ≥ 0, so that 0 < b(2)2 (X) ≠ b

(2)2 (Zn) = 0.

On the other hand, if G is the fundamental group of a closed orientable surface S of genus

g ≥ 1, then G is a one-relator group. We shall verify below that b(2)2 (S) = 0 = b

(2)2 (G), using

the energy criterion. This simply reflects the fact that S is a K(G,1) space. Also, b(2)0 (S) = 0,

and b(2)1 (S) = 2(g − 1), by Proposition 1.6 and Theorem 1.7.

There are also interesting formulae for the L2 Betti numbers of 3-manifold groups, whichrely on Thurston’s geometrization theorem, for hyperbolic manifolds of arbitrary dimension,symmetric spaces, etc. [Luc09]. A common theme is that L2 Betti numbers are very often 0. Inthe next section, we discuss possible applications of this fact, and then discuss a combinatorial

criterion ensuring that b(2)2 vanishes.

3. Nonpositive immersions, coherence, and local indicability

The question of whether or not b(2)2 (X) vanishes is interesting, from our point of view,

because of possible applications to coherence and local indicability via the nonpositive im-mersions property.

Definition 3.1 (Coherent group). The finitely generated group G is coherent, or locallyfinitely presented, if every finitely generated subgroup of G has a finite presentation.

There are many examples of coherent groups; it is not hard to see that surface groups arecoherent, and the same is true of 3-manifold groups (proved by Scott in [Sco73] and in un-published work of Shalen), free-by-cyclic groups [FH99], and groups for which the perimeteralgorithm terminates [MW05]. One-relator groups with sufficiently high-powered torsion are


coherent [Wis05], as are one-relator groups whose relator is a positive word [Wis03]. It is con-jectured that all one-relator groups are coherent, and this seems to be related to Theorem 2.2.There are also numerous incoherent groups; these include negatively-curved groups built us-ing a variant of the Rips construction [Wis98], some groups constructed using synthetic Morsetheory [BB97], and some Artin groups [Gor04].

Definition 3.2 (Locally indicable group). The group G is indicable if there exists an epi-morphism G↠ Z. If, for every nontrivial finitely generated subgroup H ≤ G, H is indicable,then G is locally indicable.

Note that each epimorphism G ↠ Z factors through the abelianization of G, so that wehave an epimorphism H1(G) ↠ Z. On the other hand, an epimorphism ϕ ∶ H1(G) ↠ Zinduces an epimorphism ψ ∶ G → Z, given by ψ(g) = ϕ(g[G,G]). Thus G is indicable if andonly if b1(G) > 0, and locally indicable if and only if b1(H) > 0 for every finitely generatedH ≤ G.

Example 3.3. If there exists g ∈ G such that g ≠ 1G and g has finite order, thenG is not locallyindicable. On the other hand, many classes of groups are locally indicable, for example torsion-free 1-relator groups [Bro] and fundamental groups of knot and link complements [How82].

Local indicability, and probably coherence, are related to L2 homology via the nonpositiveimmersions properties discussed in [Wis02].

Definition 3.4 (Nonpositive immersions). A combinatorial map Y → X of CW-complexesis one that maps open cells homeomorphically to open cells. The combinatorial map Y → Xis an immersion if it is locally injective. Inclusions of subcomplexes and covers are examplesof immersions.

The 2-complex X has weak nonpositive immersions if for every compact, connected Y withan immersion Y → X, either π1Y = {1} or χ(Y ) ≤ 0. If for every such Y with χ(Y ) > 0, Y iscontractible, then X has strong nonpositive immersions.

The connection to L2 homology comes from the following two observations about a com-

pact, connected 2-complex X with fundamental group G. First, recall that χ(X) = b(2)0 (X)−b(2)1 (X)+b

(2)2 (X). Hence, if b

(2)2 (X) = 0 and χ(X) ≥ 1, then b(2)0 (X) ≥ 1. But b

(2)0 (X) = ∣G∣

−1,

so that G ≅ {1}. On the other hand, if χ(X) = b0(X)−b1(X)+b2(X) ≥ 1 and b(2)2 (X) = 0, thenb1(X), being the rank of the abelianization of G, vanishes. However, ∣G∣b2(X) = b(2)2 (X) = 0,since G is finite (see Example 2.8 of [Luc09]). Hence X is aspherical and X contractible, andχ(X) ≤ 1. this implies:

Lemma 3.5. Let K be a class of finite 2-complexes such that, if X ∈ K, then every cover

and finite subcomplex of X also belongs to K. If each X ∈ K satisfies b(2)2 (X) = 0, then each

complex in K has strong (and thus weak) nonpositive immersions.

In view of Lemma 3.5, we would like to find a class C of 2-complexes, closed under im-

mersions, such that each X ∈ C satisfies b(2)2 (X) = 0. An interesting such class is the class of

complexes satisfying the energy criterion.Our interest in the nonpositive immersions property comes from the following statements

in [Wis02].


Theorem 3.6. If X has the weak nonpositive immersions property, then G = π1X is locallyindicable.

Conjecture 3.7. Let X have the strong nonpositive immersions property. Then G is coher-ent.

If X ∈ K, then X is also aspherical, by the observations above. Indeed, the tower liftconstruction in [Wis02] shows that any cellular map S2 → X factors through an immersion

T →X. But b2(T ) = 0, since b(2)2 (T ) = 0 and χ(T ) > 0. Thus S2 → T →X is nullhomotopic.

4. The energy criterion

Wise gives two conditions on X that guarantee that b(2)2 (X) = 0. These are the thin cactus

property, which is well-adapted to the situation in which X satisfies various small-cancellationconditions, and the energy criterion, which is of a somewhat different character, and whichis our focus.

Definition 4.1 (Grading). A grading of X is a cellular map f ∶ (X, X0)→ (R,Z) such that,

for each 2-cell c of X, there exists n ∈ Z for which f(c) = [n,n + 1].

There are many natural ways to define a grading, depending on G and X. For example,given a 0-cell x ∈ X0, one can define a map f ∶ (X1, X0) → (R,Z) by f(y) = dX1(x, y).Extending such a map to the 2-cells in the required way is somewhat tricky, and may involvesubdividing the 2-cells. This is a radial grading. See Figure 1.

Figure 1. Radially grading a 2-simplex, and radially grading a hexagon bysubdividing. Black lines are 1-cells and colored lines are level sets.

Another important example of a grading arises when we already have an epimorphismϕ# ∶ G → Z arising from a cellular map ϕ ∶ X → S1. In this case, for each y ∈ X, let y be abase lift of y and let f(gy) = ϕ(y) + ϕ#(g). Figure 2 illustrates this in a simple special case.Consider the presentation

F2 ≅ ⟨a, b, c ∣ abc⟩,so that X is obtained from a 2-simplex by identifying all three 0-cells. Figure 2 shows theuniversal cover, and a grading coming from the map F2↠ Z given by a↦ 1, b↦ 1, c↦ −2.

In fact, the distance f(y) = dX1(x, y) on the 1-skeleton can always be extended to a grading,via subdivision:


Figure 2. Grading using a homomorphism F2 → Z. Edges labeled c getsubdivided by adding a new 0-cell, and each 2-cell gets subdivided by addinga new 1-cell, as indicated by the shading. Colors of 0-cells indicate their imagesin Z. The 1-cells map to the 1-cells joining the images of their endpoints, andthe 2-cells collapse to 1-cells as in Figure 1.

Lemma 4.2. There exists a grading f ∶ X → R. If ϕ ∶ Y → X is an immersion, and f ∶ X → Ris a grading, then f ○ ϕ ∶ Y → R is a grading, where ϕ is the map of universal covers inducedby ϕ.

Proof. Fix a basepoint x ∈ X0 and let f(y) = dX1(y, x) for each y ∈ X1, i.e. fix a radial

grading f ∶ X1 → R. Let R be a 2-cell. We shall first consider the case in which R is atriangle or quadrilateral, and then indicate how to subdivide larger 2-cells into triangles andquadrilaterals so as to extend f to all of X.

Mapping bigons and triangles: If R is a bigon, then either its vertices a, b are atdistance 1, in which case R maps to [f(a), f(a) + 1] in an obvious way, or we add a 0-cell cto the interior of one of the 1-cells [a, b], so that f(c) = f(a)+ 1 and R is now a triangle. Wethen proceed as below.

Let R be a triangle with 0-cells a, b, c. Suppose that a is a closest 0-cell of R to x, withf(a) = n. If f(b) = f(c) = n, then add a 0-cell d in the interior of R and join d to each ofa, b, c by adding a 1-cell to the interior of R. This makes R into the union of 3 triangles, eachcontaining a 0-cell d with f(d) = n + 1. Proceed as below.

If f(b) = f(a) = n and f(c) = n+ 1, then map the 1-cells [a, c] and [b, c] homeomorphicallyto [n,n + 1], collapse [a, b] to n, and map the interior of R to [n,n + 1] in the obvious way(imagine foliating R by intervals joining c to the interior points of [a, b] and map each ofthese intervals to [n,n + 1], or foliating R by intervals as in Figure 1 and mapping each ofthese to an interior point of [n,n + 1]). The construction is almost identical if f(a) = n andf(b) = f(c) = n + 1.

Mapping quadrilaterals: Let R = [a, b, c, d] be a quadrilateral. If f(a) = f(d) = n andf(b) = f(c) = n + 1, then there is an obvious projection to [n,n + 1]. Otherwise, it suffices tojoin b, d or a, c by a 1-cell, subdividing R into two triangles, and the proceed as above.


Subdividing: Let ∂R = [a1, a2, . . . , am], where f(a1) = n and f(ai) ≥ n for all i ∈{1, . . . ,m} and m > 4. If f(ai) = n for all i, add a 0-cell d to the interior of R and sub-divide R into triangles by joining each ai to d. Then f(d) = n + 1, and f extends to R bymapping each triangle to [n,n + 1] as above.

If f is not constant on the 0-cubes of R, let i1 ≤m be chosen so that f(ai) = n for 1 ≤ i ≤ i1−1and f(ai1) = n + 1. Subdivide R by adding a 1-cell c joining a1 to ai1 . Then R = R1 ∪c R2,where R1 and R2 satisfy ∣∂R1∣ + ∣∂R2∣ = m + 1. If 2 < i1 < m, then each of R1 and R2 hasa shorter boundary path than R, and can thus be subdivided and graded, by induction onm. Otherwise, i1 =m, and we instead subdivide by joining am to a2 by a 1-cell, grading theresulting triangle as above, and its complement in R by induction on m. See Figure 3. The

Figure 3. Subdividing to extend a radial grading to the 2-cells.

statement about immersions follows easily from the definitions. �

Given a grading f of X, two additional conditions on f constitute the energy criterion.

Definition 4.3 (Link, corner, coefficient, level). Let A be a 2-complex. For each a ∈ A0, thelink lk(a) is the graph that has a vertex v(c) for each occurrence of a as the initial or terminal0-cell of an oriented 1-cell c of A, with v(c) and v(c′) joined by an edge in lk(a) exactly whencc′ is part of the attaching map of a 2-cell. A corner (of a 2-cell) at a is an edge of lk(a). IfR is a 2-cell, a corner of R is a corner {v(c), v(c′)} at a 0-cell of R with cc′ forming part ofthe attaching map of R. The set of all corners in A is denoted C(A). A system of coefficientsof A is a uniformly bounded map C(A)→ R.

Let f ∶ A → R be a grading of A. A level-n 2-cell R is a 2-cell such that f(R) = [n,n + 1].The upper corners of R are the corners of R at 0-cells a ∈ R with f(a) = n + 1. The lowercorners of R are the corners at 0-cells a with f(a) = n. Let C↑(R) be the set of upper cornersof R, and C↓(R) the set of lower corners. Similarly, we define C↑(a) and C↓(a) to be the setof upper and lower corners at a ∈ A0. For each corner e ∈ C, let we be the coefficient at e.

Definition 4.4 (Admissible grading, visible 2-cell). Let f ∶ A → R be a grading. To beadmissible, f must satisfy the following two hypotheses on each 2-cell R:

(♣) ∑e∈C↓(R)

we ≥ 0

and

(♠) ∑e∈C↓(R)

we + ∑e∈C↑(R)

we ≤ 0.


Moreover, we impose conditions on lk(a) for each a ∈ A0. Indeed, the coefficients at the

corners at a induce an L2 “norm” on C(2)1 (lk(a)). Indeed, if ϕ is a square-summable function

on the 1-cells of lk(a), then let

∣∣ϕ∣∣22= ∑

e∈lk(a)1we∣ϕ(e)∣2.

Since lk(a) is 1-dimensional, we can define H(2)1 (lk(a)) to be the subspace of C

(2)1 (lk(a))

consisting of 1-cycles. Admissibility requires that ∣∣ − ∣∣2 is actually a norm on H(2)1 (lk(a)),

i.e. that

(♡) ∑e∈C↓(a)

we∣ϕ(e)∣2 + ∑e∈C↑(a)

we∣ϕ(e)∣2 ≥ 0.

Finally, we say that the 2-cell R is visible if

(♢) ∑e∈C↑(R)

we < 0.

Definition 4.5 (Energy criterion). The 2-complex A satisfies the energy criterion if foreach 2-cell R of A, there exists a uniformly bounded system of coefficients on C(A) and anadmissible grading f ∶ A→ R such that R is visible with respect to f .

Example 4.6 (Finitely-generated free abelian groups). Let G ≅ ⟨a1, . . . , an ∣ [ai, aj]∀ i, j⟩.Then X is the 2-skeleton of the standard tiling of Rn by n-cubes, with 0-cells at points in Zn.The map given by f(x1, . . . , xn) = xn is a grading. It is easy to check that assigning -1 to theupper corners and 1 to the lower corners makes f admissible, and every square is a visible2-cell.

Example 4.7 (A surface group). Consider a 2-complexX homeomorphic to a genus-2 surface,associated to the presentation

G ≅ ⟨a, b, c, d ∣ [a, b][d, c]⟩.Let R be a 2-cell of X, so that every 2-cell is of the form gR for some g ∈ G. We will exhibit asubdivision, a uniformly bounded system of coefficients, and an admissible grading f makingevery 2-cell in the subdivision of R visible. Since X1 is the Cayley graph, let 1G be the0-cell of R with an outgoing a edge and label the 0-cells by elements of G and the 1-cells bygenerators.

Subdivide R by joining a to each 0-cell of R except ab and 1G. Extend this subdivisionequivariantly to X. Now define a homomorphism ϕ ∶ G→ Z by ϕ(a) = 1, ϕ(b) = ϕ(c) = ϕ(d) =0. For each g ∈ X0, let f(g) = ϕ(g) be the net number of a-labeled 1-cells in a shortest pathfrom 1G to g. As shown in Figure 4, R is the union of six triangles, T1, . . . , T6. T1, . . . , T5have one upper corner and two lower corners, while T6 has two upper corners and one lowercorner; the grading extends to R as in Lemma 4.2.

Each 0-cell x is contained in 8 copies of R, labeled R1, . . . ,R8. Let T ji be the copy of Ti in

Rj that contains x.Assign to each lower corner the coefficient 1. Then ♣ is automatically satisfied for each

2-cell. To each upper corner at x in the 2-cell R1 in which x appears as the terminus of an

a-edge, assign the coefficient −2. Then in each T ji with 1 ≤ i ≤ 5, the coefficient sum is exactly


Figure 4. The arrowed edges are the boundary path of R, labeled by thegenerators of G. The black numbers indicate the grading on X0 arising fromthe homomorphism ϕ. The red 1-cells subdivide R into six triangles, allowingus to extend this to a grading of X. The indicated corner-coefficients makethis grading admissible and every 2-cell visible.

0. To the upper corner in T j6 , corresponding to the path ba−1, assign the coefficient 1. Then

the coefficient sum in T j6 is −2 + 1 + 0 = −1 ≤ 0. Hence ♠ is satisfied by all 2-cells.

Each 0-cell is surrounded by 6 upper corners of the first type, 1 of the second type, and11 lower corners. lk(x) is a circle, so ♡ is always satisfied, since 6(−2) + 1(1) + 11(1) = 0.Finally, each triangle either has exactly one upper corner, with coefficient -2, or two uppercorners, with total coefficient −2+1, so ♢ is satisfied by all 2-cells. Thus X satisfies the energycriterion.

Example 4.8. It is a good exercise to check that the universal cover of the complex Xassociated to Zn ≅ ⟨a ∣ an⟩ does not satisfy the energy criterion for n ≥ 1. Roughly, this isbecause any grading and system of coefficients forces some 0-cell to have the property thatall of its corners are upper, and at least one has negative coefficient. But each link is a cycle,so that ♡ fails.

We can now state and prove the energy criterion, following [Wis04] with two tiny correc-tions.

Theorem 4.9. Let X be a finite presentation complex of G. If X satisfies the energy crite-

rion, then b(2)2 (X) = 0, i.e. H

(2)2 (X,N (G)) = 0.

Remark 4.10 (Comments on the original proof). Wise uses excision for L2 homology; sincewe have not discussed how this works, we have instead avoided the issue by using localfiniteness of X instead; Wise suggests this in a marginal comment. Also, the proof in [Wis04]has a small computational error in the verification that ∑nBn ≤ N ∑S ∣ZS ∣2 for some N <∞.To fix this, one must assume, as Wise does, that the coefficients are uniformly bounded, but itseems as though one also needs a uniform bound on the number of corners in each 2-cell. Onecan either suppose, for example, that the grading f was constructed on a fixed subdivisionin which each 2-cell is already a triangle or a quadrilateral, or one can simply hypothesizethat there are finitely many G-orbits of 2-cells, and thus that there is a uniform bound onthe lengths of their boundary paths.


I am puzzled about why ♡ is stated in terms of H(2)1 (lk(a)), rather than in terms of

H1(lk(a)), since, in this context, each L2 2-cycle actually yields a (finitely supported) 1-cyclein lk(a). H2

1(2)(lk(a)) is the right object in the locally infinite case, but it is not clear to methat the argument works in that context.

Proof of Theorem 4.9. Fix a 2-cell R of X, a uniformly bounded system {we}c∈C(X) of coeffi-

cients, and a grading f ∶ X → R that is admissible with respect to these coefficients and withrespect to which R is visible.

Upper and lower norms coming from a 2-cycle: Suppose [Z] ∈ H(2)2 (X). Choose a2-cycle representing [Z], of the form

Z = ∑S∈S2(X)

ZSS,

where ∑S ∣ZS ∣2 < ∞. Now write S2 = ⊔n∈ZS2n, where S

2n is the set of 2-cells S mapping to

[n,n + 1]. LetZn = ∑

S∈S2n

ZS ,

so that Z = ∑nZn. For each n ∈ Z, letBn = ∑

S∈S2n

∑e∈C↓(S)

we∣ZS ∣2

andAn = ∑

S∈S2n

∑e∈C↑(S)

we∣ZS ∣2.

By applying ♠ and summing over the level-n 2-cells, we see that for each n ∈ Z,An +Bn ≤ 0.

Likewise, let a ∈ f−1(n + 1) be a 0-cell. Then every lower corner at a is an upper corner ina 2-cell S ∈ S2

n, soAn = ∑

a∈f−1(n+1)∑S∈S2

n

∑e∈C↓(a)∩C↑(S)

we∣ZS ∣2

and, similarly,Bn = ∑

a∈f−1(n)∑S∈S2

n

∑e∈C↑(a)∩C↓(S)

we∣ZS ∣2.

Comparing An and Bn+1: In this step of the proof, Wise uses excision for L2 homology.Since we have not defined L2-homology of pairs, we will avoid this and assume instead thatX is locally finite (this covers the case when X is a finite presentation complex of G, forexample).

Let a ∈ X0. The L2 2-cycle Z gives rise to an element Z ∈H(2)1 (lk(a)) as follows. For eache ∈ C(a), let Ze = ZS , where S is the 2-cell at a containing the corner e. Let

Z = ∑e∈C(a)

Ze.

Since X contains finitely many 2-cells that contain a, there are finitely many corners at aand hence Z ∈ C1(lk(a)), so that in particular Z is square-summable. There are no 2-cells,


so Z is not a boundary. On the other hand, ∂Z = 0 since ∂Z = 0. Thus Z is the uniquerepresentative of a homology class in H1(lk(a)). Applying ♡ to Z yields, where S(e) is the2-cell containing the corner e,

∑e∈C↓(a)

we∣ZS(e)∣2 + ∑e∈C↑(a)

we∣ZS(e)∣2 ≥ 0.

Summing over the level-(n+1) 0-cells and substituting the above expressions for An,Bn yields

(1) An +Bn+1 ≥ 0.

Obtaining a contradiction when [Z] ≠ 0: Let K <∞ be a constant such that ∣we∣ ≤Kfor all corners e. Let L <∞ be a uniform bound on the number of corners in a 2-cell, whichexists since there are finitely many orbits of 2-cells.

Bn = ∑S∈S2

n

∑e∈C↓(S)

we∣ZS ∣2

≤ K ∑S∈S2

n

∣C↓(S)∣∣ZS ∣2

≤ KL ∑S∈S2

n

∣ZS ∣2.

Hence

∑n

Bn ≤ ∑S∈S2

∣ZS ∣2.

We shall show that, if Z ≠ 0, then ∑nBn = ∞, and this contradicts the fact that Z issquare-summable.

Let m ∈ Z be the level of the visible 2-cell R. If ZR ≠ 0, then for all n > m, we have Bn ≥Bm+1 > 0, and hence ∑n≥mBn =∞. Indeed, by Equation (1) and the fact that An +Bn ≤ 0,

Bn+1 ≥ −An ≥ Bn.

On the other hand, by the definition of a visible 2-cell, Am < 0. Thus Bm+1 > 0 and Bn ≥ Bm+1for all n >m, and the proof is complete. �

Lemma 8.2 of [Wis04] says that the property of satisfying the energy criterion is inheritedby immersions:

Lemma 4.11. Let X be a finite 2-complex such that X satisfies the energy criterion. Suppose

that Y →X is a combinatorial immersion of 2-complexes. Then b(2)2 (Y ) = 0.

Proof. Let ϕ ∶ Y → X be the combinatorial map induced by ϕ. Let S be a 2-cell of Y sothat ϕ(S) = R be a 2-cell of X. Note that ϕ induces an injection on each link. Suppose that

f ∶ X → R is a grading, admissible with respect to the system {we} of coefficients, such thatR is visible with respect to f .

For each corner f of Y , let e = ϕ(f) be the induced corner of X. Letting vf = we yields a

uniformly bounded system {vf} of coefficients on the corners of Y . The map f ○ ϕ ∶ Y → R isan admissible grading with respect to these coefficients, and S is visible with respect to thisgrading. Indeed, ϕ sends upper (lower) corners with respect to f ○ ϕ to upper (lower) cornerswith respect to f . �


Assembling the results gives:

Corollary 4.12. If X satisfies the energy criterion, then X has nonpositive immersions.Hence G is locally indicable, X is aspherical, and, if Conjecture 3.7 is true, G is coherent.

As an aside, it is interesting to wonder under what hypotheses it is true that X satisfies

the energy criterion provided b(2)2 (X) = 0. I suspect from some examples that the two con-

ditions are equivalent when X is compact and contains at least one 2-cell. It seems easy toconstruct radial gradings and uniformly bounded coefficients satisfying ♣,♠,♢ under thesecircumstances, so that one should try to construct an L2 2-cycle from a nontrivial elementof H1(lk(a)) that causes ♡ to fail, but at the moment I am not quite sure how this works ingeneral.

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