arXiv:1702.04416v2 [math.DS] 12 Oct 2017

DYNAMICAL CORRESPONDENCES OF L2 -BETTI NUMBERS
BINGBING LIANG
Abstract. We investigate dynamical analogues of the L2 -Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we show that
the L2 -Betti numbers exactly measure the failure of addition formula for dynamical invariants.

Contents
1. Introduction
2. Preliminaries
2.1. Group rings
2.2. Relative von Neuman-Lück rank
2.3. Amenable and sofic groups
2.4. Relative mean dimension and relative mean rank
3. L2 -Betti number, j-th mean rank, and j-th mean dimension
4. Main Results
References

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5
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8
11
15

1. Introduction
There are a couple of connections established among invariants in dynamical systems, group rings, and L2 -invariants. These connections are obtained via a type
of dynamical system called algebraic actions. Given a discrete group Γ, each ZΓmodule M can be treated as an action of Γ on the discrete abelian group M by
c of M naturally inherits an action
group automorphisms. The Pontryagin dual M
of Γ by continuous automorphisms from the module structure of M. Conversely,
by Pontryagin duality, each action of Γ on a compact Hausdorff abelian group arise
this way and thus we call such a dynamical system an “algebraic action” [36].
A surprising fact is that one can recover certain algebraic information about M
c However,
by taking advantage of purely dynamical information about Γ y M.
Date: March 1, 2017.
2010 Mathematics Subject Classification. Primary 37B99, 16D10, 55N35, 22D25.
Key words and phrases. Sofic group, mean dimension, mean rank, L2 -Betti number, dimensionflatness.
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BINGBING LIANG

c For
the dynamical information itself does not use the algebraic structure of M.
example, Li and Thom showed that, in the setting of amenable group actions, the
c coincides with the L2 -torsion of M (see [29]). One ingredient of
entropy of Γ y M
establishing this connection is Peters’ algebraic characterization of entropy [33]. This
correspondence has interesting applications to the vanishing results on L2 -torsion
and Euler characteristic [7, 29]. In the same spirit, Li and the author showed that
c coincides with the von Neumann-Lück
the mean topological dimension of Γ y M
rank of M (see [26]). Establishing this correspondence relies on the study of the
mean rank as an algebraic invariant of ZΓ-modules and Lück’s result on dimensionflatness for amenable groups [32, Theorem 6.73]. Based on this connection, the
mean dimension of algebraic actions for amenable groups is well understood [26].
Mean topological dimension is a newly-introduced dynamical invariant by Gromov
[14], systematically studied by Lindenstrauss and Weiss [30], and remains to be
further explored [8]. As a dynamical analogue of the covering dimension, it is closely
related to the topological entropy, and takes a crucial role in embedding problem of
dynamical systems [15–20, 30].
On the other hand, using Lück’s extended von Neumann dimension for any module
over the group von Neumann algebra LΓ of a discrete group Γ (see [32, Chapter
6]), for any ZΓ-module M we call the von Neumann-Lück dimension for LΓ ⊗ZΓ M
as the von Neumann-Lück rank vrk(M) of M. Von Neumann-Lück dimension is a
length function on LΓ-modules [32, Theorem 6.7] and von Neumann-Lück rank is a
length function on ZΓ-modules when Γ is amenable [27, Definition 2.1] [26, Section
5.2] [28, Theorem 3.3.4].
Mean rank is also a length function on ZΓ-modules of an amenable group Γ (see
[26, Section 3]). As a dynamical analogue of the rank of abelian groups, it serves
as a bridge connecting mean dimension and von Neumann-Lück rank [26, Theorem
1.1].
Towards more general groups, Bowen and Kerr-Li developed an entropy theory
based on the idea of approximating the dynamical data by external finite models
when the acting group can be approximated by finite groups [2, 24]. The groups
admitting this approximation are the so-called sofic groups, which include residually
finite groups and amenable groups [13, 39]. The extended notion of entropy extends
the classic notion but no longer decreases when passing to a factor system. Similarly
mean dimension has been extended to the case of sofic group actions [25]. To deal
with this nonamenable phenomenon, Li and the author introduced the relative sofic
invariants, established an alternative addition formula, and used them to relate mean
dimension with von Neumann-Lück rank for sofic groups [27]. Similar approaches
also independently appears in the works of other experts. Hayes gave a formula
for this invariant in terms of a given compact model in [22]. A similar notion for
Rokhlin entropy, called outer Rokhlin entropy, was developed by Seward in [37].
Using the microstate technique, Hayes proved that von Neumann-Lück rank of a

DYNAMICAL CORRESPONDENCES OF L2 -BETTI NUMBERS

3

c
finitely presented ZΓ-module M coincides with sofic mean dimension of Γ y M
under certain conditions [21].
Via a projective resolution of any ZΓ-module M, we can treat von Neumann-Lück
(2)
rank of M as the 0-th L2 -Betti number β0 (M) of M (see Proposition 3.5). From
c of Γ y M
c corre[27, Theorem 1.3], we know the sofic mean dimension mdimΣ (M)
(2)
spondences to β0 (M) when Γ is a countable sofic group and M is countable. Here
Σ is a fixed sofic approximation sequence for Γ. For the higher L2 -Betti numbers of
M, Hanfeng Li asked the following question.
c correspond to the
Question 1.1. If Γ is sofic, what dynamical invariants of Γ y M
(2)
2
j-th L -Betti numbers βj (M) of M for j ≥ 1?

In this paper, motivated by the above question, we mainly study dynamical ana(2)
logues of the L2 -Betti numbers βj (C∗ ) of a chain complex C∗ of ZΓ-modules:
∂

∂

· · · →2 C1 →1 C0 → 0(= C−1 ).
In the spirit of Elek [11] (also for the notational convenience), we introduce the
c :=
j-th mean rank mrkj (C∗ ) of C∗ and the j-th mean dimension mdimj (C∗ ) of C
∗
HomZ (C∗ , R/Z) for any sofic group Γ (see Definition 3.1). These definitions use the
relative sofic invariants as opposed to Elek’s approach where he considered the case
that Γ is amenable and therefore there is no nonamenable phenomenon appeared.
Let Γ y X and Γ y Y be two algebraic actions, X and Y be metrizable
spaces, and π : X → Y be a Γ-equivariant continuous homomorphism. We say
π satisfies Juzvinskiı̆ formula for mean dimension if mdimΣ (X) = mdimΣ (ker π) +
mdimΣ (im π). The main result of this paper is as follows.
Theorem 1.2. Suppose that vrk(Cj ) < ∞ for some j ≥ 0. Then
(2)

(1) βj (C∗ ) = vrk(coker ∂j+1 ) − vrk(im ∂j |Cj−1);
(2)
(2) If Γ is sofic, we have βj (C∗ ) = mrkj (C∗ ). If furthermore Cj and Cj−1 are
c );
countable, we have mrkj (C∗ ) = mdimj (C
∗
(2)
(3) If im ∂j+1 = ker ∂j , we have that βj (C∗ ) = 0 if and only if
vrk(Cj−1) = vrk(coker ∂j ) + vrk(im ∂j ).
(2)

If furthermore Γ is sofic and Cj and Cj−1 are countable, we have that βj (C∗ ) =
0 if and only if ∂cj satisfies Juzvinskiı̆ formula for mean dimension.
From Theorem 1.2, the notion of j-th mean rank provides an equivalent algebraic
definition of L2 -Betti numbers from module theory. Secondly, the L2 -Betti numbers
exactly measure the failure of the additivity of dynamical invariants. In [11], Elek
introduced an analogue of the L2 -Betti numbers for amenable linear subshifts. It was
showed that Juzvinskiı̆ formula for entropy can fail when the group Γ has nonzero
Euler characteristic [10]. Hayes proved that Juzvinskiı̆ formula for entropy fails when

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BINGBING LIANG

Γ has nonzero L2 -torsion [23]. Gaboriau and Seward established some inequalities
relating Juzvinskiı̆ formula for entropy with L2 -Betti numbers [12]. Bowen and
Gutman established Juzvinskiı̆ formula for the f -invariant of finitely generated free
group actions [3].
c of
To respond to Question 1.1, we introduce the j-th mean dimension mdimj (M)
c and j-th mean rank mrk (M) of M (Definition 3.3 and Definition 3.1). As
ΓyM
j
the first application, the following corollary may shed some light on Question 1.1.
(2)

Corollary 1.3. When mrkj (M) is defined, we have mrkj (M) = βj (M). If furtherc = β (2) (M).
more M is countable, we have mdimj (M)
j
For the second application, we give a dynamical characterization of Lück’s dimension(2)
flatness. We say Γ satisfies Lück’s dimension-flatness over Z if βj (M) vanishes for
any j ≥ 1 and ZΓ-module M. It was proven that amenable groups satisfy Lück’s
dimension-flatness [32, Theorem 6.37]. We say Γ satisfies Juzvinskiı̆ formula for
von Neumann-Lück rank if vrk(M) = vrk(ker ϕ) + vrk(im ϕ) for any ZΓ-module
homomorphism ϕ : M → N of ZΓ-modules M and N. It is similarly defined when
we talk about whether Γ satisfies Juzvinskiı̆ formula for mean rank.
Corollary 1.4. Γ satisfies Lück’s dimension-flatness over Z if and only if Γ satisfies
Juzvinskiı̆ formula for von Neumann-Lück rank. If Γ is sofic, then Γ satisfies Lück’s
dimension-flatness over Z if and only if Γ satisfies Juzvinskiı̆ formula for mean rank
and mean dimension.
We remark that the first statement of the above corollary can also be proved
using standard properties of Tor functor and additivity of von Neumann-Lück dimension. In the light of results on the failure of Juzvinskiı̆ formula [21, Proposition
7.2] [23, Corollary 6.24] [12, Theorem 6.3], we show that taking subgroups respects
the property of Lück’s dimension-flatness in Proposition 4.4. As a consequence, if
(2)
βj (H) > 0 for some subgroup H of Γ and some j ≥ 1, then Γ violoates Juzvinskiı̆
formula for mean dimension.
Lück conjectured that a group is amenable if and only if it satisfies Lück’s dimensionflatness [32, Conjecture 6.48]. Bartholdi and Kielak implicitly proved this conjecture
using a new characterization of amenability [1, Theorem 1.1]. It follows that
Corollary 1.5. A countable group is amenable if and only if it satisfies Juzvinskiı̆
formula for von Neumann-Lück rank.
This paper is organized as follows. We recall some background knowledge in
Section 2. In Section 3 we introduce the j-th mean rank, j-th mean dimension,
and establish some basic properties. We prove the main results and show some
applications in Section 4.
Throughout this paper, Γ will be a countable discrete group. For any set S, we
denote by F(S) the set of all nonempty finite subsets of S. All modules are assumed

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5

to be left modules unless specified. For any d ∈ N, we write [d] for the set {1, · · · , d}
and Sym(d) for the permutation group of [d].
Acknowledgements. We are grateful to Lewis Bowen, Ben Hayes, Fabian Henneke,
Yonatan Gutman, Huichi Huang, Yang Liu, Yongle Jiang, Wolfgang Lück, Jianchao
Wu, and Xiaolei Wu for helpful discussions and comments. The author is supported
by Max Planck Institute for Mathematics in Bonn.
2. Preliminaries
2.1. Group rings. The integral group ring of Γ, denoted by ZΓ, consists of all
P
finitely supported functions f : Γ → Z. We shall write f as s∈Γ fs s, where fs ∈ Z
for all s ∈ Γ and fs = 0 for all except finitely many s ∈ Γ. The algebraic operations
on ZΓ are defined by
X

s∈Γ

fs s +

X

s∈Γ

gs s =

X

(fs + gs )s, and

s∈Γ

ÄX

fs s

s∈Γ

äÄ X

ä

gt t =

t∈Γ

X

fs gt (st).

s,t∈Γ

We similarly have the product if one of f and g sits in CΓ .
For any countable ZΓ-module M, treated as a discrete abelian group, its Pontryac consisting of all continuous group homomorphisms M → R/Z, coincides
gin dual M
c is a compact metrizable space unwith HomZ (M, R/Z). By Pontryagin duality, M
der compact-open topology. Furthermore, the ZΓ-module structure of M naturally
c by continuous automorphisms. To be precise,
induces an adjoint action Γ y M
hsχ, ui := hχ, s−1 xi

c u ∈ M, and s ∈ Γ.
for all χ ∈ M,

2.2. Relative von Neuman-Lück rank. Let ℓ2 (Γ) be the Hilbert space of square
P
summable functions f : Γ → C, i.e. s∈Γ |fs |2 < +∞. Then Γ has two canonical
commuting unitary representations on ℓ2 (Γ), namely the left regular representation
λ and the right regular representation ρ defined by
λ(s)(x) = sx, and ρ(s)(x) = xs−1
for all x ∈ ℓ2 (Γ) and s ∈ Γ. Here we treat Γ as a subset of CΓ. The (left) group
von Neumann algebra of Γ, denoted by LΓ, consists of all bounded linear operators
ℓ2 (Γ) → ℓ2 (Γ) commuting with ρ(s) for all s ∈ Γ.
Denote by δeΓ the unit vector of ℓ2 (Γ) being 1 at the identity element eΓ of Γ, and
0 everywhere else. The canonical trace on LΓ is the linear functional trLΓ : LΓ → C
given by trLΓ (T ) = hT δeΓ , δeΓ i. For each n ∈ N, the extension of trLΓ to Mn (LΓ)
P
sending (Tj,k )1≤j,k≤n to nj=1 trLΓ (Tj,j ) will still be denoted by trLΓ .
For any finitely generated projective LΓ-module P, one has P ∼
= (LΓ)1×n P for
some n ∈ N and some P ∈ Mn (LΓ) with P 2 = P . The von Neumann dimension of
P is defined as
dim′LΓ (P) := trLΓ (P ) ∈ [0, n],

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BINGBING LIANG

which does not depend on the choice of n and P . For an arbitrary LΓ-module M,
its von Neumann-Lück dimension [32, Definition 6.6] is defined as
dimLΓ(M) := sup dim′LΓ(P),
P

for P ranging over all finitely generated projective LΓ-submodules of M.
The following theorem collects the fundamental properties of the von NeumannLück dimension [32, Theorem 6.7]. Given a unital ring R, a length function on
left RΓ-modules is a function on left RΓ-modules satisfying certain conditions ([27,
Definition 2.1]).
Theorem 2.1. dimLΓ extends dim′LΓ and is a length function on LΓ-modules with
dimLΓ (LΓ) = 1.
Definition 2.2. For any ZΓ-modules M1 ⊆ M2 , the von Neumann-Lück rank of
M1 relative to M2 is defined as
vrk(M1 |M2 ) := dimLΓ (im 1 ⊗ i),
where 1 ⊗ i is the natural map LΓ ⊗ M1 → LΓ ⊗ M2 .
Note that when M1 = M2 , we have vrk(M1 ) = vrk(M1 |M2 ).
2.3. Amenable and sofic groups. The group Γ is called amenable if for any
K ∈ F(Γ) and any δ > 0 there is a F ∈ F(Γ) with |KF \ F | < δ|F |.
A sequence of maps Σ = {σi : Γ → Sym(di )}i∈N is called a sofic approximation
for Γ if it satisfies:
(1) limi→∞ |{v ∈ [di ] : σi,s σi,t (v) = σi,st (v)}|/di = 1 for all s, t ∈ Γ,
(2) limi→∞ |{v ∈ [di ] : σi,s (v) 6= σi,t (v)}|/di = 1 for all distinct s, t ∈ Γ,
(3) limi→∞ di = +∞.
The group Γ is called a sofic group if it admits a sofic approximation.
Any amenable group is sofic since one can use a sequence of asymptoticallyinvariant subsets of the amenable group, i.e. Følner sequence, to construct a sofic
approximation. Residually finite groups are also sofic since a sequence of exhausting
finite-index subgroups naturally induces a sofic approximation in which each approximating map is actually a group homomorphism. We refer the reader to [5, 6]
for more information on sofic groups.
Throughout the rest of this paper, Γ will be a countable sofic group, and Σ =
{σi : Γ → Sym(di )}i∈N will be a sofic approximation for Γ.
2.4. Relative mean dimension and relative mean rank. We first recall the
notion of the covering dimension. For any finite open cover U of a compact metrizable space Z, denote the overlapping number of U by ord(U), i.e. ord(U) =
P
maxx∈X U ∈U 1U (x) − 1. Set
D(U) = inf ord(V)
V

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7

for V ranging over all finite open covers of Z finer than U, i.e. each element of V
is contained in some element of U. Then the covering dimension of Z is defined as
supU D(U) for U ranging over all finite open covers of Z.
Let Γ act continuously on a compact metrizable space X.
Definition 2.3. Let ρ be a compatible metric on X. For any d ∈ N, there is a
compatible metric on X d defined by
Ñ

ρ2 (ϕ, ψ) =

1 X
ρ(ϕv , ψv )2
d v∈[d]

é1/2

.

Let σ be a map from Γ to Sym(d), F ∈ F(Γ), and δ > 0. The set of approximately
equivariant maps Map(ρ, F, δ, σ) is defined to be the set of all maps ϕ : [d] → X
such that ρ2 (sϕ, ϕ ◦ σ(s)) ≤ δ.
Now let Γ act on another compact metrizable space Y and π : X → Y be a
surjective Γ-equivariant continuous map. Denote by Map(π, ρ, F, δ, σ) the set of all
π ◦ ϕ for ϕ ranging in Map(ρ, F, δ, σ). Note that Map(π, ρ, F, δ, σ) is a closed subset
of Y d . For any finite open cover U of Y , denote by Ud the open cover of Y d consisting
of Πv∈[d] Uv , where each Uv sits in U. Restricting Ud to Map(π, ρ, F, δ, σ), we obtain a
finite open cover Ud |Map(π,ρ,F,δ,σ) := {U ∩ Map(π, ρ, F, δ, σ)}U ∈Ud of Map(π, ρ, F, δ, σ).
Definition 2.4. For any finite open cover U of Y we define
D(Ud |Map(π,ρ,F,δ,σi) )
.
i→∞
di
If Map(ρ, F, δ, σi ) is empty for all sufficiently large i, we set mdimΣ (π, U, ρ, F, δ) =
−∞. We define the mean topological dimension of Γ y Y relative to the extension
Γ y X as
mdimΣ (Y |X) := sup inf inf mdimΣ (π, U, ρ, F, δ),
mdimΣ (π, U, ρ, F, δ) = lim

U

F ∈F(Γ) δ>0

where U ranges over finite open covers of Y . By a similar argument as in [25,
Lemma 2.9], we know mdimΣ (Y |X) does not depend on the choice of ρ. The sofc
mean topological dimension of Γ y X is defined as
mdimΣ (X) := mdimΣ (X|X)
for π : X → X being the identity map.
Example 2.5. Let M1 ⊆ M2 be countable ZΓ-modules. Then the induced map
d → M
d is a surjective Γ-equivariant continuous map of compact metrizable
M
2
1
d |M
d ) is well-defined.
spaces. Thus mdimΣ (M
1
2
Now we recall the notion of the relative mean rank. For any ZΓ-module M,
denote by F (M) the set of finitely generated abelian subgroups of M. Let A , B ∈
F (M), F ∈ F(Γ), and σ be a map from Γ to Sym(d) for some d ∈ N. Denote by
M (A , B, F, σ) the image of A d in Md /M (B, F, σ) under the quotient map Md →

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BINGBING LIANG

Md /M (B, F, σ). Here M (B, F, σ) denotes the abelian subgroup of Md ∼
= Zd ⊗Z M
generated by the elements δv ⊗ b − δsv ⊗ sb for all v ∈ [d], b ∈ B, and s ∈ F .
Definition 2.6. Let M1 ⊆ M2 be ZΓ-modules. For any A ∈ F (M1 ), B ∈ F (M2 ),
and F ∈ F(Γ), set
rk(M (A , B, F, σi ))
.
mrkΣ (A |B, F ) = lim
i→∞
di
We define the mean rank of M1 relative to M2 as
mrkΣ (M1 |M2 ) =

sup

inf

inf

A ∈F (M1 ) F ∈F(Γ) B∈F (M2 )

mrkΣ (A |B, F ).

The sofic mean rank of M1 is then defined as
mrkΣ (M1 ) := mrkΣ (M1 |M1 ).
Applying [27, Theorem 1.1],[27, Theorem 7.2], [27, Theorem 10.1], and running a
similar argument as in the proof of [27, Proposition 8.5] for relative mean rank, we
have:
Theorem 2.7. For any ZΓ-modules M1 ⊆ M2 , we have mrkΣ (M1 |M2 ) = vrk(M1 |M2 )
and
mrkΣ (M2 ) = mrkΣ (M1 |M2 ) + mrkΣ (M2 /M1 ).
d |M
d ) = mrk (M |M ).
If furthermore M2 is countable, we have mdimΣ (M
1
2
Σ
1
2

The following proposition collects basic properties of the sofic mean rank [27,
Section 3].
Proposition 2.8. Let M1 and M2 be ZΓ-modules. The following are true.
(1) mrkΣ (ZΓ) = 1.
(2) mrkΣ (M1 |M1 ⊕ M2 ) = mrkΣ (M1 ) and mrkΣ (M1 ⊕ M2 ) = mrkΣ (M1 ) +
mrkΣ (M2 ).
(3) If M1 ⊆ M2 and M1 is the union of an increasing net of ZΓ-submodules
{M′j }j∈J , then mrkΣ (M′j |M2 ) ր mrkΣ (M1 |M2 ). If furthermore mrkΣ (M2 ) <
∞, then mrkΣ (M2 /M′j ) ց mrkΣ (M2 /M1 ).
(4) Assume that M1 ⊆ M2 , M1 is finitely generated, and M2 is the union of
an increasing net of ZΓ-submodules {M′j }j∈J of M2 containing M1 . Then
mrkΣ (M1 |M′j ) ց mrkΣ (M1 |M2 ).
3. L2 -Betti number, j-th mean rank, and j-th mean dimension
Let C∗ be a chain complex of ZΓ-modules:
∂

∂

∂

· · · →2 C1 →1 C0 →0 0(= C−1 ).
Applying the covariant tensor functor LΓ ⊗ZΓ ·, we get a chain complex LΓ ⊗ZΓ C∗
of LΓ-modules:
1⊗∂
1⊗∂
· · · −→2 LΓ ⊗ C1 −→1 LΓ ⊗ C0 → 0;

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9

applying the contravariant Pontryagin dual functor HomZ (·, R/Z) := b·, we get a
c of algebraic actions such that the maps {∂
c } are Γ-equivariant:
chain complex C
∗
j j
∂“

∂“

2
1
” ←−
” ←− 0.
· · · ←−
C
C
1
0

’ =
Since R/Z is an injective Z-module, when im ∂j+1 = ker ∂j , we have ker ∂
j+1
c
im ∂j .

Definition 3.1. For each j ≥ 0, the j-th L2 -Betti number of C∗ is defined as
(2)

βj (C∗ ) = dimLΓ Hj (LΓ ⊗ZΓ C∗ ).
If vrk(Cj ) < ∞ for all j ≥ 1 and Cj = 0 as j is large enough, we define the Euler
characteristic of C∗ as
X
(−1)j vrk(Cj ).
χ(C∗ ) :=
j≥0

When vrk(Cj ) < ∞ for some j ≥ 0 and Γ is sofic, we define the j-th mean rank of
C∗ as
mrkj (C∗ ) = mrkΣ (coker ∂j+1 ) − mrkΣ (im ∂j |Cj−1).
If furthermore Cj and Cj−1 are countable, we define the j-th mean topological dimension of C∗ as
c ) := mdim (ker ∂
’) − mdim (im ∂
c |C
’ ).
mdimj (C
∗
Σ
j+1
Σ
j
j−1

Remark 3.2.
(1) When C∗ is a chain complex of CΓ-modules, since CΓ is flat
(2)
as a ZΓ-module, we have βj (C∗ ) = dimLΓ Hj (LΓ ⊗CΓ C∗ ), which extends
the definition of L2 -Betti numbers for chain complexes of CΓ-modules [32,
Definition1.16, Theorem 6.24].
(2) By Theorem 2.7, we know that the j-th mean rank and j-th mean dimension
are well defined.
(3) Wall gave some criteria when a chain complex of ZΓ-modules can be realized
as the chain complex of a Γ-CW complex [38, Theorem 2].
Let M be a ZΓ-module. A projective resolution of M is an exact sequence of
ZΓ-modules
∂2
∂1
∂0
· · · −→
C1 −→
C0 −→
M −→ 0
in which each Cj is a projective ZΓ-module. Denote by C∗ its deleted projective
resolution
∂1
∂2
C0 −→ 0,
C1 −→
· · · −→
which is a chain complex of ZΓ-modules. We similarly have the notion of free
resolution. Apply the notion of free module, we know that any ZΓ-module admits
a free resolution [35, Proposition 10.32].

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BINGBING LIANG

Definition 3.3. For each j ≥ 0, we define the j-th L2 -Betti number of M as
(2)

(2)

βj (M) := βj (C∗ ).
(2)

The j-th L2 -Betti number βj (Γ) of Γ is defined as the j-th L2 -Betti number of the
trivial ZΓ-module Z.
If vrk(Cj ) < ∞ for all j ≥ 1 and Cj = 0 as j is large enough, we define the Euler
characteristic of M as
χ(M) := χ(C∗ ).
When vrk(Cj ) < ∞ for some j ≥ 0 and Γ is sofic, we define the j-th mean rank of
M as
mrkj (M) := mrkj (C∗ ).
If furthermore M is countable, we can choose C∗ such that each Cj for j ≥ 0 is
countable and define the j-th mean topological dimension of M as
c := mdim (C
c ).
mdimj (M)
j
∗
(2)

Remark 3.4. In fact, βj (M) is the von Neumann-Lück dimension of torZΓ
j (LΓ, M)
(see [35, Page 836] for definition). Based on the Comparison Theorem for projective
resolutions [35, Theorem 10.46], any two projective resolutions of M are homotopy
(2)
equivalent, we know that βj (M) does not depend on the choice of projective resolutions [35, Corollary 10.51]. We refer the reader to [4, Chapter VIII] for discussions
on when a ZΓ-module admits a “small” projective resolution.
(2)

Proposition 3.5. For any ZΓ-module M, we have β0 (M) = vrk(M) and mrk0 (M) =
c = mdim (M).
c
mrkΣ (M). When M is countable, we have mdim0 (M)
Σ
Proof. From the exactness, we have
LΓ ⊗ C0 / im 1 ⊗ ∂1 = LΓ ⊗ C0 /ker 1 ⊗ ∂0 ∼
= im 1 ⊗ ∂0 = LΓ ⊗ M
and
C0 / im ∂1 = C0 /ker ∂0 ∼
= im ∂0 = M.
So by definition, we have
(2)

β0 (M) = dimLΓ (LΓ ⊗ C0 / im 1 ⊗ ∂1 ) = dimLΓ (LΓ ⊗ M) = vrk(M)
and
mrk0 (M) = mrkΣ (coker ∂1 ) = mrkΣ (C0 / im ∂1 ) = mrkΣ (M).
c So when M is countable, we
From the exactness, we have ker ∂c1 = im ∂c0 ∼
= M.
have
c = mdim (ker ∂
c ) = mdim (M).
c
mdim0 (M)
Σ
1
Σ


DYNAMICAL CORRESPONDENCES OF L2 -BETTI NUMBERS

11

Example 3.6. Let Γ = F2 be the free group with generators a and b. Set f =
(a − 1, b − 1)T ∈ (ZΓ)2×1 . Then M := (ZΓ)1×1 /(ZΓ)1×2 f has the following free
resolution:
R(f )
0 → (ZΓ)1×2 −→ (ZΓ)1×1 → M → 0,
where R(f ) sends x to xf . Note that M ∼
= Z as the ZΓ-modules for the trivial
(2)
ZΓ-module Z. By [32, Lemma 6.36], we know LΓ ⊗ZΓ Z = 0. Thus β0 (M) = 0.
(2)
(2)
It follows that β1 (M) = dimLΓ ker 1 ⊗ R(f ) = β0 (M) − χ(M) = 0 − (1 − 2) = 1.
ε
This example is essentially the same as 0 → ker ε → ZΓ → Z → 0, where ε is
the argumentation map. See [9, Chapter IV, Theorem 2.12] for a characterization
of when ker ε is a projective ZΓ-module.
4. Main Results
Corollary 1.3 follows when we apply Theorem 1.2 to a deleted projective resolution
of a ZΓ-module M.
Proof of Theorem 1.2. (1) Note that for any ZΓ-module homomorphism ϕ : M →
N and the inclusion map i : im ϕ → N, we have im 1 ⊗ ϕ = im 1 ⊗ i. Thus
dimLΓ (im 1 ⊗ ϕ) = vrk(im ϕ|N). For ZΓ-modules im ∂j+1 ⊆ Cj , we have
vrk(Cj ) = vrk(im ∂j+1 |Cj ) + vrk(coker ∂j+1 ).
Since the function dimLΓ (·) is additive, we have
(2)

βj (C∗ ) = dimLΓ(ker 1 ⊗ ∂j ) − dimLΓ(im 1 ⊗ ∂j+1 )
= (dimLΓ (LΓ ⊗ Cj ) − dimLΓ(im 1 ⊗ ∂j )) − dimLΓ(im 1 ⊗ ∂j+1 )
= (vrk(im ∂j+1 |Cj ) + vrk(coker ∂j+1 )) − dimLΓ (im 1 ⊗ ∂j ) − dimLΓ (im 1 ⊗ ∂j+1 )
= vrk(coker ∂j+1 ) − vrk(im ∂j |Cj−1).
(2) For any subgroup H of a discrete abelian group G, denote by H ⊥ the elements
÷ Thus
“ which vanish on H. By Pontryagin duality, we have H ⊥ ∼
of G
= G/H.
⊥ ∼ ¤
’ = {χ ∈ C
¤
” :χ◦∂
ker ∂
j+1
j
j+1 = 0} = (im ∂j+1 ) = Cj / im ∂j+1 = coker ∂j+1 .

÷
’ ) = mdim (im
’ ). Thus by TheoBy definition, we have mdimΣ (im ∂cj |C
∂j |C
j−1
Σ
j−1
rem 2.7, the equalities follow from (1).
(3) By the addition formula of von Neumann-Lück dimension, we first have

vrk(Cj−1 ) = vrk(im ∂j |Cj−1) + vrk(coker ∂j ).
When C∗ is exact at Cj , we have coker ∂j+1 = im ∂j . Thus the first statement
follows from (1). The second statement follows from Pontryagin duality and the
first statement.

The follow proposition is an immediate consequence of Theorem 1.2, which can
also be proved directly.

12

BINGBING LIANG

Proposition 4.1. Let C∗ be a chain complex of ZΓ-modules: 0 → Ck → · · · →
C0 → 0 for some k ∈ N. If vrk(Cj ) < ∞ for all j ≥ 1, then
X

X

(2)

(−1)j βj (C∗ ) = χ(C∗ ) =

0≤j≤k

(−1)j mrkj (C∗ ).

0≤j≤k

In particular, when C∗ is exact, we have
χ(C∗ ) =

X

(−1)j (mrkΣ (im ∂j ) − mrkΣ (im ∂j |Cj−1)).

0≤j≤k

The following lemma reduces the Juzvinskiı̆ formula for mean rank to the finitely
generated case.
Lemma 4.2. Suppose that mrkΣ (M1 |M2 ) = mrkΣ (M1 ) holds when M2 is a finitely
generated free ZΓ-module and M1 is a finitely generated ZΓ-submodule of M2 . Then
mrkΣ (M1 |M2 ) = mrkΣ (M1 ) holds for any ZΓ-modules M1 ⊆ M2 . In particular, Γ
satisfies Juzvinskiı̆ formula for mean rank if and only if mrkΣ (M1 |M2 ) = mrkΣ (M1 )
holds when M2 is a finitely generated free ZΓ-module and M1 is a finitely generated
ZΓ-submodule of M2 .
Proof. Case 1. Both M1 and M2 are finitely generated.
Write M2 as M2 = (ZΓ)n /N and M1 as M1 = M′1 /N for some n ∈ N and some
ZΓ-submodules N ⊆ M′1 ⊆ (ZΓ)n . Write N as the union of an increasing net of
finitely generated submodules {Nj }j∈J of N. Note that for each j, by Proposition
2.8, we have
mrkΣ (Nj ) = mrkΣ (Nj |Nj ) ≥ mrkΣ (Nj |M′1 ) ≥ mrkΣ (Nj |(ZΓ)n ) = mrkΣ (Nj ).
Thus mrkΣ (Nj |(ZΓ)n ) = mrkΣ (Nj |M′1 ) for all j. Moreover, by Proposition 2.8,
mrkΣ (N|(ZΓ)n ) = sup mrkΣ (Nj |(ZΓ)n ) = sup mrkΣ (Nj |M′1 ) = mrkΣ (N|M′1 ).
j∈J

j∈J

Then by Theorem 2.7 and Proposition 2.8, we have
mrkΣ (M1 |M2 )
= mrkΣ ((ZΓ)n /N) − mrkΣ ((ZΓ)n /M′1 )
= (n − mrkΣ (N|(ZΓ)n )) − (n − mrkΣ (M′1 |(ZΓ)n ))
= mrkΣ (M′1 ) − mrkΣ (N|M′1 )
= mrkΣ (M′1 /N) = mrkΣ (M1 ).
Case 2. M1 is finitely generated.
Write M2 as the union of an increasing net of finitely generated submodules
{M′j }j∈J of M2 . By Proposition 2.8 and the conclusion of Case 1, we have
mrkΣ (M1 |M2 ) = inf mrkΣ (M1 |M′j ) = inf mrkΣ (M1 ) = mrkΣ (M1 ).
j∈J

j∈J

DYNAMICAL CORRESPONDENCES OF L2 -BETTI NUMBERS

13

Now we consider the general case. Write M1 as the union of an increasing net
of finitely generated submodules {M′j }j∈J of M1 . Applying Proposition 2.8 and the
conclusion of Case 2, we have
mrkΣ (M1 |M2 ) = sup mrkΣ (M′j |M2 ) = sup mrkΣ (M′j |M1 ) = mrkΣ (M1 ).
j∈J

j∈J

Suppose that Γ satisfies Juzvinskiı̆ formula for mean rank. Let M2 be a finitely
generated free ZΓ-module and M1 be finitely generated ZΓ-submodule of M2 . Consider the quotient map M2 → M2 /M1 . Then the conclusion follows immediately by
Theorem 2.7. The converse direction also follows immediately by Theorem 2.7. 
Remark 4.3. Let L be a function as in [27, Lemma 7.7] satisfying all the properties
(i)-(v). Then the corresponding result in Lemma 4.2 still holds without change
of proof. In particular, the corresponding statement holds for von Neumann-Lück
rank.
Proof of Corollary 1.4. The second statement follows from Theorem 2.7 and the
first statement. Suppose Γ satisfies Lück’s dimension-flatness, in particular, for
(2)
any finitely presented ZΓ-module M, we have β1 (M) = 0. Write M as M =
(ZΓ)n /(ZΓ)m f for some f ∈ Mm,n (ZΓ). Then a projective resolution of M can be
∂

∂

2
1
· · · −→
(ZΓ)m −→
(ZΓ)n → M → 0,

where ∂1 = R(f ). Then by Theorem 1.2, we have
vrk(im ∂1 ) − vrk(im ∂1 |(ZΓ)n )
= vrk(coker ∂2 ) − vrk(im ∂1 |(ZΓ)n )
(2)

= β1 (M) = 0.
By Remark 4.3, we have Γ satisfies Juzvinskiı̆ formula for von Neumann-Lück rank.
(2)
For the “if” part, by Theorem 1.2, we first have β1 (M) = 0 for any finitely
presented ZΓ-module M. Since both dimLΓ (·) and torZΓ
j (LΓ, ·) commutes with the
(2)
colimits [32, Theorem 6.7] [35, Proposition 10.99], we have β1 (M) = 0 for any
ZΓ-module M.
Let 0 → N → F → M → 0 be an exact sequence of ZΓ-modules such that
F is free. Then a projective resolution of N induces a projective resolution of M.
(2)
(2)
(2)
So β2 (M) = β1 (N) = 0. Inductively we have βj (M) = 0 for all j ≥ 1 and
ZΓ-module M.

The following proposition was implicitly proven in [32, Conjecture 6.48]. For
convenience, we give a proof.
Proposition 4.4. Let Γ be a discrete group (not necessarily sofic) and H be a
(2)
(2)
subgroup of Γ. Then for any ZH-module M, we have βj (ZΓ ⊗ZH M) = βj (M).
In particular, taking subgroups respects the property of Lück’s dimension-flatness.

14

BINGBING LIANG

Proof. Let C∗ → M be a free resolution of M. Since ZΓ is a flat ZH-module, we get
a free resolution ZΓ ⊗ZH C∗ → ZΓ ⊗ZH M of the ZΓ-module ZΓ ⊗ZH M. Since the
induction functor LΓ ⊗LH · is flat [32, Theorem 6.29 (1)], we have
LΓ ⊗LH Hj (LH ⊗ZH C∗ ) ∼
= Hj (LΓ ⊗LH (LH ⊗ZH C∗ )) ∼
= Hj (LΓ ⊗ZΓ (ZΓ ⊗ZH C∗ )).
Thus by [32, Theorem 6.29 (2)], we have
(2)

βj (ZΓ ⊗ZH M)
= dimLΓ Hj (LΓ ⊗ZΓ (ZΓ ⊗ZH C∗ ))
= dimLΓ (LΓ ⊗LH Hj (LH ⊗ZH C∗ ))
(2)

= dimLH Hj (LH ⊗ZH C∗ ) = βj (M).

As a consequence of Corollary 1.4 and Proposition 4.4, we have:
Corollary 4.5. If a subgroup H of a sofic group Γ violoates Lück’s dimensionflatness, then Γ violoates Juzvinskiı̆ formula for mean dimension. In particular, if
(2)
βj (H) > 0 for some j ≥ 1, then Γ violoates Juzvinskiı̆ formula for mean rank and
mean dimension.
We refer the reader to [34, Section 5] for some discussions on L2 -Betti numbers of
subgroups. Lück’s dimension-flatness for amenable groups [32, Theorem 6.37] can
be interpreted in terms of relative sofic mean rank.
Corollary 4.6. Amenable groups satisfy Lück’s dimension-flatness.
Proof. By [27, Theorem 5.1], we know mrkΣ (M1 |M2 ) = mrkΣ (M1 ) holds for any
ZΓ-modules M1 ⊆ M2 . By Theorem 2.7, Γ satisfies Juzvinskiı̆ formula for mean
rank. Thus by Corollary 1.4, Γ satisfies Lück’s dimension-flatness.

Corollary 4.7. Let Γ be a residually finite group and {Γi }i be a sequence of finiteindexed decreasing normal subgroups of Γ with the intersection {eΓ }. Let C∗ be a
chain complex of finitely generated free ZΓ-modules. Then
mrkj (C∗ ) = lim

i→∞

rk(Hj (Γi \ C∗ ))
.
|Γ/Γi |

Proof. Since residually finite groups satisfy Lück’s approximation formula for L2 Betti numbers [31, Theorem 0.1], by the similar argument as in [32, Lemma 13.4],
we have
rk(Hj (Γi \ C∗ ))
(2)
βj (C∗ ) = lim
.
i→∞
|Γ/Γi |
Then the conclusion follows from Theorem 1.2.


DYNAMICAL CORRESPONDENCES OF L2 -BETTI NUMBERS

15

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E-mail address: bliang@mpim-bonn.mpg.de

