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Stability of Homomorphisms, Coverings and
Cocycles I: Equivalence
Michael Chapman
New York University
Alexander Lubotzky
Weizmann Institute of Science
Research Article
Keywords:
Posted Date: June 20th, 2024
DOI: https://doi.org/10.21203/rs.3.rs-4518267/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
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Additional Declarations: No competing interests reported.
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES I:
EQUIVALENCE
MICHAEL CHAPMAN AND ALEXANDER LUBOTZKY
Abstract. This paper is motivated by recent developments in group stability, high dimensional expan-
sion, local testability of error correcting codes and topological property testing. In Part I, we formulate
and motivate three stability problems:
•Homomorphism stability: Are almost homomorphisms close to homomorphisms?
•Covering stability: Are almost coverings of a cell complex close to genuine coverings of it?
•Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles?
We then prove that these three problems are equivalent.
In Part II of this paper [CL23], we present examples of stable (and unstable) complexes, discuss various
applications of our new perspective, and provide a plethora of open problems and further research direc-
tions. In another companion paper [CP23], we study the stability rates of random simplicial complexes
in the Linial–Meshulam model.
1. Introduction
Property testing, a deep and popular area of theoretical computer science, is the study of algorithmic local
to global phenomena. A property tester is a randomized algorithm which aims to infer global properties
of an object by applying a few random local checks on it (see Section 2). Property testing plays a major
role in some of the most celebrated results in theoretical computer science. These include the PCP theorem
[ALM+98,Din07,RS07], good locally testable codes [DEL+22b,DEL+22a,PK22], and the recent complexity
theoretic refutation of Connes’ embedding problem [JNV+21].
Classically, functional properties were studied, such as linearity [BLR93] and being a low-degree polynomial
[BFL91], as well as properties of graphs such as k-colorability [AK02]. In recent years, group theorists began
studying whether being a permutation solution to a system of group equations is a testable property (cf.
[GR09,AP15,BLM23,BLM22]). For example, given a pair of almost commuting permutations, are they close to
a commuting pair? By associating between systems of group equations and group presentations, this problem
can be framed as an instance of homomorphism testing. Namely, in the spirit of Ulam’s stability problem
[Ula60], this is the study of whether an approximate group homomorphism is close to a genuine one. This
framework generalizes the linearity test of [BLR93], as well as the local testability of error correcting codes (cf.
[DEL+22b]), as we discuss in Section 3.1.
Even more recently, Dinur–Meshulam [DM22] initiated the study of topological property testing. In their
paper, they study whether being a topological covering is a testable property. Namely, are near coverings of a
given complex close to actual coverings of it? Furthermore, they relate this problem to the first cohomology of
the underlying complex, and a new high dimensional expansion parameter of it.
1
2 M. CHAPMAN AND A. LUBOTZKY
In this paper, we give an alternative setup for testing coverings, and consequently a different cohomological
viewpoint from the one used in [DM22].1In our setup, the parameter acquired by the cohomological viewpoint
is equivalent to the covering testing parameter (as oppose to [DM22] in which they only bound one another).
Moreover, both the cohomological and covering parameters are related to the homomorphism testing parameter
of the fundamental group of the underlying complex. We intentionally chose to view these stability problems
through the property testing lens. One can present these problems in a purely algebraic manner, but we
believe property testing perspective is important for future applications, as well as to understand better the
mechanisms involved. At the end of this paper, we discuss a slight generalization of our framework, in which
our main Theorems 1.1,1.2 and 1.3 still hold.
The second part of this paper [CL23] provides several elementary examples of stable complexes, namely
complexes where the local perspective implies a global structure – almost coverings of it are close to genuine
coverings, and thus almost cocycles are close to cocycles and almost homomorphisms of its fundamental group
are close to actual homomorphisms of the group. In addition, large cosystols of a complex are related to property
(T) of its fundamental group. Furthermore, a family of bounded 2-dimensional coboundary expanders with F2
coefficients is provided, resolving a special case of a problem due to Gromov. Lastly, we discuss several potential
avenues of enquiry, including novel ways to tackle (but, not yet solve) the existance of non-sofic groups problem
(See Problem 3.6).
Let us now define our objects of study. As oppose to the body of the paper, here we give a concise presentation
of the notions of interest — homomorphism stability, covering stability and cocycle stability — in a purely
algebraic manner, with no additional motivation. For a more thorough and motivational presentation of these
notions, see Sections 3,4and 5. In the rest of this section, ρ:R≥0→R≥0is a rate function (Definition 2.1),
namely it is non-decreasing and satisfies ρ(ε)ε→0
−−−→ 0.
Homomrphism stability. Let Γ ∼
=⟨S|R⟩be a finite group presentation, i.e., Sand Rare finite sets. Let
Sym(n) be the symmetric group acting on [n] = {1, ..., n}. Every map f:S→Sym(n) uniquely extends to a
homomorphism from the free group f:F(S)→Sym(n). This homomorphism factors through Γ if and only if
for every r∈Rwe have f(r) = Id. Hence, our local measurement for being a homomorphism would be how
much is this condition violated. Namely, the homomorphism local defect of fis
defhom(f) = P
i∈[n]
r∈R
[f(r).i =i]≤ε,
where the probability is the uniform one.2Given permutations σ∈Sym(n) and τ∈Sym(N) where N≥n, the
normalized Hamming distance with errors between them is
dh(σ, τ) = 1 −|{i∈[n]|σ(i) = τ(i)}|
N.
This allows us to measure the distance between maps f:S→Sym(n) and g:S→Sym(N) as the expected
value of the distances between their evaluation points,3i.e.,
(1.1) dh(f, g) = E
s∈S[dh(f(s), g(s))].
1We describe the difference in Section 4.4.
2Other probability distributions over Rcan and should be discussed, though in this paper we focus on the uniform one. For
more on that, see Section 3.1, Section 7and the second part of this paper [CL23].
3Again, there are reasons to take other distributions over Sthat are not the uniform one. See Section 7and the second part of
this paper [CL23] for more on that.
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 3
The homomorphism global defect of fis its distance to the closest genuine homomorphism, namely
Defhom(f) = inf{dh(f, φ)|N≥n, φ:S→Sym(N),defhom(φ) = 0}.
The presentation ⟨S|R⟩is said to be ρ-homomorphism stable if Defhom(f)≤ρ(def hom (f)) for every map
f:S→Sym(n).
Covering stability. Recall that a polygonal complex is a 2-dimensional cell complex whose 2-cells are polygons.
The most popular example is that of 2-dimensional simplicial complexes, in which the pasted polygons are all
triangles. Another popular example is that of 2-dimensional cube complexes (cf. [Sag95]), in which all pasted
polygons are quadrilaterals. In our setup, polygons are not of any unique or bounded length.
A combinatorial map f:Y → X between two complexes is a function that maps i-cells of Yto i-cells of X
in an incident preserving manner. Hence, by defining the map on G(Y), the underlying graph of Y, there is at
most one way of extending it to all paths in G(Y), and thus to extend it to the polygons of Y. A combinatorial
map ffrom a graph Gto a complex Xis a genuine covering, if one can add polygons to Gand extend f
accordingly so that fbecomes a topological covering.
Assume fwas already a covering of the underlying graph of X. By the path lifting property of topological
coverings (See Proposition 1.30, page 60, in [Hat02]), for every closed path πin G(X) which originates at a
vertex x∈V(X), and for every vertex x′in the fiber f−1(x)⊆V(G), one can lift πto a path π′in Gwhich
originates at x′and satisfies f(π′) = π. Even if the original path πwas closed, the lifted path π′may be either
open or closed. Now, fis a genuine covering if and only if for every polygon πin Xand every x′∈f−1(x) the
lifted path π′is closed. Hence, we can define the covering local defect of fto be
defcover (f) = P
π∈−→
P(X)
x′∈f−1(x)The lift of πto x′is open,
where −→
P(X) is the set of oriented polygons in X.
There is a natural measure of distance between graphs, which is called the (normalized) edit distance. In
Section 4.1 we provide a rigorous definition (4.1), but in essence it measures how many edges need to be
changed/deleted/added to move from the smaller graph to the larger one. This metric can also be defined on the
category of X-labeled graphs, namely graphs Ywith a fixed combinatorial map into X. Thus, the covering global
defect Defcover (f) of f:Y → X would be its normalized edit distance to the closest genuine covering φ:Z → X
in the category of X-labeled graphs. The complex Xis ρ-covering stable if Defcover (f)≤ρ(defcover (f)) for
every such f.
Cocycle stability. Let Xbe a polygonal complex. An anti-symmetric map from −→
E(X), the oriented edges of
X, to Sym(n) is called a 1-cochain with permutation coefficients. In a similar manner to combinatorial maps,
these 1-cochains extend in a unique way to paths in the underlying graph G(X). A 1-cochain αis said to be a
1-cocycle, if for every (oriented) polygon πwe have α(π) = Id. Hence, we can define the cocycle local defect of
αto be
defcocyc(α) = P
i∈[n]
π∈−→
P(X)
[α(π).i =i].
We already defined in (1.1) a measure of distance between maps from a fixed set to (potentially varying)
permutation groups. Therefore, we have a measure of distance between 1-cochains. Then, the cocycle global
defect Defcocyc (α) is the distance of αto the closest 1-cocycle. As before, Xis ρ-cocycle stable if Defcover (α)≤
4 M. CHAPMAN AND A. LUBOTZKY
ρ(defcover (α)) for every 1-cochain α.
Our results. Given a polygonal complex Xtogether with a base point ∗ ∈ X, the fundamental group π1(X,∗)
is the collection of loops based at ∗up to homotopy equivalence. Furthermore, if ∗is chosen to be a vertex
of the underlying graph G(X), then by retracting any spanning tree of G(X) we acquire a presentation of
π1(X,∗) (See Definition 6.5). On the other hand, for any group presentation Γ = ⟨S|R⟩, one can construct its
presentation complex X⟨S|R⟩, whose fundamental group is isomorphic to Γ (See Definition 6.4). The main goal
(of this part) of the paper is to prove the following equivalences.
Theorem 1.1. Let Xbe a connected polygonal complex. Then, the following are equivalent:
• X is ρ-cocycle stable.
• X is ρ-covering stable.
Note that the rate ρis the same in both.
Theorem 1.2. Let Γ∼
=⟨S|R⟩be a group presentation with |S|,|R|<∞. Let X⟨S|R⟩be the presentation complex
associated with ⟨S|R⟩. Then, the following are equivalent:
• X⟨S|R⟩is ρ-cocycle stable.
• ⟨S|R⟩is ρ-homomorphism stable.
Again, ρis the exact same rate.
Theorem 1.3. Let Xbe a connected polygonal complex, and ∗a vertex of X. Let Tbe a spanning tree of
G(X), and let π1(X,∗)∼
=⟨S|R⟩be the presentation of the fundamental group of Xassociated with retracting T
to the basepoint ∗. Then
(1) If ⟨S|R⟩is ρ-homomorphism stable, then Xis ρ-cocycle stable.
(2) If Xis ρ-cocycle stable, then ⟨S|R⟩is |−→
E(X)| · ρ-homomorphism stable, where |−→
E(X)|is the number
of edges in X.
Though not as tight as Theorems 1.1 and 1.2, Theorem 1.3 still implies a qualitative equivalence, and in
many contexts – see for example the discussion in Section 3.2 – it is enough.
Our notion of homomorphism stability is usually called flexible pointwise stability in permutations in the
literature (cf. [BL20]), and it is a thoroughly studied topic. Hence, using clause (1) of Theorem 1.3, there
are off the shelf examples of complexes with known stability rates. For example, by [LLM19], triangulations
of compact surfaces of genus g≥2 are ρ-stable with rate ρ(ε) = Θ(−εlog ε). While, by [BM21], when g= 1
the surface is ρ-stable with ρbeing Ω(√ε) and O(ε1
4). In the second part of this paper [CL23], we provide
elementary examples of complexes with linear stability rate.
Remark 1.4.As oppose to Theorem 1.1, Theorems 1.2 and 1.3 can be framed in much greater generality than
presented here. Our proof method applies to any metric coefficient group — not only permutations equipped
with the Hamming metric. We thus relate flexible Hilbert–Schmidt stability [HS18,BL20], Frobenius norm
stability [DCGLT20,LO20] and so on to a stability criterion for cocycles. See Corollary 6.6.
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 5
Part I of the paper is organized as follows: In Section 2we define all property testing related notions. In
Section 3we define and motivate homomorphism stability of group presentations. In Section 4we define and
motivate covering stability of polygonal complexes. In Section 5we define and motivate cocycle stability of
polygonal complexes. The motivational parts of Sections 3,4and 5are more involved, and can be skipped by
the non-expert or first time reader. Section 6is devoted to the proofs of Theorems 1.1,1.2 and 1.3. In Section
7we provide a generalization of our framework, in which Theorems 1.1,1.2 and 1.3 still hold.
Acknowledgements. Michael Chapman acknowledges with gratitude the Simons Society of Fellows and is
supported by a grant from the Simons Foundation (N. 965535). Alex Lubotzky is supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 (N. 882751), and by a research grant from
the Center for New Scientists at the Weizmann Institute of Science.
2. Property testing
The goal of a property tester is to decide whether a given function satisfies a specific property by querying
it at a few random evaluation points. Such a tester cannot decide with certainty that the function satisfies the
property, since it may require reading all the values of the function. Hence, a property tester is good given that
it satisfies the following: As the rejection probability of the test gets smaller, the closer the function in hand is
to another function which has the property. We now make this paragraph precise. For a thorough introduction
to the theory, see [Gol17].
Let Dand Rbe collections of finite sets. Let Fbe the collection of functions whose domain is from Dand
range is from R, namely
F={f:D→R|D∈D, R ∈R}.
Let f1:D1→R1and f2:D2→R2be two functions from F. The Hamming distance with errors between them
is defined as follows:
(2.1) dH(f1, f2) = |D2\D1|+|D1\D2|+|{x∈D1∩D2|f1(x)=f2(x)}|.
When D1=D2, this notion agrees with the usual Hamming distance between functions. If for every f:D→R
and x /∈Dwe define f(x) = error, and we assume that error /∈R′for any R′∈R, then we can equivalently
define
dH(f1, f2) = |{x∈D1∪D2|f1(x)=f2(x)}|,
which is the usual way the Hamming distance is defined. Though the Hamming distance with errors is a good
metric, it outputs natural numbers and cannot tend to zero without being zero. Hence, we usually normalize it
in some way. The most common normalization is the following, which we call the normalized Hamming distance
with errors:
(2.2) dh(f1, f2) = dH(f1, f2)
|D1∪D2|=P
x∈D1∪D2
[f1(x)=f2(x)],
where Px∈Ais the uniform probability on A. As we further discuss in Section 7, and specifically in Claim 7.1,
in certain cases other probability distributions are more natural. But, we stick to the uniform distribution for
most of Part I of the paper.
Let Pbe a subset of F. We think of Pas the collection of functions in Fthat satisfy a certain property. The
P-global defect of a function f∈Fis
DefP(f) = inf{dh(f, φ)|φ∈P}.
6 M. CHAPMAN AND A. LUBOTZKY
Atester for Pis a randomized algorithm Tthat receives as input a black box version4of a function f:D→R
in Fand needs to decide whether f∈P. The way it operates is as follows:
•Sampling phase:Trandomly chooses an evaluation point x1∈Dand sends it to f. The function f
answers with f(x1). According to this value, Teither asks for another evaluation point x2∈D(which
may depend on f(x1)), or moves to the next phase. Tis allowed to repeat this process for at most q
rounds, where qis independent of f.5The maximum number of sampled points qis called the query
complexity of T.6
•Decision phase: According to the outputs f(x1), ..., f (xq), the tester Tdecides whether to accept
(namely, it decides that f∈P) or reject (namely, it decides that f /∈P).
The P-local defect of a function f∈Fis its rejection probability in T, namely
(2.3) defP(f) = P[Trejects f],
where the probability runs over all possible sampling phases.
Definition 2.1. A non-decreasing function ρ:R≥0→R≥0satisfying ρ(ε)ε→0
−−−→ 0 will be called a rate function.
Definition 2.2. The tester Tis complete if
∀f∈F: defP(f) = 0 ⇐⇒ f∈P.
Namely, functions that satisfy the property are always accepted, and all other functions are rejected with some
positive probability.
Let ρbe a rate function as in Definition 2.1. The tester Tis said to be ρ-stable 7, if
∀f∈F: DefP(f)≤ρ(defP(f)).
3. Homomorphism stability of group presentations
In 1940, Ulam raised the following problem:
Problem 3.1 (Ulam’s homomorphism stability problem [Ula60]).Given an almost homomorphism between
two groups, is it close to a genuine homomorphism between them?
There are many (non-equivalent) ways to formulate Ulam’s problem (cf. [Kaz82,BOT13,GLMR23,GH17,
BLT19,DCGLT20,BC22]). In this section we study a finite group presentations variant of it.8
For a positive integer n, let Sym(n) be the symmetric group acting on [n] = {1, ..., n}. Given permutations
σ∈Sym(n) and τ∈Sym(N) where N≥n, the normalized Hamming distance with errors between them (as
defined in (2.2)) is
(3.1) dh(σ, τ) = 1 −|{i∈[n]|σ(i) = τ(i)}|
N=P
i∈[N][σ(i)=τ(i)].
4The way Tinteracts with fis by sending to it an evaluation point x∈Dand receiving back the value f(x)∈R.
5In certain contexts, and for some applications, qmay be allowed to grow with f. Mainly, to make the stability rate ρ(See
Definition 2.2) better using parallel repetition (cf. [Raz98,Raz10]), or when fis encoding an exponentially larger function in a
succinct way (cf. [BFL91]). Actually, this flexibility is crucial for the applications of efficient stability mentioned in Section 6.4 of
[CL23].
6The term locality is also commonly used for q(cf. [DEL+22b]).
7This notion is also called ρ-robustness and ρ-soundness in the literature.
8Our notion of homomorphism stability is usually referred to in the literature as pointwise flexible stability in permutations (see
[BL20]).
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 7
Let Γ ∼
=⟨S|R⟩be a finite group presentation, and F(S) the free group with basis S. Let f:S→Sym(n)
be a function. By the universal property of free groups, fhas a unique extension to a homomorphism from
F(S) to Sym(n), which we denote by fas well. This ffactors through Γ if and only if for every r∈Rwe have
f(r) = Id.Hence, the collection of functions f:S→Sym(n) that induce a homomorphism from Γ is
Hom(Γ,Sym) = {f:S→Sym(n)|n∈N,∀r∈R:f(r) = Id}.
This suggests the following tester for deciding whether f:S→Sym(n) induces a homomorphism from Γ:
Algorithm 1 Homomorphism tester
Input: f:S→Sym(n).
Output: Accept or Reject.
1: Pick r∈Runifromly at random.
2: Pick i∈[n] uniformly at random.
3: If f(r).i =i, then return Accept.
4: Otherwise, return Reject.
Remark 3.2.Instead of sampling r∈Runiformly at random in Algorithm 1, one can use any fully supported
probability measure on R. This slight generalization is important to the discussion in Section 3.1, and is
described in Section 7. Other than these sections, we would not need this generalized version. Furthermore, it
is natural in some instances to measure the Hamming distance in a weighted way by choosing a non-unifrom
distribution on S(See Section 7and [CVY23,EK16]). But again, we will use only the uniform measure for the
Hamming distances in this part of the paper.
To fit Algorithm 1to the setup of Section 2, we need the input function fto be in the form f:S×[n]→[n]
with the extra condition that f(s, ·) : [n]→[n] is a permutation for every s∈S. Namely,
F={f:S×[n]→[n]|n∈N,∀s∈S:f(s, ·)∈Sym(n)},
P= Hom(Γ,Sym),
∀f1, f2∈F:dh(f1, f2) = E
s∈S[dh(f1(s, ·), f2(s, ·))].
Hence, the homomorphism local defect of f:S→Sym(n), which is the rejection probability in the above test,
is
defhom(f) = E
r∈R[dh(f(r),Id)].
Moreover, the homomorphism global defect of f:S→Sym(n), which is its distance to the collection Hom(Γ,Sym),
is
Defhom(f) = dh(f, Hom(Γ,Sym)) = inf{dh(f, φ)|φ∈Hom(Γ,Sym)}.
Definition 3.3. Let ρbe a rate function, as in Definition 2.1. A finite presentation Γ ∼
=⟨S|R⟩is said to be
ρ-homomorphism stable if for every f:S→Sym(n), we have
Defhom(f)≤ρ(defhom(f)).
The goal of the following Sections, 3.1 and 3.2, is to motivate the notion of ρ-homomorphism stability.
8 M. CHAPMAN AND A. LUBOTZKY
3.1. Locally testable codes. In this section, we relate homomorphism stability to the well studied topic of
locally testable codes.
The classical example of a homomorphism tester is the Blum–Luby–Rubinfeld (BLR) linearity test [BLR93].
In it, the input is a function f:Fn
2→F2, where F2={0,1}is the field with two elements, and the goal is
to decide whether fis linear. To that end, two points x, y ∈Fn
2are sampled uniformly at random, and fis
evaluated at three points x, y and x+y. The tester accepts if and only if f(x) + f(y) = f(x+y). This test is
ρ-stable for the identity rate function ρ(ε) = ε(See Section 1.6 in [O’D21] for a proof).
The BLR linearity test is a special case of Algorithm 1. To see that, note that F2∼
=Sym(2). Moreover,
Fn
2∼
=⟨{sx}x∈Fn
2| {sx·sy·s−1
x+y}x,y∈Fn
2⟩, which is the multiplication table presentation of Fn
2. So, the linearity
test is Algorithm 1under the restriction that the range of fis Sym(2) and not any Sym(N). In [BC22], it
is proved that the multiplication table presentation of any finite group is ρ-homomorphism stable with linear
rate ρ(ε) = 3000ε. This generalizes [BLR93], though with worse parameters.
Alinear binary error correcting code – or just code from now on – is a linear subspace of a finite vector space
over F2. E.g., the Hadamard code is the subspace of linear functions from Fn
2to F2, out of all functions from Fn
2
to F2. The linearity test is a complete and ρ-stable tester, in the sense of Definition 2.2, for checking whether a
given function is a Hadamard code word. Moreover, its query complexity (which was defined in the Sampling
phase of the tester in Section 2) is 3. Codes that have a tester which is complete, ρ-stable and with constant
query complexity, are called locally testable.
Every matrix A∈Mm×n(F2) defines the code Ker(A)⊆Fn
2.9Together with a probability distribution µover
[m], it also defines a tester:
Algorithm 2 Matrix tester
Input: A column vector v∈Fn
2.
Output: Accept or Reject.
1: Sample i∈[m] according to µ.
2: Let Ri(A) be the ith row of A. Then, if Ri(A)·v= 0, return Accept.
3: Otherwise, return Reject.
Algorithm 2is complete given that µis fully supported. If each row of Acontains at most qnon-zero entries,
for some universal constant q, then this tester has bounded query complexity.10 So, the main obstacle is to
choose Asuch that the tester is ρ-stable for some desired rate function ρ.11 Algorithm 2is clearly a generalization
of the BLR linearity test. Furthermore, every tester for a linear locally testable code can be assumed to be in
this form [BSHR03]. But, the matrix tester is a special case of Algorithm 1:12 The code Ker(A) can be viewed
as the collection of homomorphism inducing maps from S={xj}j∈[n]to Sym(2) ∼
=F2. The relations in this
case are R=nQxAij
joi∈[m], where the product is always ordered by the index. This demonstrates how general
Algorithm 1is: It encapsulates local testability of error correcting codes as a special case. This viewpoint on
Algorithm 1raises fascinating new problems which we further discuss in the open problem section of the second
part of this paper [CL23] (see also [CVY23]).
9Such matrices are usually called parity-check matrices.
10Codes with a parity matrix whose rows are of bounded weight are known in the literature as LDPC codes (cf. [Gal63,SS96]).
11Every code would be ρ-stable for some ρ. The goal is to find codes with a sufficiently good rate of local testabillity. See, e.g.,
our choice of parameters in the efficient stability section in Part II of this paper [CL23].
12As mentioned in Remark 3.2, in Algorithm 1we chose a relation uniformly at random. To mimic Algorithm 2, we need to
generalize the homomorphism tester so it may sample a relation according to some distribution other than the uniform one.
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 9
3.2. Group soficity. In this section, we relate homomorphism stability to the well studied notion of sofic
groups. As oppose to the general philosophy of this paper, in this section the exact rate ρis not important.
Namely, homomorphism stability is seen as a qualitative and not quantitative property. This viewpoint is of
interest when the finitely presented group Γ ∼
=⟨S|R⟩is infinite.
Fact 3.4 ([AP15]).Let ⟨S|R⟩,⟨S′|R′⟩be two finite presentations of the same group Γ. Assume ⟨S|R⟩is ρ-
homomorphism stable for some rate function ρ. Then ⟨S′|R′⟩is (C·ρ)-homomorphism stable for some constant
C∈R>0that depends only on the two presentations.
Hence, we say that a group Γ is homomorphism stable if it has some finite presentation Γ ∼
=⟨S|R⟩which is
ρ-homomorphism stable for some rate function ρ. Namely, by disregarding the exact rate ρ, homomorphism
stability may be viewed as a group property and not a presentation specific property.
Definition 3.5. A finitely presented group Γ ∼
=⟨S|R⟩is said to be sofic if there is a sequence of functions
fn:S→Sym(n) such that
defhom(fn)n→∞
−−−→ 0,
and for every w /∈ ⟨⟨R⟩⟩,13
lim inf(dh(fn(w),Id)) ≥1
2.
Problem 3.6 (Gromov [Gro99], Weiss [Wei00]).Are there non-sofic groups?
The following is an easy to prove, yet quite insightful, observation.
Proposition 3.7 (Glebsky-Rivera [GR09]).If Γis homomorphism stable and sofic, then it is residually finite.
Soficity is a group approximation property. There are other well studied group approximation properties, such
as hyperlinearity, property MF and Lp-approximations (cf. [CL13,DCGLT20]). The only known examples of
non-approximable groups were constructed using the (analogous) observation of Proposition 3.7 (See [DCGLT20,
LO20]). Though more sophisticated methods for constructing non-sofic groups were recently suggested (cf.
[BB19,Dog23]), they still require the proof of homomorphism stability of certain groups.14 We further discuss
group soficity in Part II of this paper [CL23], and even suggest a new paradigm for potentially resolving Problem
3.6 in the positive.
4. Covering stability of polygonal complexes
Let us move now to another stability problem.
4.1. Graphs a la Bass–Serre.
Definition 4.1. A graph (a la Bass–Serre, see [Ser80,Kol21]) Xconsists of the following data:
(1) A set V=V(X) of vertices.
13⟨⟨R⟩⟩ is the normal subgroup generated by R.
14There are also suggested methods for constructing non-sofic groups that do not use homomorphism stability, at least not in
such a direct manner. E.g., transforming the answer reduction part of the compression theorem in [JNV+21] into a linear constraint
system game (cf. [Slo19]) would imply the existence of a non-sofic group. Problem 3.6 is still a major open problem, with many
suggested tackling angles. We chose to emphasize its relation to homomorphism stability.
10 M. CHAPMAN AND A. LUBOTZKY
(2) A set −→
E=−→
E(X) of directed edges, together with functions τ, ι :−→
E→Vand an involution :−→
E→−→
E
satisfying
∀e∈−→
E:τ(e) = ι(e).
The function τis the terminal point (or end point) function, the function ιis the initial point (or origin
point) function and is the reverse edge function. Namely, an edge ecan be visualized as xe
−→ y,
where x=ι(e) and y=τ(e). Moreover, we have y¯e
−→ x.
Remark 4.2.Note the following:
•A graph (a la Bass-Serre) is essentially an undirected multi-graph. To make this statement precise,
define [e] = {e, ¯e}and E(X) = {[e]|e∈−→
E(X)}. Then (V(X), E(X)) is a multi-graph where the
endopints of [e]∈E(X) are ι(e) and τ(e).
Furthermore, multi-graphs can be viewed as 1-dimensional CW complexes (cf. [Hat02]) by defining
V(X) to be the 0-dimensional cells, E(X) to be the 1-dimensional cells, and with gluing maps induced
by τand ι. Hence, we can view graphs a la Bass-Serre as topological spaces.
•We may abuse notation and write xy for the edge xe
−→ y, though there may be several edges with the
exact same origin and endpoint.
Definition 4.3. Apath πin a graph Xis a sequence of edges e1...eℓsuch that each ei∈−→
Eand for every
1≤i≤ℓ−1, we have τ(ei) = ι(ei+1 ). By denoting xi:= τ(ei)∈Vand x0:= ι(e1)∈V, we can graphically
represent the path πby
x0e1
−→ x1e2
−→ ... eℓ
−→ xℓ.
The integer ℓ(π) = ℓis the length of the path. The path πis non-backtracking (or reduced) if ei+1 = ¯eifor
every 1 ≤i≤ℓ−1. For a path π, its inverse or reverse orientation is
¯π=xℓ
¯eℓ
−→ ... ¯e1
−→ x0.
The path πis closed if x0=xℓ, and open otherwise. When πis closed, it has ℓshifts ek...eℓe1...ek−1and ℓ
inverses ¯ek−1...¯e1¯eℓ...¯ekwhich are also closed paths in X. We call all these shifts and inverses the orientations
of π, and denote the collection of all of them by [π]. The path πis cyclically reduced if all its orientations are
non-backtracking.
Definition 4.4 (Combinatorial maps between graphs).Acombinatorial map f:Y → X between two graphs
is a function that maps vertices of Yto vertices of Xand edges of Yto edges of X, and is compatible with the
structural data of the graph. Namely, it preserves initial points, terminal points and edge flips:
∀e∈−→
E(Y) : τX(f(e)) = f(τY(e)), ιX(f(e)) = f(ιY(e)), f(e) = f(e).
Remark 4.5.Given a combinatorial map f:Y → X, we can extend it to paths in Y. Namely, if π=e1...eℓis
a path in Y, then f(π) = f(e1)...f(eℓ) is a path in X. Moreover, combinatorial maps induce a continuous map
between the graphs as topological spaces.
Recall that a continuous function f:Y → X between two topological spaces is a (topological) covering if for
every point x∈ X there is an open neighborhood Uof xsuch that f−1(U) is a disjoint union of open sets,
each of which homeomorphic to Uvia f. If Xis (path) connected, then the degree of the covering, which is
the cardinality of |f−1(x)|for any of the points x∈ X, is well defined (cf. Chapter 1.3 in [Hat02]). When
|f−1(x)|=n < ∞, we say that fis a degree ncovering, or just n-covering.
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 11
Definition 4.6. A combinatorial map f:Y → X between graphs is a (combinatorial) covering if it is a
topological covering.
Remark 4.7.Graph coverings15 is an intensely studied object. They were used both as a random model for
graphs [AL02,ALMR01,MSS13,Pud15] and as a building block in a recent construction of good locally testable
codes and good quantum codes [PK22].
Fact 4.8. Let Xand Ybe finite graphs. Let f:Y → X be a combinatorial map between them. Then, fis a
covering if and only if for every vertex y∈V(Y), the star of yis mapped bijectively to the star of f(y). Namely,
if Ey(Y) = {e∈−→
E(Y)|τ(e) = y}, then f|Ey(Y)is a bijection onto Ef(y)(X).
A combinatorial map between graphs f:A → X is said to be an embedding if it is a set theoretic injection.
Asubgraph of Xis the image of some embedding into it. The edit distance between a graph Xand a subgraph
of it Ais
dEdit(X,A) = |E(X)| − |E(A)|,
namely, this is the number of edges needed to be deleted to move from Xto A. Given two graphs Xand Z,
not necessarily embedded in one another, we can define the edit distance between them using the largest graph
Awhich embeds into both Xand Z.16 The edit distance between Xand Zwould be the maximum between
their distances to A. Namely,
(4.1) dEdit(X,Z) = max(dEdit (X,A), dEdit (Z,A)) = max(|E(X)|,|E(Z)|)− |E(A)|.
The normalized version of the edit distance will be
(4.2) dedit(X,Z) = 1 −|E(A)|
max(|E(X)|,|E(Z)|).
Remark 4.9.There is an encoding of graphs for which the edit distance (4.1) agrees with the Hamming distance
(2.1) (and similarly for the normalized versions): The vertex set V(X) will always be {1, ..., |V(X)|}. The edge
set E(X) will always be {1, ..., |E(X)|}. Then, the function ι×τ:E(X)→V(X)×V(X), which outputs for each
edge its origin and terminus, encodes the graph (note that we essentially chose an orientation for every edge
[e] = {e, ¯e}). There are many encodings of the same graph (one can permute V(X) and E(X) arbitrarily, as
well as choose different orientations of the edges). Hence, the edit distance between two graphs is the Hamming
distance between the two most compatible encodings of them.
One may argue that we did not choose the most natural encoding of graphs. A multi-graph can be encoded
by its adjacency matrix (see (5.3)), and the resulting edit distance will be slightly different — it would be the
sum of dEdit(X,A) and dEdit (Z,A) instead of their maximum as in (4.1). But, up to a factor of 2, they are the
same.
Given a graph X, an X-labeled graph is a graph Ywith a combinatorial map ΦY:Y → X which we call the
labeling. A morphism between X-labeled graphs is a combinatorial map f:Y → Z that commutes with the
labelings, namely ΦY= ΦZ◦f. Embeddings and subgraphs of X-labeled graphs can be defined in an analogous
way to before, and hence the edit distance is defined on this category.17
15Graph coverings are sometimes called graph liftings in the literature. This notion is a bit confusing because of the path lifting
and homotopy lifting lemmas (which are used for example in the proof sketch of Fact 4.13). Hence, we stick with coverings for these
objects.
16The largest in this case is according to the number of edges.
17Note that by choosing a weighting system won the edges of X, we get a weighted edit distance on X-labeled graphs by paying
w(Φ(e)) when changing e. This is shortly discussed in Section 7.
12 M. CHAPMAN AND A. LUBOTZKY
π1
e0
e1
e2
e3
e4
π2
e′
1
e′
2
e′
3
e′
4
e′
5
s1
s2
s3
Figure 4.1. Examples of polygons. π1=e1e2e3e4is a length 4 polygon. We can also construct
length 5 polygons of the form π=e1e2e0e3e4or π′=e1e2¯e0e3e4.π2=e′
1e′
2e′
3e′
4e′
5is a length 5
polygon. Lastly, every word of length ℓin the free group on {s1, s2, s3}defines a length ℓpolygon
on the bouquet of circles.
4.2. Polygonal complexes. As mentioned above, graphs are 1-dimensional CW complexes. We now wish
to add 2-dimensional cells to the mixture in a combinatorial way that will allow us to study them through a
property testing lens.
Definition 4.10. Apolygonal complex Xis a graph with extra data: A set −→
P=−→
P(X) of cyclically reduced
closed paths in X. The elements of −→
Pare called (oriented) polygons. Similar to the way we always include
both eand ¯eas edges in −→
E(X), we assume that all the orientations of a polygon must be in −→
P. Namely, if
π=e1...eℓ∈−→
P, then
∀1≤k≤ℓ:ek...eℓe1...ek−1∈−→
Pand ¯ek−1...¯e1¯eℓ...¯ek∈−→
P .
We denote the set of un-oriented polygons by P(X) = {[π]|π∈−→
P(X)}.
For visual examples of polygons see Figure 4.1. As oppose to edges, where there may be multiple edges with
the same origin and end point, we assume that there are no two polygons with the same pasted path. Namely,
a polygon is uniquely defined by its closed path boundary.
Remark 4.11.Note the following:
•Polygonal complexes are graphs with added data. To be able to distinguish between the graph structure
and the whole complex X, we denote by G(X) the underlying graph of X.
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 13
•Polygonal complexes are 2-dimensional CW complexes (hence, topological spaces): Their 1-skeleton
is induced by the underlying graph G(X). For the 2-dimensional cells, we paste a disc for every un-
oriented polygon [π], with the gluing map being according to some representative π. We therefore use
the standard notation for i-dimensional cells of a CW complex when suitable:
(4.3) X(0) = V(X),X(1) = E(X),X(2) = P(X),
and
(4.4) −→
X(1) = −→
E(X),−→
X(2) = −→
P(X)
for the oriented versions.
Definition 4.12. Let X,Ybe two polygonal complexs. A combinatorial map f:G(Y)→G(X) is called a
combinatorial map between polygonal complexes, if for every π∈−→
P(Y), we have f(π)∈−→
P(X).
The notions of embeddings, subcomplexes and coverings generalize to the polygonal complex setup in the
natural way.
4.3. Almost covers. Let Gbe a graph and Xafinite polygonal complex. Let f:G → G(X) be a finite
covering. Let π=xe1
−→ x1e2
−→ ... eℓ−1
−−−→ xℓ−1
eℓ
−→ xbe a polygon in X. By the path lifting property of covering
maps (See Proposition 1.30, page 60, in [Hat02]), for every x′∈f−1(x) there exists a unique path π′in Gthat
begins at x′and satisfies f(π′) = π. The lifted path π′may be closed or open.
Fact 4.13. Let f:G → G(X)be a finite covering. Then, all the lifts of all the polygons in −→
P(X)are closed if
and only if polygons can be added to Gsuch that fbecomes a covering map of X(namely, Gis the 1-skeleton
of a covering of Xvia f).
Proof idea. This is essentially the correspondence between (conjugation orbits of) subgroups of the fundamental
group π1(X,∗) and (connected) coverings of X(See Theorem 1.38, page 67, in [Hat02]). □
We call a function f:G → G(X) that can be completed to a covering of X(as in Fact 4.13) a genuine covering
of X. This suggests the following tester for whether a graph covering f:G → G(X) is a genuine covering of
X:18
Algorithm 3 Covering tester
Input: f:G → G(X),a finite covering of the 1-skeleton of X.
Output: Accept or Reject.
1: Pick [π]∈P(X) uniformly at random. Then, pick a representative πof [π] uniformly at random.
2: Let xbe the starting vertex of π. Pick x′∈f−1(x) uniformly at random.
3: If the lift π′of πto x′is a closed path in G, return Accept.
4: Otherwise, return Reject.
Algorithm 3may be fitted to the setup of Section 2using an extension of the encoding system suggested in
Remark 4.9. In this case:
18In Algorithm 3we sample a polygon uniformly at random. One could have sampled a polygon using a different distribution.
More on that in Section 7.
14 M. CHAPMAN AND A. LUBOTZKY
•Fis the collection of all graph coverings f:G → G(X);
•Pis the collection of all genuine coverings of X;
•Given f1:G1→G(X) and f2:G2→G(X), the normalized Hamming distance (2.2) between them is the
normalized edit distance (4.2) between them as G(X)-labeled graphs. Hence, the covering global defect
of fis the normalized distance of Gto the closest genuine covering of Xin the category of G(X)-labeled
graphs. Namely,
Defcover (f) = infndedit(G, G(Y))φ:Ycovering
−−−−−→ Xo.
•As before, the covering local defect of fis the rejection probability in the covering tester (Algorithm 3),
namely
defcover (f) = P
[π]∈P(X)
x′∈f−1(x)The lift of πto x′is open.
Definition 4.14. Let ρbe a rate function, as in Definition 2.1. A polygonal complex is ρ-covering stable if for
every graph Gand covering map f:G → G(X), we have
Defcover (f)≤ρ(defcover (f)).
In the next section, we discuss our motivation to study ρ-covering stability, as well as the difference between
our covering tester (Algorithm 3) and the Dinur–Meshulam one (Algorithm 4).
4.4. The Dinur–Meshulam framework. In [JNV+22], Ji–Natarajan–Vidick–Wright–Yuen proved a homo-
morphism stability result, in unitaries, that has a very good trade-off between presentation length and stability
rate of the studied groups (see [CVY23]). In an ongoing project [BCLV24], we try to prove a permutation
analogue of their result. Covering stability, as in Definition 4.14, arises naturally in our proof strategy. In
[DM22], Dinur–Meshulam studied a similar problem. The main difference between the Dinur–Meshulam setup
and ours is that they think of the degree parameter nas a constant. Namely, in their alternative to Algorithm
3, instead of sampling a polygon [π=x→... →x]and a starting position from x′∈f−1(x), they only sample
a polygon. Then, they check that its lift to every vertex in the fiber f−1(x) is closed.
Algorithm 4 Dinur-Meshulam covering tester
Input: f:G → G(X),a finite covering of the 1-skeleton of X.
Output: Accept or Reject.
1: Pick [π]∈P(X) uniformly at random.
2: Let xbe the starting vertex of some uniformly chosen representative πof [π].
3: If for every x′∈f−1(x) the lift π′of πto x′is a closed path in G, return Accept.
4: Otherwise, return Reject.
In Section 6.1, we give a one to one correspondence between 1-cochains of a polygonal complex Xand
coverings of its 1-skeleton. To encapsulate the Dinur–Meshulam Algorithm, one can use this equivalence and
the notations of Section 5, but with a different metric on Sym(n). They use the discrete distance
∀σ, τ ∈Sym(n) : dD(σ, τ) = (1σ=τ,
0σ=τ,
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 15
while we are using the normalized Hamming distance on Sym(n). This change may seem minor, but it allows
one to prove beautiful results in their setup (See Theorem 1.2 in [DD23]) that seem very difficult to prove in
our setting. The motivation to insist on our setting is coming from the fact that it can lead to a solution to
Problem 3.6. For more details, see the open problem section of Part II of this paper [CL23].
There is another strengthening of Algorithm 3which is perpendicular to the Dinur–Meshulam one. Instead
of sampling a single polygon with a single starting position (as done in Algorithm 3), one can sample for every
polygon [π=x→... →x]∈P(X) a uniformly random starting position x′
π∈f−1(x). Then, the tester checks
for each polygon and sampled starting position whether the lift of πto x′
πis closed or not. Namely:
Algorithm 5 L∞covering tester
Input: f:G → G(X),a finite covering of the 1-skeleton of X.
Output: Accept or Reject.
1: For every [π]∈P(X) let xπbe the starting vertex of a uniformly chosen representative πof [π].
2: Sample for every [π]∈P(X) a lift x′
π∈f−1(xπ).
3: If for every πand sampled x′
π, the lift π′of πto x′
πis closed in G, return Accept.
4: Otherwise, return Reject.
When thinking of the complex Xas a constant, this tester is essentially equivalent to Algorithm 3. To
encapsulate the difference between Algorithm 5and Algorithm 3, one can replace the expectations over polygons
in our analysis with a maximum over polygons. The same change can be done to Algorithms 1and 6, and we
call these the L∞-analogues of the testers.
5. Cocycle stability of polygonal complexes
In this section, we define the last stability problem of interest (for this paper).
5.1. Non-commutative cohomology. Let Xbe a polygonal complex. Recall our notation −→
X(i) for the
oriented i-cells of Xand X(i) for the unoriented versions, which were defined in (4.3) and (4.4). Moreover,
for i= 1 or 2, recall that ¯cis the reverse orientation of the cell c∈−→
X(i). Let Γ be a non trivial group and
d: Γ ×Γ→R≥0a bi-invariant metric on Γ.
The i-cochains of Xwith Γcoefficients are the anti-symmetric assignments of elements of Γ to oriented i-cells.
Namely,
Ci(X,Γ) = {α:−→
X(i)→Γ| ∀c∈−→
X(i): α(¯c) = α(c)−1}.
Since we did not define orientations on vertices, there are no anti-symmetricity conditions on 0-cochains. Also,
for every 2-cochain α:−→
P(X)→Γ and an orientation πof a polygon [π], we assume that α(π) is conjugate to
α(π′), where π′is a shift of π.
The coboundary maps are defined as follows. For a 0-cochain α:V(X)→Γ, its coboundary δα is the
1-cochain
∀xe
−→ y∈−→
E(X) : δα(e) = α(x)−1α(y).
For a 1-cochain α:−→
E(X)→Γ, its coboundary δα is the 2-cochain
∀π=e1...eℓ∈−→
P(X) : δα(π) = α(e1)...α(eℓ).
Remark 5.1.Every map from −→
E(X) to a group Γ can be extended to paths in G(X) in the natural way: For
π=x0e1
−→ ... eℓ
−→ xℓ, we let α(π) = α(e1)...α(eℓ). Note that δα(π) is exactly α(π) in the extended sense, but
restricted to the polygons of X.
16 M. CHAPMAN AND A. LUBOTZKY
Next, we prove exactness, namely, that for α∈C0(X,Γ), δ2αis the constant identity function. For a polygon
π=x0e1
−→ x1e2
−→ ... eℓ
−→ xℓ=x0, we have
δ2α(π) = δα(e1)...δα(eℓ) = α(x0)−1α(x1)α(x1)−1α(x2)...α(xℓ−1)−1α(xℓ) = α(x0)−1α(xℓ) = Id.
For any two i-cochains with Γ coefficients αand β, the distance between them is
(5.1) d(α, β) = E
[x]∈X (i)E
x∈[x][d(α(x), β(x))].
The norm of an i-cochain with Γ coefficients αis its distance to the constant identity cochain, namely
(5.2) ∥α∥=E
[x]∈X (i)E
x∈[x][d(α(x),Id)].19
An i-cocycle is an i-cochain αfor which δα is the constant identity function. We denote the collection of
i-cocycles by Zi(X,Γ). A 0-cochain β:V(X)→Γ is said to be a 0-coboundary if it is constant. Namely, for
every x, y ∈V(X) we have β(x) = β(y). A 1-cochain α:−→
E(X)→Γ is said to be a 1-coboundary if it is in the
image of the coboundary operator δ:C0(X,Γ) →C1(X,Γ). Namely, there exists a 0-cochain β:V(X)→Γ
such that for every xy ∈−→
E(X), α(xy) = δβ(xy) = β(x)−1β(y). We denote by Bi(X,Γ) the collection of
i-coboundaries of X. Note that, without assuming further assumptions on Γ, the only indices for which we
define Zi(X,Γ) and Bi(X,Γ) are i= 0 or 1. We say that the ith cohomology of Xwith Γcoefficients vanishes
if every i-cocycle of Xis an i-coboundary.
5.2. Cohomology with permutation coefficients. Though all the cohomological definitions we listed were
defined for a general group Γ, in this paper we focus on the case where Γ is a finite permutation group equipped
with the normalized Hamming distance (3.1).20 Moreover, since the normalized Hamming distance (with errors)
can compare permutations of different sizes, we will study them in a collective manner. Let
Ci(X,Sym) =
∞
G
n=2
Ci(X,Sym(n)) , Zi(X,Sym) =
∞
G
n=2
Zi(X,Sym(n)) ,
and
Bi(X,Sym) =
∞
G
n=2
Bi(X,Sym(n))
be the i-cochains with permutation coefficients, the i-cocycles with permutation coefficients and the i-coboundaries
with permutation coefficients, respectively. Thus, dhdefines a metric on Ci(X,Sym) as in (5.1), and thus also
a norm on it as in equation (5.2).
As the title of the section suggests, we now devise a tester for deciding whether a given i-cochain with
permutaition coeffcients is an i-cocycle. Note that, by definition, αis a cocycle if and only if ∥δα∥= 0. How
robust is this property? Namely, if δα has a small norm, does it imply that αis close to a cocycle? The following
tester assumes exactly that:
19It is common to associate weights to the i-cells of the complex, and then to sample them in (5.1) and (5.2) according to their
weight instead of uniformly at random. This is often used in the contemporary study of high dimensional expanders (cf. [EK16]),
and is related to Remark 3.2. We present this in more detail in Section 7.
20As mentioned in Section 4.4, to get the Dinur–Meshulam framework you replace the Hamming metric with the discrete metric.
STABILITY OF HOMOMORPHISMS, COVERINGS AND COCYCLES 17
Algorithm 6 i-cocycle tester, for i= 0 or 1
Input: α:−→
X(i)→Sym(n) an i-cochain.
Output: Accept or Reject.
1: Pick an un-oriented i-cell [π]∈ X(i+1) uniformly at random. Then, pick a representative πof [π] uniformly
at random.
2: Pick an index i∈ {1, ..., n}uniformly at random.
3: If δα(π).i =i, then return Accept.
4: Otherwise, return Reject.
Algorithm 6can be framed in the setup of Section 2in a similar way to Algorithm 1. Namely, the input
function is of the form α:X(i)×[n]→[n] with the restriction that for every i-cell e,α(e, ·): [n]→[n] is a
permutation. Hence:
•Fis the i-cochains Ci(X,Sym);
•Pis the i-cocycles Zi(X,Sym);
•The normalized Hamming distance between two i-cochains α, β ∈Ci(X,Sym) is
dh(α, β) = E
[e]∈X (i)[dh(α(e), β(e))].
Hence, the cocycle global defect of αis
Defcocyc(α) = dh(α, Zi(X,Sym))
= inf{dh(α, φ)|N∈N, φ:X(i)→Sym(N),∥δφ∥= 0}.
•The rejection probability of αin the