A Constructive Proof that Many Groups with Non-Torsion 2-Cohomology are Not Matricially Stable

A discrete group is matricially stable if every function from the group to a complex unitary group that is"almost multiplicative"in the point-operator norm topology is"close"to a genuine unitary representation. It follows from a recent result due to Dadarlat that all amenable, groups with non-torsion integral 2-cohomology are not matricially stable, but the proof does not lead to explicit examples of asymptotic representations that are not perturbable to genuine representations. The purpose of this paper is to give an explicit formula, in terms of cohomological data, for asymptotic representations that are not perturbable to genuine representations for a class of groups that contains all finitely generated groups with a non-torsion 2-cohomology class that corresponds to a central extension where the middle group is residually finite. This class includes polycyclic groups with non-torsion 2-cohomology.


Introduction
An asymptotic representation of a discrete group Γ is a sequence of functions ρ n : Γ → U (k n ) so that for all g, h ∈ Γ we have ||ρ n (gh)−ρ n (g)ρ n (h)|| → 0 as n goes to infinity, where || • || is operator norm.We say an asymptotic representation is perturbable to a genuine representation if there is a sequence of representations ρn : Γ → U (k n ) so that for all g ∈ Γ we have ||ρ n (g) − ρn (g)|| → 0 as n goes to infinity.Recall that a countable discrete group, Γ, is matricially stable if every asymptotic representation of Γ is perturbable to a genuine representation of Γ [5].
In [22] Voiculescu shows that Z 2 is not matricially stable by constructing an explicit sequence of pairs of unitaries that commute asymptotically in operator norm, but remain far from pairs of unitaries that commute.In [15] Kazhdan independently uses the same sequence of pairs of unitaries to show that a particular surface group is not Ulam stable, where Ulam stability is defined similarly to matricial stability, but the poitwise convergence is replaced by uniform convergence.Kazhdan also connects his argument to the 2-cohomology of the group.In [5] Eilers, Shulman, and Sørensen, give explicit asymptotic representations that are not perturbable to genuine representations for non-cyclic torsion-free finitely generated 2-step nilpotent, groups and several other groups.
In [3] Dadarlat shows that a large class of countable discrete groups with nonvanishing even rational cohomology are not matricially stable, including amenable, and hence polycyclic, groups with nonvanishing rational even cohomology.In [4] he connects this obstruction on the level of 2-cohomology to the "winding number argument" used by Kazhdan.However, the proof in [3] uses Voiculescu's theorem so it cannot lead to an explicit construction of an asymptotic representation that is not perturbable to a genuine representation.In this paper we will give an alternate proof that a group with a 2-homology class that pairs nontrivially with a 2-cohomology class, x satisfying the an additional condition is not matricially stable, and give a formula, in terms of cohomological data.The following result is what we aim to prove: Theorem 1.1.Suppose that Γ is a countable discrete group and x ∈ H 2 (Γ; Z) is a cohomology class represented as a central extension: ι If x is not in the kernel of the map h : H 2 (Γ; Z) → Hom(H 2 (Γ; Z), Z) induced by the Kronecker pairing and there is a sequence of finite index subgroups Γ n ≤ Γ so that ι(Z) ∩ Γ n = {e} then Γ is not matricially stable.The sequence of functions ρ n we will define in Proposition 3.17 is an asymptotic representation of Γ that cannot be perturbed to a genuine representation.In fact the asymptotic representation may not be perturbed to any representation let alone a unitary one.
In particular if Γ is residually finite and x is not in the kernel of the map h : H 2 (Γ; Z) → Hom(H 2 (Γ; Z), Z) then we can create an explicit formula for an asymptotic representation that cannot be perturbed to a genuine representation in terms of finite quotients of Γ and cocycle representatives of x.If Γ is finitely generated the condition that x is not in the kernel of h is equivalent to the condition that x non-torsion.In particular it follows that any virtually polycyclic group with non-torsion 2-cohomology is not matricially stable and an explicit formula for the relevant asymptotic representation can be found in terms of cohomological data.
Our construction is similar to another construction of projective representations of subgroups of Z 2 ⋊ SL 2 (Z) that come from factoring cocycles through finite quotients; see the proof of Corollary B in [13].
Three virtues of our proof compared to Dadarlat's broader result are as follows.First, our proof leads to a formula for asymptotic representations that cannot be perturbed to genuine representations (Proposition 3.17).We use this formula to construct new examples of asymptotic representations that cannot be perturbed to genuine representations in Section 5. Second, our proof is relatively elementary and uses only basic group cohomology, instead of employing techniques used in the Novikov conjecture.Third, because we do not use these techniques, we do not require the existence of a γ-element.
This paper is organized as follows.In Section 2 we will review relevant background information.In Section 3 we prove the main result (Theorem 1.1), and find a formula for an asymptotic representation that cannot be perturbed to a genuine representation for a group satisfies the assumptions of the main theorems (Proposition 3.17).We show that the main results apply to virtually polycyclic groups with non-torsion 2-cohomology (Corollary 3.23), and hence non-cyclic finitely generated torsion-free nilpotent groups.In Section 4 we give an alternate proof non-cyclic torsion-free finitely generated nilpotent groups satisfy the cohomological conditions required for the main result, which is useful for computing examples.In Section 5 we show how to recognize the cohomological conditions from generators and relations of a finitely generated group (Theorem 5.7).In Section 6 we give an abstract motivation for our formula, showing that it can be made from an induced representation of a finite quotient of a central extension of the original group (Theorem 6.1).In Section 7 we illustrate our formula for the following groups: Z 2 to show our methods can recover Voiculescu's matrices, a 3-step nilpotent group, and the polycyclic group Z 2 ⋊ Z where the action of Z on Z 2 is induced by "Arnold's cat map." 2. Background Information 2.1.Group Homology and Cohomology.There are many ways to characterize group homology and cohomology, but to us the most useful will be to describe them as the homology and cohomology of an explicit chain complex described below.We will only use homology with coefficients in Z and cohomology with trivial action in this paper.For more about this construction see [1,Chapter II.3].
Definition 2.1.Let Γ be a discrete group.We define C n (Γ) to be the free abelian group generated by elements of Γ n .We may write an element of Γ n as [a Often we will just write ∂ where the domain is clear from context.The group homology of Γ is the homology group the chain complex (C Definition 2.2.If A is any abelian group then the cohomology of Γ with coefficients in A is the cohomology of (C • , ∂ • ) with coefficients 1 in A.
To be more explicit, we use the notation and note that this is isomorphic to the group of functions from Γ n to A.
Then we define Then we define H n (Γ; A) := ker(∂ n )/ im(∂ n−1 ).As with homology we will suppress the n in ∂ n if the dimension is obvious from context. 1 In general A may be taken to be a left Z[Γ] module, but we will only consider the case where the action of Γ on A is trivial here, so we may consider A to only have the structure of an abelian group.
2 Actually [1] defines the coboundary map to be (−1) n+1 times the adjoint, but this does not change the image or kernel boundary of the maps so it leads to an equivalent definition of the cohomology groups.
Suppose that f is a group homomorphism from Γ 1 to Γ 2 .This induces a map f # from C n (Γ 1 ) to C n (Γ 2 ) and a map f * : is a homomorphism of abelian groups there is a map g # from C n (Γ; A 1 ) → C n (Γ; A 2 ).This map descends to a well-defined map g * : H n (Γ; A 1 ) → H n (Γ; A 2 ).All maps defined in this paragraph are functorial.
Because C n (Γ; A) is isomorphic to Hom(C n (Γ), A) there is a natural bilinear map, called the Kronecker pairing, from This descends to a well-defined bilinear map from H n (Γ; A) × H n (Γ) → A. We will use the notation ⟨•, •⟩ for both maps.
An explicit isomorphism from the the right side of the equation to H 2 (Γ) as described above can be defined as follows.Suppose that then by [1,chapter II.5 Exercise 4], where ȧi and ḃi are the images in Γ of a i and b i and Given an explicit central extension we may find a cocycle representative of the corresponding element of H 2 (Γ; A) as follows.Pick θ to be a set theoretic section from Γ to Γ. Then viewing A as a subset of Γ define By [1,Equation IV.3.3] this is a cocycle representative of the cohomology class corresponding to this central extension.

Polycyclic and Nilpotent Groups
A sequence of subgroups obeying this condition is called a polycyclic sequence of subgroups.We may pick this sequence so that each quotient is nontrivial.We may pick a i to be a representative of a generator of Γ i /Γ i+1 .We call these generators a polycyclic sequence for Γ, and they generate Γ. Definition 2.7.A group is called virtually polycyclic if it has a finite index polycyclic subgroup.
In this case we may assume that there is a normal finite index polycyclic subgroup.This may be constructed by intersecting over each conjugates of the subgroup and using the fact that a subgroup of a polycyclic group is polycyclic [21,Proposition 9.3.7].By [10,Theorem 3] polycyclic groups are residually finite.From this it follows that virtually polycyclic groups are residually finite as well.
Proposition 2.8.Suppose that Γ is virtually polycyclic, and is an extension of Γ.Then Γ is virtually polycyclic as well.
Proof.Let P ⊆ Γ be a finite index normal polycyclic subgroup of Γ.Since ι(Z) ⊆ φ −1 (P ) we have that Thus we may show that φ −1 (P ) is polycyclic.If {P i } m+1 i=1 is a polycyclic sequence of subgroups for P then we have for i ≤ m and P m+1 /{1} ∼ = Z.Thus these make a polycyclic sequence of subgroups.□ By [21,Proposition 9.3.4]all finitely generated nilpotent groups are polycyclic.Let Γ be a torsion-free finitely generated nilpotent group.Definition 2.9.A Mal'cev basis for a Torsion-Free finitely generated nilpotent group is an m-tuple of elements, (a 1 , . . .a m ) ∈ Γ m so that obeys the following conditions • For all g ∈ Γ, g can be written uniquely as g = a x 1 1 • • • a xm m for some (x 1 , . . ., x m ) ∈ Z m .We call this presentation the canonical form of g.
• The subgroups Γ i = ⟨a i , . . ., a m ⟩ form a central series for Γ.
Every finitely generated torsion-free nilpotent group has a Mal'cev basis by [12,Lemma 8.23].It also follows that Γ i make a polycyclic sequence of subgroups.
2.4.Rational 2-Cohomology of a Torsion-Free Finitely Generated Nilpotent Group.We will need the following result.
Proof.By a result of Pickel [20] we have that H • (Γ; Q) can be calculated in terms of the of the cohomology of an associated rational Lie algebra.By a result of Ado explained on [2,   The classical example due to Voiculescu in [22] for an asymptotic representation of Z 2 that is not perturbable to a genuine representation comes in the form: where u n and v n are n × n matrices such that .
The argument that we summarize here was first applied to this problem by Exel and Loring in [6], and had previously, idependently been used by Kazhdan in [15].It can be computed that u It is not difficult to show that the fact that this gets arbitrarily close to id C n in operator norm implies asymptotic multiplicativity of the associated representation.A sketch of the argument that this asymptotic representation cannot be perturbed follows.
The path n we make a contradiction as follows.We define It can be shown that h(t, s) ̸ = 0 for all s, t ∈ [0, 1].It follows that h is a homotopy from the path p to the trivial loop centered at 1.This is a contradiction since p has nonzero winding number.
A more general statement of this type of invariant can be found in [5, Theorem 3.9] or in [4].Essentially the relation u n v n u −1 n v −1 n can be replaced with another product of commutators.Note that the winding number of p could also be computed by calculating Tr log , where log is defined to be a power-series centered at 1.We will phrase our analogous argument in terms of computing trace of log instead of computing the winding number directly, but it is inspired by this more classical argument.Kazhdan develops this example independently to show that a certain surface group is not uniformly stable [15].He develops them as a representation of a central extension of the group in question, thereby connecting asymptotic representations to 2-cohomology.

Main Results
In Subsection 3.1 we prove some analytic lemmas that we use later.In subsection 3.2 we develop a pairing between between almost multiplicative functions from Γ to M n and 2-chains (Definition 3.3).We show that if an almost multiplicative function is close enough to a genuine unitary representation its pairing with a 2-cycle is zero (Theorem 3.7).In subsection 3.3 we introduce a "finite type" condition on cohomology classes (Definition 3.9), and give some alternate characterizations of the definition (Proposition 3.14).In subsection 3.4 we develop a formula for an asymptotic representation (Proposition 3.17), and show that if the right cohomological conditions hold it is well-defined and cannot be perturbed to a genuine representation (Theorem 1.1).We show that polycyclic groups with non-torsion 2-cohomology meet this condition (Corollary 3.23).
(1) We compute Applying the triangle inequality and submultiplicativity to the first term gives us the desired inequality. ( Then if log is defined as a power series centered at 1 we have that is well defined for all t ∈ [0, 1] by and in 2πiZ by the same argument above.
Because this expression depends continuously on t we must have that it is constant in t.Plugging in t = 0 we must have that the expression is uniquely zero.□

A Homological Version of the Winding Number Argument.
The idea of this section is to find a pairing between maps from Γ to M n that are "almost multaplicative" and elements of 2-cycles in C 2 (Γ).In general how "close" to being multiplicative depends on the specific element of C 2 (Γ) we pair with.
where log is defined as a power series centered at 1.This is clearly Z-linear in the second entry, in the sense that when the right side is well-defined.Due to potential cancellation, the support of c 1 + c 2 may be smaller than the support of c 1 union the support of c 2 .It is also "linear" in the first entry in the sense that if one side if this equality is well-defined then so is the other because Proof.Let F be the boundary support of c and let C 1 (F ) be the subgroup of C 1 (Γ) spanned by elements of the form [g] where g ∈ F .Define a homomorphism φ : . This is welldefined because C 1 (F ) is a free abelian group and det(ρ(g)) If c is a 2-cycle on Γ with boundary support F and ρ 0 and ρ 1 are maps from Γ to GL n (C) that are 1-almost multiplicative on the support of c we say that ρ 0 and ρ 1 are homotopy equivalent on the boundary support of c if the following conditions are met.There is a family of functions continuous in t so that ρ 0 (g) = ρ 0 (g) and ρ 1 (g) = ρ 1 (g) for all g ∈ F and for all t, ρ t is 1-almost multiplicative on the support of c. Proof.
(1) Define ρ t to be Then for each g in the boundary support of c we must have Applying Lemma 3.1 part 2 this gives us Then for (a i , b i ) in the support of c we have The last step is using Lemma 3.1 part 1 with "N "=3, "M "=2, and "ε"= 1−ε 12 .Applying this to t = 1 we get that Proof.This follows from Proposition 3.6, taking the limit as ε → 0. □ This is sufficient for our purposes, but for conceptual clarity it would be nice to show that this pairing depends only on homology class, not on the choice of cycle representative.We do not have a result that is quite this strong, but we can show that it is "eventually true" for asymptotic homomoprhisms.
For large enough n we have that ρ n is multiplicative enough that we may apply Lemma 3.2.Thus 2πi( ( (ρ n , c) Now since the complex unitary group is path connected we can make a path For large enough n this will be less than small, so Because this is a discrete space the values cannot depend on t.We conclude that 2πi( ( (ρ n , c) Remark 3.11.We may assume that the subgroups in Definition 3.9 are normal and decreasing.To achieve normality replace Γ k with the kernel of the action of Γ on Γ/Γ k .To achieve a decreasing sequence replace Γ k with the cumulative intersection of Γ k .
To develop our formula we will develop an alternate characterization of finite type cohomology classes that can be expressed in terms of the cohomology cochain complex.
Let Γ be a discrete group and let Q be a finite quotient of Γ.Call q the quotient map from Γ to Q and f n the canonical map from Z to Z/nZ.Then q induces a cochain map That this is a cocycle is easy to check: This is not a coboundary because Then if Q n = (Z/nZ) 2 , and let q n : Z 2 → Q n be the obvious quotient map.
We have that σ, and hence [σ], is of n-Q n type.To show this note that the same formula used for σ defines a 2-cochain, σ ′ ∈ C 2 (Q n ; Z/nZ).The same computations that show that σ is a cocycle also show that σ ′ is a cocycle, and clearly , where f n is the quotient map from Z to Z/nZ.
Let the central extension corresponding to [σ] be as follows: The following are equivalent: (1) [σ] is of finite type; (2) there are infinitely many n ∈ N so that there is a finite quotient of Qn of Γ so that ι(1) has order n in the quotient; (3) there are infinitely many n ∈ N so that Γ has a finite quotient Proof.(1)=⇒(2) Let Γ k be a sequence of subgroups as in Definition 3.9 and assume that they are normal as in Remark 3.11.For ℓ ∈ N pick Γ k so that ι(1) ℓ! ̸ ∈ Γ k .Then let n be the order of ι(1) in Γ/Γ k .Call Qn = Γ/Γ k .Note that n > ℓ so letting ℓ → ∞ we get the desired family of subgroups for infinitely many distinct n ∈ N.
( This theorem motivates a definition that extends the finite type concept to cocycle representatives of the cohomology class.Definition 3.15.If σ is a Z-valued 2-cocycle on Γ, we say that σ is of finite type if for infinitely many n ∈ N there is a finite quotient Q n of Γ so that that σ is of n-Q n type.
The cocycle defined in Example 3.13 is finite type.From Proposition 3.14 it follows that its cohomology class is as well.
To show this let q : Γ → Q be the quotient map and let f : Z → Z/nZ be the usual map.We can take ω Thus ω is of n-Q type.However, a finite type cohomology class does not obviously have a finite type representative, because the choice of representative of [σ] might depend on n.

3.4.
Constructing Asymptotic Representations From Cocycles.We start by defining our formula for an asymptotic representation.Proposition 3.17.Suppose that Γ is a discrete group and [σ] ∈ H 2 (Γ; Z).Let Q n be a finite quotient of Γ so that [σ] is of n-Q n type; this exists by Remark 3.16.Let σ n be a representative of [σ] so that σ n is of n-Q n type.Let α n be a 1-cochain so that Treat ḡ as a basis element for V n where g ∈ Γ and ḡ is its image in Q n .Then there is a well-defined function ρ n : Γ → U (V n ) that obeys the formula ρ n (g 1 )ḡ 2 = χn (g 1 , g 2 )ḡ 1 ḡ2 .
Proof.We will show that ρ n is well-defined.Suppose that ḡ2 = ḡ′ 2 .If we show that mod n we will have shown that ρ n is well-defined because χn only depends on σ n up to equivalence mod n.If σn is σ n reduced mod n then we have by assumption that σn = q # (σ ′ n ) where q is the quotient map from Γ to Q and .
Note that ρ n (g) maps the orthonormal basis {h : h ∈ Q n } to another orthonormal basis, so it is a unitary.□ Definition 3.18.Suppose ρ : Γ → U (k) and χ is an We will show that ρ n is a projective representation in Lemma 3.20.
In the case that α n = 0 our formula reduces to what is known as the projective left regular representation for Q n and χ n , for example see [19] page 2. The proof of Corollary B in [13] also uses a cohomology class that "behaves well" with respect to finite quotients, to make projective representations.
An alternate justification for the formula is as follows.Suppose that there is a finite quotient of a central extension of Γ as follows and a set theoretic section θ of the extension.Let π n be the induced representation of Qn from the character on ι ′ (Z/nZ) that takes ι ′ (1) → exp(2πi/n).Then ρ n = π n • qn • θ.Deriving the formula from here is technical.The discussion at the start of Chapter 3.3 in [14] explains how one should expect a projective representation to come from a splitting and representation of Γ as described above.
Remark 3.19.The existence technically only uses the fact that σ(g 1 , g 2 ) depends only on g 1 and the reduction of g 2 in Q n , rather than the image of both g 1 and g 2 in Q n .We will use this fact in examples to reduce the asymptotics of the dimension of the asymptotic representation.Lemma 3.20.Let Γ, n, σ, Q n , and ρ n be as above.Define χ n ∈ C 2 (Γ; S 1 ) by Then ρ n obeys the formula Proof.Define σn (g 1 , g 2 ) = α n (g 1 ) + σ n (g 1 , g 2 ).We compute The second equality follows from the fact that σ n is a cocycle.Exponentiating both sides of (3.3) we get χn (g 1 , g 2 g 3 ) −1 χn (g 1 g 2 , g 3 ) χn (g 2 , g 3 Next we claim that To check this it suffices to compute that Using this we can compute Here the last step follows from (3.4) applied to g 1 , g 2 and g −1 2 g −1 2 g 3 .□ Now we are ready to prove Theorem 1.1 Proof.First note that by Proposition 3.14 and Remark 3.16 there are infinitely many n so the formula given in Proposition 3.17 is well-defined.Now we will show that ρ n is asymptotically multiplicative.Noting that since σ does not depend on n we have that χ n (g 1 , g 2 ), defined as in Lemma 3.20, goes to 1 as n goes to infinity.Thus Lemma 3.20 implies asymptotic multiplicativity.Now we will show that for large enough n, ρ n is not close to any genuine representation of Γ on a particular finite subset of Γ. From the fact that [σ] ̸ ∈ ker(h) there is some 2-cycle c ∈ C 2 (Γ) written Then we compute that x j Tr(log(χ j (a j b j ) −1 ) id Vn ) by Lemma 3.20 By Theorem 3.7 it follows that ρ n cannot be within

Torsion-Free Finitely Generated Nilpotent Groups
The purpose of this section is to provide an alternate proof that torsionfree finitely generated nilpotent groups fit the conditions of Theorem 1.1 (Theorem 4.2).While this follows from Corollary 3.23 the alternate proof gives rise to a simple formula for the asymptotic representation.Proposition 4.1.Suppose that Γ is a torsion-free finitely generated nilpotent group with a Mal'cev basis (a 1 , . . ., a m ) and a central extension as follows: Then if ãi is a lift of a i for i ∈ {1, . . ., m} and ãm+1 = ι(1), (ã 1 , . . ., ãm+1 ) is a Mal'cev basis for Γ.
Proof.Let θ be the set theoretic section for φ defined by θ(a First we claim that any element g ∈ Γ can be written in the form ãx To see this note that g = (θ • φ(g))ã x m+1 m+1 .Then the claim follows from the definition of θ.Next to show that this is unique we suppose that ãx we get that x i = y i for i ̸ = m + 1.Then equality for i = m + 1 follows canceling the other terms and noting that ãm+1 is non torsion.Next we define Γi = ⟨ã i , . . ., ãm+1 ⟩ and similarly Γ i = ⟨a i , . . ., a m ⟩.By our assumptions Γ i form a central series for Γ.Note that for i ≤ m For i = m + 1 we have [ Γ, Γm+1 ] = {e} = Γm+2 .This shows that Γi is a central series which completes our proof.□ Theorem 4.2.Suppose that Γ is a torsion-free finitely generated nilpotent group that is not Z or trivial.Then Γ has a cohomology class that meets the conditions of Theorem 1.1, and the asympotic representation can be expressed as follows: Γ can be viewed as Z m with the multiplication as follows x * y = (η 1 (x, y), . . ., η m (x, y)) where η 1 , . . ., η n are rational 3 polynomials in x = (x 1 , . . ., x m ) and y = (y 1 , . . ., y m ).In addition we have a non-torsion cocycle σ(x, y) that is also a rational polynomial in the entries.Then the underlying vector space is (C n ) ⊗m .Then for n co-prime to the denominators of coefficient in the η i and σ we have where we have the convention that e j+n = e j . Proof Any element in Γ can be written uniquely as ãx where z i is a function of x 1 , . . ., x m+1 , y 1 , . . .y m+1 .Hall has shown that these functions are rational polynomials [8,Theorem 6.5].They can be computed by methods here [7].Then if we pick a section θ : m+1 .Since θ(gh) is in canonical form and has no power of ãm+1 it follows that σ ′ (g, h) must be a rational polynomial of the powers in the canonical forms of g and h.Note that by writing elements of Γ in canonical form we get that Γ can be viewed as Z m with a multiplication given by polynomial formulas of the entries.Thus if we take n to be co-prime to the denominator of each polynomial in the multiplication for Γ and the denominator of σ ′ we may define a quotient Q n of Γ by reducing each entry of Γ mod n.Then we may also reduce the formula for σ ′ mod n showing that σ ′ is of n-Q n type.To justify the formula note that ℓ 2 (Q n ) ∼ = (C n ) ⊗n and the isomorphism sends (y 1 , . . ., y m ) → e y 1 ⊗ • • • ⊗ e ym .The applying Proposition 3.17 we get the formula mentioned here, with σ n = σ, and so α n = 0. □ 3 The polynomials will always take integer values if given integer inputs, but in general the coefficients may not be integers.
Remark 4.3.Additionally we may modify this construction to get an asymptotic representation of dimension n m−1 instead of dimension n m .Note that the power of ãm+1 in the product (ã ) can be computed by using the relations to put all terms in order.If we leave the ãym m term at the end until the last step, we notice that we may have to switch the positions of the ãm and ãm+1 but these commute by because the extension is central.From this it follows that σ does not depend on y m .Thus we may replace our quotient Q n with Q ′ n = Q n /⟨q n (a m )⟩ and by Remark 3.19 we may use the formula for ρ n except with Q ′ n instead of Q n .The rest of the proof for asymptotic multiplicativity and non-perturbability flows the same way.An alternative explanation is that because σ(x, y) does not depend on y m the formula for ρ n commutes with the projection where p n is defined by the formula p n e i = j 1 √ n e j .Thus r n ρ n r n defines an n m−1 -dimensional asymptotic representation.

Generators and Relations
In this section we will show how to recognize conditions of Theorem 1.1 by looking at a presentation of a group.This can be helpful for computing the cocycle.It also relates our method to the method of other authors such as [22], [15], and [5].
Note that there must be some cocycle that satisfies the first condition in Theorem 1.1 if the 2-homology group is non-torsion.Note that because of Theorem 2.3 it is possible to view the 2-homology as a group of relations, so it should be possible to test this condition by looking at relations of the group.
Definition 5.1.We say the relation r We say that powers of r 1 are centrally irredundant relative to r 2 , . . .if r n 1 is not in the normal subgroup of F m generated by r 2 , . . .and [γ 1 , r 1 ], . . ., [γ m , r 1 ] for any n ∈ Z + .Example 5.2.If Z 2 is written as ⟨a, b|aba −1 b −1 = 1⟩ then the relation aba −1 b −1 is both homogeneous and powers of it are centrally irredundant.
Note that if r 1 meets both these conditions then it corresponds to a non torsion element of H 2 (Γ, Z) by Theorem 2.3.Proposition 5.3.If Γ = ⟨g 1 , . . ., g m |r 1 , . ..⟩where r 1 is homogeneous and powers of r 1 are centrally irredundant relative to the other relations then there is a central extension and mapping ι : 1 → r1 ∈ Γ and φ : gi → g i where r1 is the image of r 1 in the map from F m to Γ defined by mapping γ i → gi .
Proof.The map ι is injective because powers of r 1 are centrally irredundant.
Because r 1 is one of the defining relations of Γ, we have im(ι) ⊆ ker(φ).Note that im ι is central in Γ by the construction of Γ.The map φ is well defined because every relationship defining Γ is true of the generators of Γ as well.
If w is some word in the generators that is in ker(φ) then w pulls back to some word in F m that is in the smallest normal subgroup containing r 1 , . ... Since every other relation is trivial in Γ, this means w is in the smallest normal subgroup containing r1 .However since r1 is central, this group is precisely im(ι).
The map φ is surjective because it hits every generator of Γ. □ Definition 5.4.We call the central extension in Proposition 5.3 the dual extension of r 1 relative to the other relations.Proof.Pick a section θ of the extension so that θ(1) = 1.Then σ is defined by the formula To compute the 2-homology class corresponding to r 1 we need write r 1 as then by [1, chapter II.5 Exercise 4] the class corresponding to r 1 can be written as where ȧi and ḃi are the images in Γ of a i and b i and Call ãi = θ( ȧi ) and bi = θ( ḃi ).Here ãi and bi must be the image of a i and b i in the map from F m to Γ times some power of r1 .Since r1 is central in Γ this power can be ignored while computing

Note that
Then multiplying these in reverse order, starting from i = N and ending at i = 1 we get r−⟨σ,c⟩ Here the last equality is because I 0 = I N = 1 since I N is the relation r 1 .□ Theorem 5.7.Suppose that r 1 is a homogeneous relation on a group Γ = ⟨g 1 , . . ., g m |r 1 , . ..⟩ so that powers of r 1 are centrally irredundant relative to the other relations.If is the dual extension of r 1 relative to this presentation and there is a family of finite index subgroups Proof.This follows immediately from Proposition 5.6.□

Motivating Our Formula
The purpose of this section is to explain the motivation behind the formula in Proposition 3.17.For clarity recall the statement of the proposition: Treat ḡ as a basis element for V n where g ∈ Γ and ḡ is its image in Q n .
Then there is a well-defined function ρ n : Γ → U (V n ) that obeys the formula .
In terms of central extensions this means we have the following (6.1) Here we are taking θ to be the section from Γ to Γ corresponding to the cocycle σ and θ ′ n to be a section from Q n to Qn .Neither is a group homomorphism.Consider the character on Z/nZ defined by 1 → exp 2πi n .Then if π n is the representation induced by this character and the inclusion ι ′ we will show that ρ n = π n • qn • θ.We may view this induced representation as the left regular representation of Q n "twisted" by the cocycle σ n which corresponds to a Z/nZ cocycle on Q n because σ n is of n-Q n type.The α n in the formula is there because the right to left maps in the diagram may not commute with the horizontal maps.We cannot always pick θ to commute with the horizontal maps in this way, because if we did θ would need to depend on n and our argument for asymptotic multiplicativity would not hold.The following theorem summarizes this result.Theorem 6.1.Suppose that Γ meets the conditions of Theorem 1.1.Then our formula for an asymptotic representation that cannot be perturbed to a genuine representation can be obtained as follows.Consider the central extension corresponding to [σ]: Let S be the set of n ∈ N so that Γ has a finite quotient Qn where ι(1) has order n.Henceforth assume n ∈ S. Let θ be a set theoretic section from Γ to Γ, and let qn be the quotient map from Γ to Qn .If π n is the induced representation on Qn by the character on ⟨q n (ι(1))⟩ defined by qn ((ι(1)) → exp 2πi n .Then ρ n = π n • qn • θ equivalent to the formula given in Proposition 3.17.
Proof.Then let ω n be the character on ⟨q n (ι(1))⟩ defined by qn (ι(1)) → exp 2πi n .By [16,Exercise 2.3.16],one construction for the induced representation comes from the space n i=1 where one element g i is a representative of a different one of the n cosets in Qn /⟨ι(1)⟩ for each i.Here V n is the space as described in Proposition 3.17.
Then the induced representation is defined by π n (g)g i = ω n (q n (ι(1)) k )g j where g j and k are determined by the formula gg i = g j qn (ι(1)) k .This determines g j uniquely and k up to equivalence mod n.If we pick θ ′ n to be the map taking an element in Q n to the g i in the coset corresponding to it we get that g j = θ ′ n (φ ′ n (gg i )) and qn (ι(1) Then we may take θ n to be another section from Γ to Γ defined so that qn To show that such a section exists note that for all g ∈ Γ qn • θ(g) = qn • ι(1) k θ ′ n • q n (g) for some k ∈ Z.We then define θ n (g) = θ(g)ι(1) −k .We define the Γ cocycles σ, σ n and the Γ 1-cochain α n as follows: We claim that σ n is of n-Q n type.To show this qn (ι(1)) σn(g,h) = qn (θ n (gh)θ n (h) −1 θ n (g) −1 ) = θ ′ n (q n (gh))θ ′ n (q n (h)) −1 θ ′ n (q n (g)) −1 Since the order of qn (ι(1)) is n it follows that σ n (g, h) reduced mod n can be computed the same way as q # n applied to the cocycle corresponding to θ ′ n .Then note that by [1, Chapter IV.3 Equation 3.11] we have that σ n = σ + ∂α n .Note that V n is isomorphic to the vector space for π n by mapping h → θ ′ n (h).So we have To simplify this expression we note first that This gives us the formula Supposing that h = q n (a) we can rewrite this as In the notation of Proposition 3.17 the q n (ga) would be written in the form ḡā and ḡ → θ ′ n (g) describes the isomorphism of V n with the vector space on which the induced representation is defined.Thus this recovers the formula from Proposition 3.17.□

Examples
7.1.Z 2 Revisited.In this subsection we will apply our results to Z 2 , the simplest nontrivial example.We will compare the result of our algorithm to the classical results and show that we get Voiculescu's matrices tensored against another representation.Using remark 4.Moreover, since σ is a polynomial with integer coefficients it follows that σ is Z/nZ-compatible with (Z/nZ) 2 .Then applying Theorem 4.2 we get that ρ n acts on Note that ρ 1 n is precisely the asymptotic representation given by Voiculescu's matrices while ρ 2 n is a genuine representation.This is unsurprising because Remark 4.3 allows us to reduce the dimension by "ignoring" the second tensor coordinate and the resulting formula is precisely ρ 1 n .
7.2.A 3-Step Nilpotent Group.Consider the group Γ generated by a 1 , . . ., a 5 with the following relations a 3 a 2 = a 2 a 3 a 5 a i a j = a j a i for all other pairs {i, j}.
We first state a simplified version of our formula, then we explain how to compute it.We first compute the general version, then explain how it simplifies in this case.Our asymptotic representation is defined for n co-prime to 6 and sends generators to the following C n spanned e i with i ∈ Z and e i+n = e i : Here j 3 is the polynomial 1 6 j(j − 1)(j − 2).This group can be concretely realized as Z 5 with multiplication given by (x 1 , . . ., x 5 ) * (y 1 , . . .y 5 ) = (η 1 (x, y), . . ., η 5 (x, y)) where ).In general these polynomials may be calculated by methods given in [7].We will explain how to verify that these polynomials by computer.There is a full description of the code in the appendix, but we will summarize the main steps here.
(2) Calling a i the element with a 1 in the ith entry, and zeroes elsewhere, verify that a x 1 1 • • • a x 5 5 = (x 1 , . . ., x 5 ) under the operation * .
) Verify that the formula computed in the previous step is both a left, and a right inverse to (x 1 , . . ., x 5 ).( 5) Verify that a 1 , . . ., a 5 satisfies the relations of the group.In order to compute a non-torsion cocycle we will develop one as a central extension.We will do this by "blowing up" the relation [a 4 , a 1 ] = e.Thus we get a group Γ generated by ã1 , . . ., ã6 with the relations ã2 ã1 = ã1 ã2 ã3 ã3 ã1 = ã1 ã3 ã2 In this case the relation [b 3 , b 4 ] = 1 follows from the other three relations so we cannot construct a central extension by "blowing it up."We will explain why this issue does not arise in our example below.In general an algorithm for finding when relations of this form make a nilpotent group where each "a i " has infinite order is described in [21, Proposition 9.8.3,Proposition 9.9.1].
Then Γ can be identified with Z 6 with the multiplication is given by (x 1 , . . ., x 6 ) * (y 1 , . . ., y 6 ) = (γ 1 (x, y), . . ., γ 6 (x, y)) where We have verified the fact that these polynomials give rise to a group operation satisfying the relations of the group with similar code to what we used to verify these things for Γ.Since the element (0, 0, 0, 0, 0, 1) has infinite order in the group determined by these polynomials it follows that ã6 has infinite order as well.
From this it follows that a cocycle corresponding to the central extension is given by For any n co-prime to 6 we may define with multiplication given by (x 1 , . . ., x n ) * (y 1 , . . ., y n ) = (η 1 (x, y), . . ., η5 (x, y)) where ηi is η i with each coefficient reduced mod n. 4 Then reducing each coefficient of σ mod n we get σ.Note that the fact that σ is a Z/nZvalued cocycle on Q n implies that σ is of Q n -n type.Then we can use the formula from Remark 3.22 to define our asymptotic representation.Note that ℓ2 (Q n ) ∼ = ℓ 2 (Z/nZ) ⊗5 .Treat {e i } i=0,...,n−1 as a basis for ℓ 2 (Z/nZ).For ease of notation we will extend e i to be well-defined for all i ∈ Z by the formula e i+n = e i .Thus we get ρ n (x 1 , . . ., x 5 )e y 1 ⊗e y 2 ⊗• • •⊗e y 5 = exp 2πiσ(x, y) n e η 1 (x,y) ⊗e η 2 (x,y) ⊗• • •⊗e η 5 (x,y) .
Theorem 1.1 guarantees that this formula is well-defined.As in Remark 4.3 we can "ignore" the last tensor coordinate since σ(x, y) does not depend on y 5 and neither does η i for i < 5.This gives us an n 4 dimensional asymptotic representation.In this particular case we may go much further.It turns out that σ depends only on x and y 1 so we can ignore every tensor coordinate except the first.This gives us the asymptotic representation ρ ′ n we introduced in the start of the section.

7.3.
A Polycyclic Group.Let Γ = Z 2 ⋊ Z where the action of Z on Z 2 is given by "Arnold's Cat Map:" The generators are a 1 , a 2 , a 3 with the relations A simplified version of our asymptotic representation is given on C n ⊗ C n with basis {e j ⊗ e k } j,k∈Z and the convention that e j+n = e j .With this notation the generators are sent to the following operators As in the last chapter we will explain how to compute the asymptotic representation given by Proposition 3.17, then explain how to derive the simpler formula ρ ′ n .We will compute a non-torsion cocycle in H 2 (Γ; Z).We explain our reasoning about how to find the cocycle without formal proof then show formally that it obeys the cocycle condition.The idea is as in the previous section to find a central extension of Γ, compute the multiplication in the middle group of the central extension.We may consider the following presentation of Γ.Each element in the group can be written uniquely as a x 1 1 a x 2 2 a x 3 3 and this element will be sent to the corresponding product of matrices.We may make an extension by "blowing up" the relation for all i.
Using these relations we may write any element of Γ uniquely in the form b x 1 1 • • • b x 4 4 with x i ∈ Z.As the reader is warned in Remark 7.1 it is not always the case that "blowing up" a relation like this leads to a sensible extension.In this case when we verify the cocycle condition we will also have verified that this makes a sensible extension.
We will describe an element of Γ implicitly by a pair (v, k) with v = (v 1 , v 2 ) ∈ Z 2 and k ∈ Z then by the definition of the semi-direct product the multiplication is given implicitly by (v 1 , k 1 ) * (v 2 , k 2 ) = (v 1 + T k 1 v 2 , k 1 + k 2 ).Our goal is to implicitly describe Γ similarly.To that end we will describe an element of Γ as a triple (v, k, d) with v = (v 1 , v 2 ) ∈ Z 2 and k, d ∈ Z.This represents the element b To that end we make the following write a closed form for the analogue of S k in the linear terms.We have done enough to motivate our definition of the cocycle.It comes from keeping track of each of the "b 3 terms" when computing multiplication in Γ.Note an element of Γ as g i = (v i , k i ) = ((v 1 i , v 2 i ), k i ) so that v i ∈ Z 2 represent the element a Then we define our cocycle Note that σ 4 is subtracted unlike the others.Here σ 1 comes from the "(5.1) terms," σ 2 comes from the "(5.4) terms," and σ 3 − σ 4 comes from the "(5.2) and (5.3) terms."Before we compute ∂σ we observe the following identities about S k Now we compute ∂σ piece by piece.We will let g i ∈ Γ be represented as the pair (v i , k i ).Then we compute ∂σ 1 (g 1 , g 2 , g 3 ) =α(−T Finally ∂σ 4 (g 1 , g 2 , g 3 ) =γ((T − 1) −1 (−(T Note that since S k 1 commutes with the map u ⊗ w → w ⊗ u by construction so we have that Now we have ∂σ(g 1 , g 2 , g 3 ) = 2 i=1 α(−(T ⊗T )(u i ⊗w i )+2u i ⊗w i +w i ⊗u i )+β(u i ⊗w i +w i ⊗u i ) .
Next we see from the definition of T and α that α((T ⊗ T )u i ⊗ w i ) = (2u Then since S 0 = 0 and T 0 − 1 = 0 we have that σ 1 is the only one of the forms to pair nontrivially with c.Thus we see ⟨σ, c⟩ = 1. We next investigate finite quotients of Γ.If n, m ∈ N + so that the order of T reduced mod n in GL 2 (Z/nZ) divides m then a finite quotient of Γ can be of the form (Z/nZ) 2 ⋊ Z/mZ with the action described by the reduction of T mod n.Our goal is to find finite quotients Q n of this form so that σ is n-Q n compatible.To do this note that the pair (S k±1 , T k±1 ) can be determined from the pair (S k , T k ) and the entries of (S k±1 , T k±1 ) are polynomials in the entries of (S k , T k ).These polynomials may be reduced mod n.So it follows that if we pick m (depending on n) so that S m ≡ S 0 mod n and T m ≡ 1 mod n we have that (S k , T k ) ≡ (S k+m , T k+m ) mod n for all k, by induction on |k|.It follows that the order of the order of T reduced mod n in GL 2 (Z/nZ).Thus for odd n we define The first line of code is needed to make sage treat z 1 , . . ., z 6 as algebraic expressions.The next two lines compute x * (y * z) and (x * y) * z respectively.The for loop checks that these expressions are equivalent in each coordinate.It is easy to see by inspecting the polynomials that (1, 0, 0, 0, 0) x 1 = (x 1 , 0, 0, 0, 0), (0, 1, 0, 0, 0) x 2 = (0, x 2 , 0, 0, 0), and so on.The following code, assumes this fact and checks that the relation (x 1 , . . ., x 5 ) = a x 1 1 • • • a x 5 5 makes sense with a i corresponding to the vector that has a 1 in the ith place and zeroes elsewhere: Next we must check the existence of inverses: The first line of code computes what the inverse of (x 1 , . . ., x 5 ) must be if it exists.The next two lines verify that this is in fact, both a left and right inverse, respectively.Finally, we need to check that these relations satisfy the the presentation of the group.The first few of the computations look like this These lines verify the relations a 2 a 1 = a 1 a 2 a 3 , a 3 a 1 = a 1 a 3 a 2 4 , and a 5 a 1 = a 1 a 5 , respectively.The rest of the relations may be checked with similar code.

Theorem 3 . 8 .
Let c = ∂d be a 2-boundary in C 2 (Γ) and let ρ n : Γ → U (N n ) be an asymptotic homomorphism.Then for large enough n we have ( ( (ρ n , c) ) ) = 0 Proof.By linearity we may reduce the case that c = ∂[g 1 |g 2 |g 3 ].In this case we have that

Example 5 . 5 .Proposition 5 . 6 .
If Z 2 ∼ = ⟨a, b|[a, b] = 1⟩ is given the same presentation as in Example 5.2 then the dual extension to the relation aba −1 b −1 can be expressed as the group H = ⟨a, b, c|c = [a, b], [a, c] = [b, c] = 1⟩ commonly known as the discrete Heisenberg group.The map ι is given by ι(1) = c and φ is determined by φ(a) = a and φ(b) = b Suppose that r 1 is a homogeneous relation on a group Γ = ⟨g 1 , . . ., g m |r 1 , . ..⟩ so that powers of r 1 are centrally irredundant relative to the other relations.If [σ] is the 2-cohomology class corresponding to the dual extension of r 1 and [c] is the 2-homology class corresponding to r 1 then ⟨σ, c⟩ = 1.

n bj e a+j and ρ 2 n
Then pick the basis {e j ⊗ e k } with e j defined for all j ∈ Z by the formula e j+n = e j .Using the formula for Theorem 4.2 we get that ρ n (a, b)e j ⊗ e k = exp 2πi n bj e a+j ⊗ e b+k .Note that we may write ρ n (a, b) = ρ 1 n (a, b) ⊗ ρ 2 n (a, b) where ρ 1 n (a, b)e j = exp 2πi (a, b)e j = e b+j .
5 ) = id C n .And sends the element written uniquely in the form a

[a 1 , a 2 ] = 1 .
Then we may consider Γ to be the group generated by b 1 , . . ., b 4 and the relations b 2 b 1 = b 4 b 3 b 1 = b 2 1 b 2 b 3 b 3 b 2 = b 1 b 2 b 3 b 4 b i = b i b 4
ι φWe say that [σ] is finite type if Γ has a sequence of finite index subgroups {Γ k } k∈N so that Corollary 3.23.Suppose that Γ is a virtually polycyclic group with nontorsion 2-cohomology.Then Γ meets the conditions of Theorem 1.1, and is thus not matricially stable.