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 $\Gamma$ is a sequence of functions $\rho_n:\Gamma\rightarrow U(N_n)$ so that for all $g,h\in\Gamma$ we have $||\rho_n(gh)-\rho_n(g)\rho_n(h)||\rightarrow0$ as $n$ goes to infinity, where $||\cdot||$ is operator norm. We say an asymptotic representation is perturbable to a genuine representation if there is a sequence of representations $\Tilde{\rho}_n:\Gamma\rightarrow U(N_n)$ so that for all $g\in\Gamma$ we have $||\rho_n(g)-\Tilde{\rho}_n(g)||\rightarrow0$ as $n$ goes to infinity. Recall that a countable discrete group, $\Gamma$, is matricially stable if every asymptotic representation of $\Gamma$ is perturbable to a genuine representation of $\Gamma$ [4].

In [16] 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 [9] Kazhdan independently uses the same sequence of pairs of unitaries to show that a particular surface group is not stable in the uniform sense. Kazhdan also connects his argument to the 2-cohomology of the group. In [4] 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. However, the proof does not lead to an explicit construction of an asymptotic representation that is not perturbable to a genuine representation.
In this paper we will show that a group with a 2-homology class that pairs nontrivially with a 2-cohomology class, $x$ corresponding to a central extension with a residually finite group in the middle is not matricially stable. In general every element of $x\in H^2(\Gamma;\Z)$ corresponds to a central extension

1\arrow{r} & \Z
\arrow{r}{} &
\Tilde{\Gamma} \arrow{r}{}
& \Gamma\arrow{r}{}& 1.
\end{tikzcd}$$

If $\tilde\Gamma$ is residually finite (or more generally if there is a sequence of finite index subgroups $\Gamma_n\le\tilde\Gamma$ so that $\Z\cap\bigcap\Gamma_n=\{e\}$) and $x$ is not in the kernel of the map $h:H^2(\Gamma;\Z)\rightarrow\Hom(H_2(\Gamma;\Z))$ 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 $\Gamma$ and cocycle representatives of $x$ (Theorems <ref> and <ref>). If $\Gamma$ 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 will follow that any virtually polycyclic group with non-torsion 2-cohomology is not matricially stable.

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 <ref>). We use this formula to construct new examples of asymptotic representations that cannot be perturbed to genuine representations in Section 7. 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 $\gamma$-element.

This paper is organized as follows. In Section 2 we will review relevant background information. In Section 3 we prove the main results (Theorems <ref> and <ref>), and 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 <ref>). We show that the main results apply to virtually polycyclic groups with non-torsion 2-cohomology (Corollary <ref>), 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 from Section 3, 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 <ref>). 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 <ref>). 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 $\Z^2\rtimes\Z$ where the action of $\Z$ on $\Z^2$ is induced by “Arnold's cat map.”

§ BACKGROUND INFORMATION

§.§ 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 are only use homology with coefficients in $\Z$ and cohomology with trivial action in this paper. For more about this construction see <cit.>.

Let $\Gamma$ be a discrete group. We define $C_n(\Gamma)$ to be the free abelian group generated by elements of $\Gamma^n$. We may write an element of $\Gamma^n$ as $[a_1|a_2|\cdots|a_n]$ with $a_i\in\Gamma$. We thus write a typical element of $C_n(\Gamma)$ as

with $x_i\in\Z$ and $a_{ij}\in\Gamma$. We define the boundary map $\partial_n$ from $C_n(\Gamma)\rightarrow C_{n-1}(\Gamma)$ by

Often we will just write $\partial$ where the domain is clear from context. The group homology of $\Gamma$ is the homology group the chain complex $(C_\bullet,\partial_\bullet)$.
For readers who are less familiar with homological algebra we will be more explicit. It happens that $\im(\partial_n)\subseteq\ker(\partial_{n+1})$ so we define the $n$th homology group of $\Gamma$ to be

An element of $\ker\partial_n$ is called an $n$-cycle.

If $A$ is any abelian group then the cohomology of $\Gamma$ with coefficients in $A$ is the cohomology of $(C_\bullet,\partial_\bullet)$ with coefficients[In general $A$ may be taken to be a left $\Z[\Gamma]$  module, but we will only consider the case where the action of $\Gamma$ on $A$ is trivial here, so we may consider $A$ to only have the structure of an abelian group.] in $A$.

We will again be more explicit for readers who are less familiar with homological algebra. Let

and note that this is isomorphic to the set of functions from $\Gamma^n$ to $A$ because an element in $\Hom(C_n(\Gamma);A)$ is defined by what it does to each generator in $C_n(\Gamma)$ which we identify with elements of $\Gamma^n$. Then we define $\partial^n:C^n(\Gamma;A)\rightarrow C^{n+1}(\Gamma;A)$ to be the adjoint[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.] of $\partial_{n+1}$. When considering $C^n(\Gamma;A)$ to be the the set of functions from $\Gamma^n$ to $A$ the map follows the formula

$$\sigma(a_2,\ldots, a_{n+1})+\sum_{i=1}^n(-1)^i\sigma(a_1,\ldots, a_{i-1},a_ia_{i+1},a_{i+2},\ldots, a_{n+1})+(-1)^{n+1}\sigma(a_1,\ldots,a_n).$$

Then we define

An element of $\ker(\partial^n)$ is called an $n$-cocycle. As with homology we will suppress the $n$ in $\partial^n$ if the dimension is obvious from context.

Suppose that $f$ is a group homomorphism from $\Gamma_1$ to $\Gamma_2$. This induces a map $f_\#$ from $C_n(\Gamma_1)$ to $C_n(\Gamma_2)$ by the formula

The same formula descends to a well-defined map $f_*:H_n(\Gamma_1)\rightarrow H_n(\Gamma_2)$. Similarly the adjoint of $f_\#$ from $C^n(\Gamma_2)\rightarrow C^n(\Gamma_1)$ called $f^\#$ and can be defined by the formula

$$(f^\#(\sigma))(a_1,\ldots a_n)=\sigma(f(a_1),\ldots, f(a_n)).$$

As in the homological case this descends to a well-defined map from $f^*:H^2(\Gamma_2;A)\rightarrow H^2(\Gamma_1;A)$. Similarly if $g:A_1\rightarrow A_2$ is a homomorphism of abelian groups there is a map $g_\#$ from $C^n(\Gamma;A_1)\rightarrow C^n(\Gamma;A_2)$ by the formula $\sigma\mapsto g\circ\sigma$. This formula descends to a well-defined map $g_*:H^n(\Gamma;A_1)\rightarrow H_n(\Gamma;A_2)$. All maps defined in this paragraph are functorial in the sense that $(f\circ g)_*=f_*\circ g_*$ and $(f\circ g)^*=g^*\circ f^*$ and the same is true if the $*$ is replaced by a $\#$.

Because $C^n(\Gamma;A)$ is isomorphic to $\Hom(C_n(\Gamma),A)$ there is a natural bilinear map from $C^n(\Gamma;A)\times C_n(\Gamma)\rightarrow A$ defined by

This descends to a well-defined bilinear map from $H^n(\Gamma;A)\times H_n(\Gamma)\rightarrow A$. We will use the notation $\langle\cdot,\cdot\rangle$ for both maps.

§.§ 2-(Co)homology

Since we will be more concerned with these groups than homology and cohomology groups in general it will be helpful to describe these clearly. In addition each has an alternative characterization that is useful to us, so we will describe these alternate characterizations here. First since the formulas for the boundary maps are messy in general we will write formulas for $\partial_2$, $\partial_3$, $\partial^1$, and $\partial^2$ that are neater:

\begin{align*}
\partial_2[g_1|g_2]&=[g_1]-[g_1g_2]+[g_2]\\
\partial_3[g_1|g_2|g_3]&=-[g_1|g_2]+[g_1|g_2g_3]-[g_1g_2|g_3]+[g_2|g_3]\\
\end{align*}

The following result gives an alternative characterization of the 2-homology of a group.

Suppose that $\Gamma\simeq F/R$ where $F$ is a free group. Then

An explicit isomorphism from the the right side of the equation to $H_2(\Gamma)$ as described above can be defined as follows. Suppose that

$$r_1=\prod_{i=1}^N[{a}_i,{b}_i]\in R\cap[F,F]$$

then by <cit.>,

\[r_1\mapsto \sum_{i=1}^N([I_{i-1}|\dot a_i]+[I_{i-1}\dot a_i|\dot b_i]-[I_{i-1}\dot a_i\dot b_i\dot a_i^{-1}|\dot a_i]-[I_i|\dot b_i])+\im(\partial_3)\]

where $\dot a_i$ and $\dot b_i$ are the images in $\Gamma$ of ${a}_i$ and ${b}_i$ and $I_i=[\dot a_1,\dot b_1]\cdots[\dot a_i,\dot b_i]$.

The 2-cohomology also has an alternative characterization.

If $\Gamma$ is a discrete group and $A$ is an abelian group, then central extension of $\Gamma$ by $A$ is a short exact sequence

1\arrow{r} & A
\arrow{r}{} &
\Tilde{\Gamma} \arrow{r}{}
& \Gamma\arrow{r}{}& 1
\end{tikzcd}$$

where the image of $A$ in $\tilde{\Gamma}$ is central in $\tilde{\Gamma}$. We say two central extensions are equivalent if we can make a commutative diagram as follows:

1\arrow{r} & A
\arrow{r}{}\arrow{d}{\Id_A} &
\Tilde{\Gamma}_1\arrow{d}{\simeq} \arrow{r}{}
& \Gamma\arrow{r}{}\arrow{d}{\Id_\Gamma}& 1
\\
1\arrow{r} & A
\arrow{r}{} &
\Tilde{\Gamma}_2 \arrow{r}{}
& \Gamma\arrow{r}{}& 1.
\end{tikzcd}$$

As a set $H^2(\Gamma;A)$ is in bijection with the equivalence classes of central extension of $\Gamma$ by $A$.

Given an explicit central extension we may find a cocycle representative of the corresponding element of $H^2(\Gamma;A)$ as follows. Pick $\theta$ to be a set theoretic section from $\Gamma$ to $\Tilde{\Gamma}$. Then viewing $A$ as a subset of $\Tilde{\Gamma}$ define

$$\sigma(g,h)=\theta(g)\theta(h)\theta(gh)^{-1}\in A.$$

By <cit.> this is a cocycle representative of the cohomology class corresponding to this central extension.

§.§ Polycyclic and Nilpotent Groups

A group $\Gamma$ is called polycyclic if there is a sequence of subgroups

$$\Gamma=\Gamma_1\ge \Gamma_2\ge\cdots\ge\Gamma_m\ge\Gamma_{m+1}=\{1\}$$

so that $\Gamma_{i}\triangleright\Gamma_{i+1}$ and $\Gamma_i/\Gamma_{i+1}$ is cyclic. 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 $\Gamma_i/\Gamma_{i+1}$. We call these generators a polycyclic sequence for $\Gamma$.

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 group. 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 <cit.>. By <cit.> polycyclic groups are residually finite. From this it follows that virtually polycyclic groups are residually finite as well.

Suppose that $\Gamma$ is virtually polycyclic, and

1\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1
\end{tikzcd}$$

is an extension of $\Gamma$. Then $\tilde{\Gamma}$ is virtually polycyclic as well.

Let $P\subseteq\Gamma$ be a finite index normal polycyclic subgroup of $\Gamma$. Since $\iota(\Z)\subseteq\varphi^{-1}(P)$ we have that

Thus we may show that $\varphi^{-1}(P)$ is polycyclic. If $\{P_i\}_{i=1}^{m+1}$ is a polycyclic sequence of subgroups for $P$ then we have for $i\le m$

and $P_{m+1}/\{1\}\simeq\Z$. Thus these make a polycyclic sequence of subgroups.

By <cit.> all finitely generated nilpotent groups are all polycyclic. Let $\Gamma$ be a torsion-free finitely generated nilpotent group.

A Mal'cev basis for a Torsion-Free finitely generated nilpotent group is an $m$-tuple of elements, $(a_1,\ldots a_m)\in\Gamma^m$ so that obeys the following conditions

* For all $g\in\Gamma$, $g$ can be written uniquely as $g=a_1^{x_1}\cdots a_m^{x_m}$ for some $(x_1,\ldots, x_m)\in\Z^m$.  We call this presentation the canonical form of $g$.

* The subgroups $\Gamma_i=\langle a_i,\ldots, a_m\rangle$ form a central series for $\Gamma$.

Every finitely generated torsion-free nilpotent group has a Mal'cev basis by <cit.>. It also follows that $\Gamma_i$ make a polycyclic sequence.

§.§ Rational 2-Cohomology of a Torsion-Free Finitely Generated Nilpotent Group

We will need the following result.

Let $\Gamma$ be a torsion-free finitely generated nilpotent group that is neither $\Z$ nor the trivial group. Then $H^2(\Gamma;\Q)\not\simeq\{0\}$.

By a result of Pickel [13] we have that $H^\bullet(\Gamma;\Q)$ can be calculated in terms of the of the cohomology of an associated rational Lie algebra. By a result of Ado explained on <cit.> the 2-cohomology of the algebra in nonzero.

§.§ Voiculescu's Matrices and the Winding Number Argument

The classical example due to Voiculescu in [16] 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 $n\times n$ matrices such that

\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\
\end{bmatrix}\mbox{ and }
\exp\left(\frac{2\pi i}{n}\right)&0&0&\cdots&0\\
0&\exp\left(\frac{4\pi i}{n}\right)&0&\cdots&0\\
0&0&\exp\left(\frac{6\pi i}{n}\right)&\cdots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
\end{bmatrix}.

It can be computed that $u_nv_nu_n^{-1}v_n^{-1}=\exp\left(\frac{-2\pi i}{n}\right)\Id_n$. It is not difficult to show that the fact that this gets arbitrarily close to $\Id_n$ 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 $p(t)=\det\left(t\exp\left(\frac{-2\pi i}{n}\right)\Id_n+(1-t)\Id_n\right)$ is a path in $\C^\times$ with winding number $-1$. Suppose for contradiction that $u_n$ is close enough[Within 1 is sufficient for the argument to hold.] in operator norm to a $u_n'$ and $v_n$ is close enough to $v_n'$ so that $v_n'u_n'=u_n'v_n'$ we make a contradiction as follows. We define

\begin{align*}
\hat{u}_n(s)&=su_n+(1-s)u_n'\\
\hat{v}_n(s)&=sv_n+(1-s)v_n'
\end{align*}

It can be shown that $h(t,s)\ne0$ for all $s,t\in[0,1]$. 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 <cit.>. Essentially the relation $u_nv_nu_n^{-1}v_n^{-1}$ can be replaced with another product of commutators. Note that the winding number of $p$ could also be computed by calculating $\Tr\left(\log\left(\frac{-2\pi i}{n}\Id_n\right)\right)/(2\pi i)$. 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 [9]. 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 $\Gamma$ to $M_n$ and 2-chains (Definition <ref>). 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 <ref>). In subsection 3.3 we introduce a “finite type” condition on cohomology classes (Definition <ref>), and give some alternate characterizations of the definition (Proposition <ref>). In subsection 3.4. we develop a formula for an asymptotic representation (Proposition <ref>), and show that if the right cohomological conditions hold it is well-defined and cannot be perturbed to a genuine representation (Theorem <ref>). We show that if $[\sigma]$ pairs non-trivially with a 2-cycle and corresponds to a residually finite central extension then our formula for an asymptotic representation that is not perturbable to a genuine representation can be applied (Theorem <ref>) and show that polycyclic groups with non-torsion 2-cohomology meet this condition (Corollary <ref>).

§.§ Analytic Lemmas

We will use the following elementary results. While we prove them for any $C^*$-algebra, we are only using them on matrix algebras here.

Let $A$ be a $C^*$-algebra. Then the following hold:

* If $a_i,b_i\in A$ for $i\in\{1,\ldots,N\}$ so that for all $i$, $||a_i-b_i||<\varepsilon$ and $||a_i||,||b_i||<M$ we have that

* If $a\in A$ and $u$ is a unitary in $A$ so that $||a-u||\le\frac12$ we have

In particular $||a^{-1}||<2$.

(1) We compute

\begin{align*}
\sum_{j=1}^N\left(\prod_{i=1}^{j-1} a_i\right)(a_j-b_j)\prod_{i=j+1}^Nb_i&=\sum_{j=1}^N\left(\prod_{i=1}^{j}a_i\right)\prod_{i=j+1}^Nb_i-\sum_{j=1}^N\left(\prod_{i=1}^{j-1}a_i\right)\prod_{i=j}^Nb_i\\
\\
\end{align*}

Applying the triangle inequality and submultiplicativity to the first term gives us the desired inequality.

(2) Let $b=u^*a$. Note that $||b-1||=||a-u||<1$ so by <cit.> we have that

\begin{align*}
\end{align*}

We use that $||a-u||<\frac12$ in the last step.

Let $m_1,m_2\in U(n)$ so that $||m_i-\Id_{\C^n}||<\frac12$. Then if $\log$ is defined as a power series centered at 1 we have that

First note that

$$\Tr(\log(m_1m_2))-\Tr(\log(m_1))-\Tr(\log(m_2))\in2\pi i\Z.$$

is well defined for all $t\in[0,1]$ by Lemma <ref> and is in $2\pi i\Z$ 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 $\Gamma$ to $U(n)$ that are “almost multaplicative” and elements of $H_2(\Gamma)$. In general how “close” to being multiplicative depends on how close the specific element of $H_2(\Gamma)$ we pair with and the representative we pick for the element.

Suppose that $c\in C_2(\Gamma)$ is expressed by the formula

$$c=\sum_{i=1}^N x_i[a_i|b_i]$$

so that $(a_i,b_i)=(a_j,b_j)$ implies $i=j$. The support of $c$ is the set of ordered pairs $\{(a_i,b_i)\}_{i=1}^N$. The boundary support of $c$ is the set of elements of $\Gamma$, $\{a_i,b_i,a_ib_i\}_{i=1}^N$. Then we say $\rho:\Gamma\rightarrow M_n(\C)$ is $\varepsilon$-almost multiplicative on the support of $c$ if $\rho(a_i),\rho(b_i)\in\Gl_n(\C)$ and

for each $i$. In this case, if additionally $\varepsilon\le1$, we define

$$\langle \rho,c\rangle=\frac{1}{2\pi i}\sum_{j=1}^Nx_i\Tr(\log(\rho(a_jb_j)\rho(b_j)^{-1}\rho(a_j)^{-1}))$$

where $\log$ is defined as a power series centered at 1.

This is clearly $\Z$-linear in the second entry. It is also “linear” in the first entry in the sense that

If $\partial c=0$ then

Let $F$ be the boundary support of $c$ and let $C_1(F)$ be the subgroup of $C_1(\Gamma)$ spanned by elements of the form $[g]$ where $g\in F$. Define a homomorphism $\varphi:C_1(F)\rightarrow \C^\times$ by taking $[g]\mapsto\det(\rho(g))$. This is well-defined because $C_1(F)$ is a free abelian group and in $\C^\times$ since $\rho(F)\subset\Gl_n(\C)$. Then we see that

$$\exp(2\pi i\langle\rho,c\rangle)=\prod_{j=1}^N\det((\rho(a_jb_j)\rho(b_j)^{-1}\rho(a_j)^{-1})^{x_i})=\varphi(-\partial c)=\varphi(0)=1.$$

If $c$ is a 2-cycle on $\Gamma$ with boundary support $F$ and $\rho_0$ and $\rho_1$ are maps from $\Gamma$ to $\Gl_n(\C)$ that are 1-almost multiplicative on the support of $c$ we say that $\rho_0$ and $\rho_1$ are homotopy equivalent on the boundary support of $c$ if the following conditions are met. There is a family of functions

$$\rho^t:\Gamma\rightarrow M_n(\C)$$

continuous in $t$ so that $\rho^0(g)=\rho_0(g)$ and $\rho^1(g)=\rho_1(g)$ for all $g\in F$ and for all $t$, $\rho^t$ is 1-almost multiplicative on the support of $c$.

Let $c$ be a 2-cycle on $\Gamma$ and let $0<\varepsilon<1$. Let $\rho_0:\Gamma\rightarrow U(n)$ be $\varepsilon$ multiplicative and $\rho_1:\Gamma\rightarrow\Gl_n(\C)$.

* If
for all $g$ in the boundary support of $c$ then $\rho_0$ and $\rho_1$ are homotopy equivalent in the boundary support of $c$ and $\rho_1$ is 1-almost multiplicative on the support of $c$.

* If $\rho_0$ and $\rho_1$ are homotopy equivalent in the boundary support of $c$ then $\langle\rho_0,c\rangle=\langle\rho_1,c\rangle$.

(1) Define $\rho^t$ to be

Then for each $g$ in the boundary support of $c$ we must have

Applying Lemma <ref> part 2 this gives us

Then for $(a_i,b_i)$ in the support of $c$ we have

\begin{align*}
\end{align*}

The last step is using Lemma <ref> part 1.

(2) The function

is a continuous function from $[0,1]$ to $\Z$ so it must be constant.

If $\rho_0$ is a genuine unitary representation, $\rho_1$ is a function from $\Gamma$ to $M_n(\C)$ $c$ is a 2-cocycle on $\Gamma$ and

for all $g$ in the boundary support of $c$ then $\rho_1$ is 1-multiplicative on the support of $c$ and $\langle\rho_1,c\rangle=0$.

This is sufficient for our purposes, but for conceptual clarity we would like to show that this pairing depends only on homology class, not on the choice of cycle representative.

Let $c=\partial d$ be a 2-boundary in $C^2(\Gamma)$ then if $\rho:\Gamma\rightarrow U(n)$ is $\frac18$-almost multiplicative on the the support of $c$ then $\langle\rho,c\rangle=0$.

By linearity, we may reduce to the case that $c=\partial[g_1,g_2,g_3]$. In this case we have that

Using Lemma <ref> we get that these expressions are close enough to $\Id_{\C^n}$ that we may apply Lemma <ref>. Thus

\begin{align*}
2\pi i\langle\rho,c\rangle=&\Tr(\log(\\
\end{align*}

Now since the complex unitary group is connected <cit.> we can make a path $u_t:[0,1]\rightarrow U(n)$ so that $u_0=\rho(g_1)$ and $u_1=\Id_{\C^n}$. Then

\begin{align*}
\end{align*}

Moreover since

we must have that

$$\Tr(\log(u_t\rho(g_2)\rho(g_3)\rho(g_2g_3)^{-1}u_t^{-1}\rho(g_2g_3)\rho(g_3)^{-1}\rho(g_2)^{-1})\in2\pi i\Z.$$

Because this is a discrete space the values cannot depend on $t$. We conclude that

\begin{align*}
2\pi i\langle\rho,c\rangle&=\Tr(\log(\rho(g_2)\rho(g_3)\rho(g_2g_3)^{-1}\rho(g_2g_3)\rho(g_3)^{-1}\rho(g_2)^{-1})\\
\end{align*}

§.§ Finite Type Cohomology

Let $\Gamma$ be a countable discrete group and $[\sigma]\in H^2(\Gamma;\Z)$ be given by the central extension

1\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1.
\end{tikzcd}$$

We say that $[\sigma]$ is finite type if $\tilde{\Gamma}$ has a sequence of finite index subgroups $\{\Gamma_k\}_{k\in\N}$ so that

Clearly if $\tilde{\Gamma}$ is residually finite $[\sigma]$ is of finite type.

We may assume that the subgroups in Definition <ref> are normal and decreasing. To achieve normality replace $\Gamma_k$ with the kernel of the action of $\tilde{\Gamma}$ on $\tilde{\Gamma}/\Gamma_k$. To achieve a decreasing sequence replace $\Gamma_k$ with the cumulative intersection of $\Gamma_k$.

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 $\Gamma$ be a discrete group and let $Q$ be a finite quotient of $\Gamma$. Call $q$ the quotient map from $\Gamma$ to $Q$ and $f^n$ the canonical map from $\Z$ to $\Z/n\Z$. Then $q$ induces a cochain map $q^\#$ from $C^\bullet(Q;\Z/n\Z)$ to $C^\bullet(\Gamma;\Z/n\Z)$ and $f$ induces a map $f_\#^n$ from $C^\bullet(\Gamma;\Z)$ to $C^\bullet(\Gamma;\Z/n\Z)$. These in turn induce maps $q^*:H^*(Q;\Z/n\Z)\rightarrow H^*(\Gamma;\Z/n\Z)$ and $f^n_*:H^*(\Gamma;\Z)\rightarrow H^*(\Gamma;\Z/n\Z)$.

If $\sigma$ is a $\Z$-valued $k$-cocycle on $\Gamma$, we say that $\sigma$ is of $n$-$Q$ type if $f^n_\#(\sigma)=q^\#(\sigma')$ for some $\Z/n\Z$-valued $k$-cocycle on $Q$, $\sigma'$. We say that a cohomology class $[\sigma]\in H^k(\Gamma;\Z)$ is $n$-$Q$ type if there is a cohomology class $[\sigma']\in H^k(Q;\Z/n\Z)$ so that $f^n_*([\sigma])=q^*([\sigma'])$.

\left[\sigma'\right]\in H^k(Q;\Z/n\Z)\arrow{d}{q^*}
\\
\left[\sigma\right]\in H^k(\Gamma;\Z)\arrow{r}{f^n_*}
& H^k(\Gamma;\Z/n\Z).
\end{tikzcd}$$

Consider $\Gamma=\Z^2$ and the cocycle $\sigma((x_1,x_2),(y_1,y_2))=x_2y_2$. That this is a cocycle is easy to check:

\begin{align*}
\partial\sigma((x_1,x_2),(y_1,y_2),(z_1,z_2))&=x_2y_1-x_2(y_1+z_1)+(x_2+y_2)z_1-y_2z_1\\
\end{align*}

This is not a coboundary because

is a 2-chain such that

Then if $Q_n=(\Z/n\Z)^2$ we have that $\sigma$, and hence $[\sigma]$, is of $n$-$Q_n$ type. To show this note that the same formula used for $\sigma$ defines a 2-cochain, $\sigma'\in C^2(Q_n;\Z/n\Z)$. The same computations that show that $\sigma$ is a cocycle also show that $\sigma'$ is a cocycle, and clearly $g_\#(\sigma)=f^\#(\sigma')$.

Suppose that $\Gamma$ is a discrete group and $[\sigma]\in H^2(\Gamma;\Z)$. Let the central extension corresponding to $[\sigma]$ be as follows:

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1.
\end{tikzcd}
\end{equation*}

The following are equivalent:

* $[\sigma]$ is of finite type;

* there are infinitely many $n\in\N$ so that there is a finite quotient of $\tilde{Q}_n$ of $\Tilde{\Gamma}$ so that $\iota(1)$ has order $n$ in the quotient;

* there are infinitely many $n\in\N$ so that $\Gamma$ has a finite quotient $Q_n$ so that $[\sigma]$ is of $n$-$Q_n$ type.

(1)$\Longrightarrow$(2) Let $\Gamma_k$ be a sequence of subgroups as in Definition <ref> and assume that they are normal as in Remark <ref>. For $\ell\in\N$ pick $\Gamma_k$ so that $\iota(1)^{\ell!}\not\in\Gamma_k$. Then let $n$ be the order of $\iota(1)$ in $\Tilde{\Gamma}/\Gamma_k$. Call $\tilde{Q}_n=\Tilde{\Gamma}/\Gamma_k$. Note that $n>\ell$ so letting $\ell\rightarrow\infty$ we get the desired family of subgroups for infinitely many distinct $n\in\N$.

(2)$\Longrightarrow$(1) Let $\Gamma_n$ be the kernel of the map from $\tilde{\Gamma}$ to $\tilde{Q}_n$. Then for any $\ell\in\Z\setminus\{0\}$ we can pick $n>|\ell|$ so that $\iota(1)^\ell\not\in\Gamma_n$.

(2)$\Longrightarrow$(3) By assumption, for infinitely many $n$ we have a diagram:

\begin{equation*}\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota}\arrow{d}{f^n} &
\Tilde{\Gamma}\arrow{d}{\Tilde{q}_n} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}\arrow{d}{q_n}& 1
\\
1\arrow{r} & \Z/n\Z
\arrow{r}{} &
\Tilde{Q}_n \arrow{r}{\varphi_n'}
& Q_n\arrow{r}{}& 1
\end{tikzcd}\tag{3.1}\end{equation*}

Then we may factor the vertical maps as follows:

\begin{equation*}\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota}\arrow{d}{f^n} &
\Tilde{\Gamma}\arrow{d}{} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}\arrow{d}{\Id}& 1
\\
1\arrow{r} & \Z/n\Z\arrow{d}{\Id}
\arrow{r}{} &
\Tilde{\Gamma}/{\langle\Tilde{r}^n\rangle}\arrow{d} \arrow{r}{}
& \Gamma\arrow{r}{}\arrow{d}{q_n}& 1\\
1\arrow{r} & \Z/n\Z
\arrow{r}{} &
\Tilde{Q}_n \arrow{r}{\varphi_n'}
& Q_n\arrow{r}{}& 1
\end{tikzcd}\tag{3.2}\end{equation*}

By <cit.>, $f^n_*([\sigma])$ corresponds to the middle row of diagram (3.2). Let $[\sigma']$ be the element of $H^2(Q_n;\Z/n\Z)$ corresponding to the bottom row of diagram (3.2). Again using <cit.>  we get that $q_n^*([\sigma'])$ corresponds to the middle row as well. It follows that $f^n_*([\sigma])=q_n^*([\sigma'])$.

(3)$\Longrightarrow$(2) Using <cit.> as we did for the other direction we get that diagram (3.2) must exist for infinitely many $n$. Composing the vertical maps we get that quotients such as diagram (3.1) must exist for infinitely many $n$ as well.

This theorem motivates a definition that extends the finite type concept to cocycle representatives of the cohomology class.

If $\sigma$ is a $\Z$-valued $k$-cocycle on $\Gamma$, we say that $\sigma$ is of finite type if for infinitely many $n\in\N$ there is a finite quotient $Q_n$ of $\Gamma$ so that that $\sigma$ is of $n$-$Q_n$ type.

The cocycle defined in Example <ref> is finite type. From Proposition <ref> it follows that its cohomology class is as well.

Obviously if $\sigma$ is of $n$-$Q$ type then so is $[\sigma]$. Conversely, if $[\sigma]\in H^k(\Gamma;\Z)$ is of $n$-$Q$ type then we have $\omega\in[\sigma]$ that is of $n$-$Q$ type. To show this we can take $\dot\omega\in g_*([\sigma])$ so that $\dot\omega\in\im(f^\#)$. Let $\dot\omega=\partial\dot\alpha+g_\#(\sigma)$ where $\dot\alpha\in C^{k-1}(\Gamma;\Z/n\Z)$. Then find $\alpha\in C^{k-1}(\Gamma;\Z)$ so that $g_\#(\alpha)=\dot\alpha$. Then letting $\omega=\partial\alpha+\sigma$ we see that $g_\#(\omega)=\dot\omega\in\im(f^\#)$. Thus $\omega$ is of $n$-$Q$ type. However, a finite type cohomology class need not have a finite type representative, because the choice of representative of $[\sigma]$ might depend on $n$.

§.§ Constructing Asymptotic Representations From Cocycles

We start by defining our formula for an asymptotic representation.

Suppose that $\Gamma$ is a discrete group and $[\sigma]\in H^2(\Gamma;\Z)$. Let $Q_n$ be a finite quotient of $\Gamma$ so that $[\sigma]$ is of $n$-$Q_n$ type. Let $\sigma_n$ be a representative of $[\sigma]$ so that $\sigma_n$ is of $n$-$Q_n$ type. Let $\alpha_n$ be a 1-cochain so that $\sigma_n+\partial\alpha_n=\sigma$. Let $\hat{\chi}_n(g_1,g_2)=\exp\left(\frac{2\pi i}{n}(\alpha_n(g_1)+\sigma_n(g_1,g_2))\right)$. Then define $V_n=\ell^2(Q_n)$. Treat $\bar{g}$ as a basis element for $V_n$ where $g\in\Gamma$ and $\bar{g}$ is its image in $Q_n$. Then there is a well-defined function $\rho_n:\Gamma\rightarrow U(V_n)$ that obeys the formula

We will show that $\rho_n$ is well-defined. Suppose that $\bar{g}_2=\bar{g_2'}$. If we show that

$$\sigma_n(g_1,g_2)\equiv\sigma_n(g_1,g_2')\mod n$$

we will have shown that $\rho_n$ is well-defined because $\hat{\chi}_n$ only depends on $\sigma_n$ up to equivalence mod $n$. If $\dot\sigma_n$ is $\sigma_n$ reduced mod $n$ then we have by assumption that $\dot\sigma_n=q^\#(\sigma_n')$ where $q$ is the quotient map from $\Gamma$ to $Q$ and $\sigma_n'$ is a 2-cocycle in $C^2(Q;\Z/n\Z)$. Thus

$$\dot\sigma_n(g_1,g_2)=\sigma_n'(\bar{g}_1,\bar g_2)=\sigma_n'(\bar g_1,\bar{g_2'})=\dot{\sigma}_n(g_1,g_2').$$

Note that $\rho_n(g)$ maps the orthonormal basis $\{h:h\in Q_n\}$ to another orthonormal basis, so it is a unitary.

The existence technically only uses the fact that $\dot\sigma$ depends only on the class of second entry in $Q_n$. We will use this fact in examples to reduce the asymptotics of the dimension of the asymptotic representation.

Let $\Gamma$, $n$, $\sigma$, $Q_n$, and $\rho_n$ be as above. Define $\chi_n\in C^2(\Gamma;S^1)$ by

$$\chi_n(g_1,g_2)=\exp\left(\frac{2\pi i }{n}\sigma(g_1,g_2)\right).$$

Then $\rho_n$ obeys the formula

Define $\hat{\sigma}_n(g_1,g_2)=\alpha_n(g_1)+\sigma_n(g_1,g_2)$. We compute

\begin{align*}
\end{align*}

The second equality follows from the fact that $\sigma_n$ is a cocycle. Exponentiating both sides of (3.3) we get

\begin{align*}
\hat{\chi}_n(g_1,g_2g_3)^{-1}\hat{\chi}_n(g_1g_2,g_3)\hat{\chi}_n(g_2,g_3)^{-1}=\chi_n(g_1,g_2)^{-1}&&(3.4)
\end{align*}

Next we claim that

To check this it suffices to compute that

\begin{align*}
\rho_n(g_1)\hat{\chi}_n(g_1,g_1^{-1}g_2)\bar{g}_1^{-1}\bar{g}_2&=\hat{\chi}_n(g_1,g_1^{-1}g_2)^{-1}\rho_n(g_1)\bar{g}_1^{-1}\bar{g}_2\\
\end{align*}

Using this we can compute

\begin{align*}
\rho_n(g_1g_2)\rho_n(g_2)^{-1}\rho_n(g_1)^{-1}\Bar{g}_3&=\rho_n(g_1g_2)\rho_n(g_2)^{-1}\hat\chi_n(g_1,g_1^{-1}g_3)^{-1}\Bar{g}_1^{-1}\Bar{g}_3\\
\end{align*}

Here the last step follows from (3.4) applied to $g_1$, $g_2$ and $g_2^{-1}g_2^{-1}g_3$.

Suppose that $\Gamma$ is a countable discrete group and $[\sigma]\in H^2(\Gamma;\Z)$ is a finite type cohomology class $[\sigma]$ is not in the kernel of the map $h:H^2(\Gamma;\Z)\rightarrow\Hom(H_2(\Gamma,\Z)$. Then $\Gamma$ is not matricially stable. The sequence of functions $\rho_n$ defined in Proposition <ref> is an asymptotic representation of $\Gamma$ that cannot be perturbed to a genuine representation.

Now we will show that $\rho_n$ is asymptotically multiplicative. Noting that since $\sigma$ does not depend on $n$ we have that $\chi_n(g_1,g_2)$, defined as in Lemma <ref>, goes to 1 as $n$ goes to infinity. Thus Lemma <ref> implies asymptotic multiplicativity.

Now we will show that for large enough $n$, $\rho_n$ is not close to any genuine representation of $\Gamma$ on a particular finite subset of $\Gamma$. From condition 1 in the statement of the theorem there is some 2-cycle $c\in C_2(\Gamma)$ written

so that

$$\langle \sigma,c\rangle\ne0.$$

Then we compute that

\begin{align*}
\langle\rho_n,c\rangle&=\frac{1}{2\pi i}\sum_{j=1}^N x_j\Tr(\log(\rho_n(a_jb_j)\rho_n(b_j)^{-1}\rho_n(a_j)^{-1}))\\
&=\frac{1}{2\pi i}\sum_{j=1}^N x_j\Tr(\log(\chi_j(a_jb_j)))&\mbox{by Lemma~\ref{chiform}}\\
&=\frac{1}{2\pi i}\sum_{j=1}^N x_j\frac{2\pi i}{n}\sigma(a_j,b_j)\Tr(\Id_{V_n})\\
\end{align*}

By Theorem <ref> it follows that $\rho_n$ cannot be within $\frac{1}{24}$ of a genuine representation on the boundary support of $c$, and thus cannot be perturbed to a genuine representation.

If $\Gamma$ is finitely generated then pairing nontrivially with a cohomology class is equivalent to being non-torsion. To see this note that from the universal coefficient theorem <cit.>  we have a short exact sequence

0\arrow{r}{}&\Ext(H_1(\Gamma),\Z)\arrow{r}{}& H^2(\Gamma;\Z)\arrow{r}{}&\Hom(H_2(\Gamma),\Z)\arrow{r}{}&0
\end{tikzcd}.$$

Clearly any torsion element cannot fit the condition since $\Hom(H_2(\Gamma),\Z)$ is a torsion-free group. To see the converse note that since $\Gamma$ is finitely generated $H_1(\Gamma)\simeq\Gamma/[\Gamma,\Gamma]$ <cit.> is finitely generated as well. Thus $\Ext(H_1(\Gamma),\Z))$ can be show to be torsion from <cit.> and the table on [11] page 331.

In many examples we will have the stronger condition that there is a particular cocycle representative $\sigma$ of $[\sigma]$ that is of finite type. In this case the formula simplifies to

where $\chi_n(g_1,g_2)=\exp\left(\frac{2\pi i}{n}\sigma(g_1,g_2)\right)$.

If $\Gamma$ is a discrete group, and $[\sigma]\in H^2(\Gamma;\Z)$ with a central extension

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{} &
\Tilde{\Gamma} \arrow{r}{}
& \Gamma\arrow{r}{}& 1
\end{tikzcd}
\end{equation*}

so that $\tilde{\Gamma}$ is residually finite, and $[\sigma]$ is not in the kernel of the map $h:H^2(\Gamma;\Z)\rightarrow\Hom(H_2(\Gamma,\Z)$ then $\Gamma$ is not matricially stable. The formula for in Proposition <ref> defines an asymptotic representation for $\Gamma$ that cannot be perturbed to a genuine representation.

This follows from Theorem <ref> and Remark <ref>.

Suppose that $\Gamma$ is a virtually polycyclic group with non-torsion 2-cohomology. Then $\Gamma$ meets the conditions of Theorem <ref>, and is thus not matricially stable.

First we note that if $[\sigma]$ is a non-torsion cohomology class it is not in the kernel of the map from $H^2(\Gamma;\Z)$ to $\Hom(H_2(\Gamma,\Z)$ by Remark <ref>. Let

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1
\end{tikzcd}
\end{equation*}

be the central extension corresponding to $[\sigma]$. We have that $\Tilde{\Gamma}$ is also virtually polycyclic by Proposition <ref>. Now because $\tilde{\Gamma}$ is virtually polycyclic it is residually finite <cit.>. Thus $[\sigma]$ is of finite type.

§ TORSION-FREE FINITELY GENERATED NILPOTENT GROUPS

The point of this section is to provide an alternate proof that torsion-free finitely generated nilpotent groups fit the conditions of Theorem <ref> (Theorem <ref>). While this follows from Corollary <ref> the alternate proof gives rise to a simple formula for the asymptotic representation (Remark <ref>).

Suppose that $\Gamma$ is a torsion-free finitely generated nilpotent group with a Mal'cev basis $(a_1,\ldots, a_m)$ and a central extension as follows:

$$1\begin{tikzcd}\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1.
\end{tikzcd}$$

Then if $\Tilde{a}_i$ is a lift of $a_i$ for $i\in\{1,\ldots,m\}$ and $\Tilde{a}_{m+1}=\iota(1)$, $(\Tilde{a}_1,\ldots,\Tilde{a}_{m+1})$ is a Mal'cev basis for $\Tilde{\Gamma}$.

Let $\theta$ be the set theoretic section for $\varphi$ defined by $\theta(a_1^{x_1}\cdots a_m^{x_m})=\Tilde{a}_1^{x_1}\cdots\Tilde{a}_m^{x_m}$. First we claim that any element $g\in\Tilde{\Gamma}$ can be written in the form $\Tilde{a}_1^{x_1}\cdots\Tilde{a}_m^{x_m}\cdot\Tilde{a}_{m+1}^{x_{m+1}}$. To see this note that $g=(\theta\circ\varphi(g))\Tilde{a}_{m+1}^{x_{m+1}}$. Then the claim follows from the definition of $\theta$. Next to show that this is unique we suppose that $\Tilde{a}_1^{x_1}\cdots \Tilde{a}_{m+1}^{x_{m+1}}=\Tilde{a}_1^{y_1}\cdots\Tilde{a}_{m+1}^{y_{m+1}}$. Noting that

$$a_1^{x_1}\cdots a_m^{x_m}=\varphi(\Tilde{a}_1^{x_1}\cdots \Tilde{a}_{m+1}^{x_{m+1}})=\varphi(\Tilde{a}_1^{y_1}\cdots \Tilde{a}_{m+1}^{y_{m+1}})=a_1^{y_1}\cdots a_m^{y_m}$$

we get that $x_i=y_i$ for $i\ne m+1$. Then equality for $i=m+1$ follows canceling the other terms and noting that $\Tilde{a}_{m+1}$ is non torsion. Next we define $\Tilde{\Gamma}_i=\langle \Tilde a_i,\ldots, \Tilde a_{m+1}\rangle$ and similarly $\Gamma_i=\langle a_i,\ldots ,a_m\rangle$. By our assumptions $\Gamma_i$ form a central series for $\Gamma$. Note that for $i\le m$

\begin{align*}
\end{align*}

For $i=m+1$ we have $[\Tilde{\Gamma},\Tilde{\Gamma}_{m+1}]=\{1\}=\Tilde{\Gamma}_{m+2}$. This shows that $\Tilde{\Gamma}_i$ is a central series which completes our proof.

Suppose that $\Gamma$ is a torsion-free finitely generated nilpotent group that is not $\Z$ or trivial. Then $\Gamma$ has a cocycle that meets the conditions of Theorem <ref>.

The rational 2-cohomology of $\Gamma$ is nontrivial by Theorem <ref>. By the universal coefficient theorem <cit.> we have a sequence

0\arrow{r}{}&\Ext(H_1(\Gamma),\Q)\arrow{r}{}& H^2(\Gamma;\Q)\arrow{r}{}&\Hom(H_2(\Gamma),\Q)\arrow{r}{}&0
\end{tikzcd}.$$

Since $\Hom(
\bullet,\Q)$ is exact $\Ext(H_1(\Gamma),\Q))\simeq\{0\}$. From this it follows that $H_2(\Gamma)$ is non-torsion, so $\Hom(H_2(\Gamma),\Z)$ is nonzero. In particular there must be a cocycle $[\sigma]$ that is not in the kernel of the map from $H^2(\Gamma;\Z)$ to $\Hom(H_2(\Gamma),\Z)$. By <cit.> we have a central extension corresponding to $[\sigma]$:

1\arrow{r} & \Z
\arrow{r}{} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1.
\end{tikzcd}$$

By Proposition <ref> we have a Mal'cev basis $(\Tilde{a}_1,\ldots,\Tilde{a}_{m+1})$ for $\tilde\Gamma$ and a Mal'cev basis $(a_i,\ldots a_m)$ for $\Gamma$ where for $i\ne m+1$ we have $\varphi(\tilde{a}_i)=a_i$. Any element in $\Tilde{\Gamma}$ can be written uniquely as $\Tilde a_1^{x_1}\cdots \tilde a_{m+1}^{x_{m+1}}$. It follows that

$$\Tilde a_1^{x_1}\cdots \Tilde a_{m+1}^{x_{m+1}}\cdot \tilde a_1^{y_1}\cdots  \tilde a_{m+1}^{y_{m+1}}=\tilde a_1^{z_1}\cdots \tilde{a}_{m+1}^{z_{m+1}}$$

where $z_i$ is a function of $x_1,\ldots, x_{m+1},y_1,\ldots y_{m+1}$. Hall has shown that these functions are rational[The polynomials will always take integer values if given integer inputs, but in general the coefficients may not be integers.] polynomials <cit.>. They can be computed by methods here [5]. Then if we pick a section $\theta:a_1^{x_1}\cdots a_m^{x_m}\mapsto\tilde{a}_1^{x_1}\cdots\tilde{a}_m^{x_m}$ by <cit.> there is a cocycle $\sigma'$ with $[\sigma]=[\sigma']$ so that

Since $\theta(gh)$ is in canonical form and has no power of $\tilde a_{m+1}$ it follows that $\sigma'(g,h)$ must be a rational polynomial of the powers in the canonical forms of $g$ and $h$. Note that by writing elements of $\Gamma$ in canonical form we get that $\Gamma$ 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 $\Gamma$ and the denominator of $\sigma'$ we may define a quotient $Q_n$ of $\Gamma$ by reducing each entry of $\Gamma$ mod $n$. Then we may also reduce the formula for $\sigma'$ mod $n$ showing that $\sigma'$ is of $n$-$Q_n$ type.

Note that we actually develop an explicit cocycle which is of finite type. Thus we may use the simpler formula in Remark <ref> to compute an asymptotic representation for $\Gamma$.

The explicit asymptotic representation we have constructed looks like this: $\Gamma$ can be viewed as $\Z^m$ with the multiplication as follows

where $\eta_1,\ldots,\eta_n$ are rational polynomials. In addition we have a non-torsion cocycle $\sigma(\mathbf{x,y})$ that is also a rational polynomial in the entries. Then the underlying vector space is $\ell^2(Q_n)\simeq(\C^n)^{\tensor m}$. In this view we have

$$\rho_n(\mathbf{x})e_{y_1}\tensor e_{y_2}\tensor\cdots\tensor e_{y_m}=\exp\left(\frac{2\pi i}{n}\sigma(\mathbf{x,y})\right)e_{\eta_1(\mathbf{x,y})}\tensor\cdots\tensor e_{\eta_m(\mathbf{x,y})}$$

where we have the convention that $e_{i+n}=e_i$.

Additionally we may use this construction to get an asymptotic representation of dimension $n^{m-1}$ instead of dimension $n^m$. Note that the power of $\tilde a_{m+1}$ in the product $(\tilde a_1^{x_1}\cdots \tilde a_m^{x_m})(\tilde a_1^{y_1}\cdots\tilde a_m^{y_m})$ can be computed by using the relations to put all terms in order. If we leave the $\tilde a_m^{y_m}$ term at the end until the last step, we notice that we may have to switch the positions of the $\tilde a_m$ and $\tilde a_{m+1}$ but these commute by because the extension is central. From this it follows that $\sigma$ does not depend on $y_m$. Thus we may replace our $Q_n$ with $Q_n'=Q_n/\langle q_n(a_m)\rangle$ and by Remark <ref> we may use the formula for $\rho_n$ except with $Q_n'$ instead of $Q_n$. The rest of the proof for asymptotic multiplicativity and non-perturbability flows the same way.

§ GENERATORS AND RELATIONS

In this section we will show how to recognize conditions of Theorem <ref> 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 [16], [9], and [4].

Note that there must be some cocycle that satisfies the first condition in Theorem <ref> if the 2-homology group is non-torsion. Note that because of Theorem <ref> 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.

Let $\Gamma=\langle g_1,\ldots, g_m|r_1,\ldots\rangle$. By convention we will let $\mathbb{F}_m=\langle\gamma_1,\ldots,\gamma_m\rangle$.

We say the relation $r_1$ is homogeneous if $r_1\in[\F_m,\F_m]$. We say that powers of $r_1$ are centrally irredundant relative to $r_2,\ldots$ if $r_1^n$ is not in the normal subgroup of $\mathbb{F}_m$ generated by $r_2,\ldots$ and $ [\gamma_1,r_1],\ldots,[\gamma_m,r_1]$ for any $n\in\Z^+$.

If $\Z^2$ is written as $\langle a,b|aba^{-1}b^{-1}=1\rangle$ 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 $H_2(\Gamma,\Z)$ by Theorem <ref>.

If $\Gamma=\langle g_1,\ldots, g_m|r_1,\ldots\rangle$ where $r_1$ is homogeneous and powers of $r_1$ are centrally irredundant relative to the other relations then there is a central extension

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1
\end{tikzcd}
\end{equation*}

by taking

$$\Tilde{\Gamma}=\langle \Tilde{g}_1,\ldots, \Tilde{g}_m|[\Tilde{g}_1,r_1],\ldots,[\Tilde{g}_m,r_1], r_2,r_3,\ldots\rangle$$

and mapping $\iota:1\mapsto \Tilde{r}_1\in\Tilde{\Gamma}$ and $\varphi:\Tilde{g}_i\mapsto g_i$ where $\Tilde{r}_1$ is the image of $r_1$ in the map from $\F_m$ to $\Tilde{\Gamma}$ defined by mapping $\gamma_i\mapsto\Tilde{g}_i$.

The map $\iota$ is injective because powers of $r_1$ are centrally irredundant.

Because $r_1$ is one of the defining relations of $\Gamma$, we have $\im(\iota)\subseteq\ker(\varphi)$.

Note that $\im\iota$ is central in $\Tilde{\Gamma}$ by the construction of $\Tilde{\Gamma}$.

The map $\varphi$ is well defined because every relationship defining $\Tilde{\Gamma}$ is true of the generators of $\Gamma$ as well.

If $w$ is some word in the generators that is in $\ker(\varphi)$ then $w$ pulls back to some word in $\F_m$ that is in the smallest normal subgroup containing $r_1,\ldots$. Since every other relation is trivial in $\Tilde{\Gamma}$, this means $w$ is in the smallest normal subgroup containing $\tilde{r}_1$. However since $\tilde{r}_1$ is central, this group is precisely $\im(\iota)$.

The map $\varphi$ is surjective because it hits every generator of $\Gamma$.

We call the central extension in Proposition <ref> the dual extension of $r_1$ relative to the other relations.

If $\Z^2\simeq\langle a,b|[a,b]=1\rangle$ is given the same presentation as in Example <ref> then the dual extension to the relation $aba^{-1}b^{-1}$ can be expressed as the group

$$H=\langle a,b,c|c=[a,b],[a,c]=[b,c]=1\rangle$$

commonly known as the discrete Heisenberg group. The map $\iota$ is given by $\iota(1)=c$ and $\varphi$ is determined by $\varphi(a)=a$ and $\varphi(b)=b$

Suppose that $r_1$ is a homogeneous relation on a group $\Gamma=\langle g_1,\ldots, g_m|r_1,\ldots\rangle$ so that powers of $r_1$ are centrally irredundant relative to the other relations. If $[\sigma]$ 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 $\langle\sigma,c\rangle=1$.

Pick a section $\theta$ of the extension so that $\theta(1)=1$. Then $\sigma$ is defined by the formula

\[\Tilde{r}_1^{-\sigma(a,b)}=\theta(ab)\theta(b)^{-1}\theta(a)^{-1}.\tag{see \cite[Chapter IV.3 equation 3.3.]{coho}}\]

To compute the 2-homology class corresponding to $r_1$ we need write $r_1$ as

then by <cit.> the class corresponding to $r_1$ can be written as

\[c=\sum_{i=1}^N([I_{i-1}|\dot a_i]+[I_{i-1}\dot a_i|\dot b_i]-[I_{i-1}\dot a_i\dot b_i\dot a_i^{-1}|\dot a_i]-[I_i|\dot b_i])\tag{5.1}\]

where $\dot a_i$ and $\dot b_i$ are the images in $\Gamma$ of ${a}_i$ and ${b}_i$ and $I_i=[\dot a_1,\dot b_1]\cdots[\dot a_i,\dot b_i]$. Call $\Tilde{a}_i=\theta(\dot a_i)$ and $\Tilde{b}_i=\theta(\dot b_i)$. Here $\Tilde{a}_i$ and $\Tilde{b}_i$ must be the image of ${a}_i$ and ${b}_i$ in the map from $\F_m$ to $\Tilde{\Gamma}$ times some power of $\Tilde{r}_1$. Since $\Tilde{r}_1$ is central in $\Tilde{\Gamma}$ this power can be ignored while computing $[\Tilde{a}_i,\Tilde{b}_i]$. Thus

Note that

$$I_{i-1}\dot{a}_i\dot b_i\dot a_i^{-1}=I_{i-1}\dot a_i\dot b_i\dot a_i^{-1}\dot b_i^{-1}b_i=I_i\dot b_i.$$

Let $c_i$ be the $i$th term in the sum of equation (5.1). We get

\begin{align*}
\Tilde{r}_1^{-\langle\sigma,c_i\rangle}=&\theta(I_i)\theta(\dot b_i)\theta(I_i\dot b_i)^{-1}\\
&\cdot\theta(I_{i-1}\dot a_i\dot b_i\dot a_i^{-1})\theta(\dot a_i)\theta(I_{i-1}\dot a_i\dot b_i)^{-1}\\
&\cdot\theta(I_{i-1}\dot a_i\dot b_i)\theta(\dot b_i)^{-1}\theta(I_{i-1}\dot a_i)^{-1}\\
&\cdot\theta(I_{i-1}\dot a_i)\theta(\dot a_i)^{-1}\theta(I_{i-1})^{-1}\\
\end{align*}

Then multiplying these in reverse order, starting from $i=N$ and ending at $i=1$ we get

\begin{align*}
\Tilde{r}_1^{-\langle\sigma,c\rangle}&=\theta(I_N)\left(\prod_{i=0}^{N-1}[\Tilde{b}_{N-i},\Tilde{a}_{N-i}]\right)\theta(I_0)^{-1}\\
\end{align*}

Here the last equality is because $I_0=I_N=1$ since $I_N$ is the relation $r_1$.

Suppose that $r_1$ is a homogeneous relation on a group $\Gamma=\langle g_1,\ldots, g_m|r_1,\ldots\rangle$ so that powers of $r_1$ are centrally irredundant relative to the other relations. If

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1
\end{tikzcd}
\end{equation*}

is the dual extension of $r_1$ relative to this presentation and there is a family of finite index subgroups $\Gamma_k\le\tilde\Gamma$ so that $\iota(\Z)\cap\bigcap_{k}\Gamma_k=\{1\}$. Then $\Gamma$ meets the conditions of Theorem <ref>.

This follows immediately from Proposition <ref>.

§ MOTIVATING OUR FORMULA

The purpose of this section is to explain the motivation behind the formula in Proposition <ref>. For clarity recall the statement of the proposition:

Suppose that $\Gamma$ is a discrete group and $[\sigma]\in H^2(\Gamma;\Z)$. Let $Q_n$ be a finite quotient of $\Gamma$ so that $[\sigma]$ is of $n$-$Q_n$ type. Let $\sigma_n$ be a representative of $[\sigma]$ so that $\sigma_n$ is of $n$-$Q_n$ type. Let $\alpha_n$ be a 1-cochain so that $\sigma_n+\partial\alpha_n=\sigma$. Let $\hat{\chi}_n(g_1,g_2)=\exp\left(\frac{2\pi i}{n}(\alpha_n(g_1)+\sigma_n(g_1,g_2))\right)$. Then define $V_n=\ell^2(Q_n)$. Treat $\bar{g}$ as a basis element for $V_n$ where $g\in\Gamma$ and $\bar{g}$ is its image in $Q_n$. Then there is a well-defined function $\rho_n:\Gamma\rightarrow U(V_n)$ that obeys the formula

In terms of central extensions this means we have the following

\begin{equation*}\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota}\arrow{d}{f^n} &
\Tilde{\Gamma}\arrow{d}{\Tilde{q}_n} \arrow[shift left]{r}{\varphi}
& \Gamma\arrow{r}{}\arrow{d}{q_n}\arrow[shift left]{l}{\theta}& 1
\\
1\arrow{r} & \Z/n\Z
\arrow{r}{\iota'} &
\Tilde{Q}_n \arrow[shift left]{r}{\varphi_n'}
& Q_n\arrow{r}{}\arrow[shift left]{l}{\theta_n'}& 1
\end{tikzcd}\tag{6.1}\end{equation*}

Here we are taking $\theta$ to be the section from $\Gamma$ to $\tilde{\Gamma}$ corresponding to the cocycle $\sigma$ and $\theta_n'$ to be a section from $Q_n$ to $\tilde Q_n$. Neither is a group homomorphism. Consider the character on $\Z/n\Z$ defined by $1\mapsto\exp\left(\frac{2\pi i}{n}\right)$. Then if $\pi_n$ is the representation induced by this character and the inclusion $\iota'$ we will show that $\rho_n=\pi_n\circ\tilde q_n\circ \theta$. We may view this induced representation as the left regular representation of $Q_n$ “twisted” by the cocycle $\sigma_n$ which corresponds to a $\Z/n\Z$ cocycle on $Q_n$ because $\sigma_n$ is of $n$-$Q_n$ type. The $\alpha_n$ 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 $\theta$ to commute with the horizontal maps in this way, because if we did $\theta$ would need to depend on $n$ and our argument for asymptotic multiplicativity would not hold. The following theorem summarizes this result.

Suppose that $\Gamma$ meets the conditions of Theorem <ref>. 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 $[\sigma]$:

\begin{equation*}
\begin{tikzcd}
1\arrow{r} & \Z
\arrow{r}{\iota} &
\Tilde{\Gamma} \arrow{r}{\varphi}
& \Gamma\arrow{r}{}& 1.
\end{tikzcd}
\end{equation*}

Let $S$ be the set of $n\in\N$ so that $\tilde\Gamma$ has a finite quotient $\tilde Q_n$ where $\iota(1)$ has order $n$. Henceforth assume $n\in S$. Let $\theta$ be a set theoretic section from $\Gamma$ to $\Tilde{\Gamma}$, and let $\tilde{q}_n$ be the quotient map from $\Tilde{\Gamma}$ to $\Tilde{Q}_n$. If $\pi_n$ is the induced representation on $\Tilde{Q}_n$ by the character on $\langle \tilde{q}_n(\iota(1))\rangle$ defined by $\tilde{q}_n((\iota(1))\mapsto\exp\left(\frac{2\pi i}{n}\right)$. Then $\rho_n=\pi_n\circ \tilde q_n\circ\theta$ equivalent to the formula given in Proposition <ref>.

Then let $\omega_n$ be the character on $\langle \tilde{q}_n(\iota(1))\rangle$ defined by $\tilde{q}_n(\iota(1))\mapsto \exp\left(\frac{2\pi i}{n}\right)$. By <cit.>, one construction for the induced representation comes from the space

$$\bigoplus_{i=1}^n g_i\C\simeq V_n$$

where one element $g_i$ is a representative of a different one of the $n$ cosets in $\tilde{Q}_n/\langle\iota(1)\rangle$ for each $i$. Here $V_n$ is the space as described in Proposition <ref>. Then the induced representation is defined by

where $g_j$ and $k$ are determined by the formula $gg_i=g_j\Tilde{q}_n(\iota(1))^k$. This determines $g_j$ uniquely and $k$ up to equivalence mod $n$. If we pick $\theta_n'$ to be the map taking an element in $Q_n$ to the $g_i$ in the coset corresponding to it we get that $g_j=\theta_n'(\varphi_n'(gg_i))$ and $\tilde{q}_n(\iota(1))^k=gg_i\theta_n'(\varphi_n'(gg_i))^{-1}$. This gives us

Then we may take $\theta_n$ to be another section from $\Gamma$ to $\Tilde{\Gamma}$ defined so that $\tilde{q}_n\circ\theta_n=\theta_n'\circ q_n$. To show that such a section exists note that for all $g\in\Gamma$ $\tilde{q}_n\circ\theta(g)=\tilde q_n\circ\iota(1)^k\theta_n'\circ q_n(g)$ for some $k\in\Z$. We then define $\theta_n(g)=\theta(g)\iota(1)^{-k}$. We define the $\Gamma$ cocycles $\sigma$, $\sigma_n$ and the $\Gamma$ 1-cochain $\alpha_n$ as follows:

\begin{align*}
\iota(1)^{-\sigma(g,h)}&=\theta(gh)\theta(h)^{-1}\theta(g)^{-1}\\
\iota(1)^{-\sigma_n(g,h)}&=\theta_n(gh)\theta_n(h)^{-1}\theta_n(g)^{-1}\\
\iota(1)^{-\alpha_n(g)}&=\theta_n(g)\theta(g)^{-1}.
\end{align*}

We claim that $\sigma_n$ is of $n$-$Q_n$ type. To show this

\begin{align*}
\tilde{q}_n(\iota(1))^{\sigma_n(g,h)}&=\tilde{q}_n(\theta_n(gh)\theta_n(h)^{-1}\theta_n(g)^{-1})\\
\end{align*}

Since the order of $\tilde{q}_n(\iota(1))$ is $n$ it follows that $\sigma_n(g,h)$ reduced mod $n$ can be computed the same way as $q_n^\#$ applied to the cocycle corresponding to $\theta_n'$. Then note that by <cit.> we have that $\sigma_n=\sigma+\partial\alpha_n$. Note that $V_n$ is isomorphic to the vector space for $\pi_n$ by mapping

So we have

\pi_n\circ\Tilde{q}_n\circ\theta(g)\cdot\theta_n'(h)=\omega_n\Big(\Tilde{q}_n\circ\theta(g)\cdot\theta_n'(h)\cdot\big(\theta_n'\circ\varphi_n'(\Tilde{q}_n\circ\theta(g)\cdot\theta_n'(h))\big)^{-1}\Big)\cdot\theta_n'\circ\varphi_n'\Big(\Tilde{q}_n\circ\theta(g)\cdot\theta_n'(h)\Big).$$

To simplify this expression we note first that

\begin{align*}
\varphi_n'\Big(\Tilde{q}_n\circ\theta(g))\cdot\theta_n'(h)\Big)&=\varphi_n'\circ\Tilde{q}_n\circ\theta(g)\cdot\varphi_n'\circ\theta_n'(h)\\
&=q_n\circ\varphi\circ\theta(g)\cdot h\\
&=q_n(g)\cdot h.
\end{align*}

\begin{align*}
\Tilde{q}_n\circ\theta(g)&=\Tilde{q}_n(\iota(1)^{\alpha_n(g)}\cdot\theta_n(g))\\
&=\Tilde{q}_n(\iota(1)^{\alpha_n(g)})\cdot\theta_n'\circ q_n(g).
\end{align*}

This gives us the formula

$$\pi_n\circ\Tilde{q}_n\circ\theta(g)\cdot\theta_n'(h)=\omega_n\Big(\Tilde{q}_n(\iota(1))^{\alpha_n(g)}\theta_n'(q_n(g))\cdot\theta_n'(h)\cdot\theta_n'(q_n(g)h)^{-1}\Big)\cdot \theta_n'\big(q_n(g)\cdot h\big).$$

Supposing that $h=q_n(a)$ we can rewrite this as

\begin{align*}
\pi_n\circ\Tilde{q}_n\circ\theta(g)\cdot\theta_n'\circ q_n(a)&=\omega_n\Big(\Tilde{q}_n(\iota(1))^{\alpha_n(g)}\theta_n'\circ q_n(g)\cdot\theta_n'\circ q_n(a)\cdot\theta_n'\circ q_n(ga)^{-1}\Big)\cdot\theta_n'\circ q_n(ga)
\\
&=\omega_n\Big(\tilde{q}_n(\iota(1)^{\alpha_n(g)})\tilde{q}_n\big(\theta_n(g)\cdot\theta_n(a)\cdot\theta_n(ga)^{-1}\big)\Big)\cdot\theta_n'\circ q_n(ga)\\
&=\omega_n(\tilde{q}_n(\iota(1)^{\alpha_n(g)}\iota(1)^{\sigma_n(g,a)})\cdot\theta_n'\circ q_n(ga)\\
&=\exp\left(\frac1n(\alpha_n(g)+\sigma_n(g,a))\right)\cdot\theta_n'\circ q_n(ga).
\end{align*}

In the notation of Proposition <ref> the $q_n(ga)$ would be written in the form $\bar{g}\bar{a}$ and $\bar{g}\mapsto\theta_n'(g)$ describes the isomorphism of $V_n$ with the vector space on which the induced representation is defined. Thus this recovers the formula from Proposition <ref>.

§ EXAMPLES

§.§ $\Z^2$ 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 <ref> we obtain Voiculescu's matrices precisely. In Example <ref> we have shown that

is a cocycle on $\Z^2$ and

is a 2-chain such that

Moreover, since $\sigma$ is a polynomial with integer coefficients it follows that $\sigma$ is $\Z/n\Z$-compatible with $(\Z/n\Z)^2$. Then applying Theorem <ref> we get that $\rho_n$ acts on $V_n=\ell^2((\Z/n\Z)^2)\simeq\C^n\tensor\C^n$. Then pick the basis $\{e_j\tensor e_k\}$ with $e_j$ defined for all $j\in\Z$ by the formula $e_{j+n}=e_j$. Using the formula for Theorem <ref> we get that

$$\rho_n(a,b)e_j\tensor e_k=\exp\left(\frac{2\pi i}{n}bj\right)e_{a+j}\tensor e_{b+k}.$$

Note that we may write $\rho_n(a,b)=\rho_n^1(a,b)\tensor\rho_n^2(a,b)$ where

$$\rho_n^1(a,b)e_j=\exp\left(\frac{2\pi i}{n}bj\right)e_{a+j}$$

Note that $\rho_n^1$ is precisely the asymptotic representation given by Voiculescu's matrices while $\rho_n^2$ is a genuine representation. This is unsurprising because Remark <ref> allows us to reduce the dimension.

§.§ A 3-Step Nilpotent Group

Consider the group $\Gamma$ generated by $a_1,\ldots ,a_5$ with the following relations

\begin{align*}
a_ia_j&=a_ja_i&\mbox{for all other pairs $\{i,j\}$.}
\end{align*}

Our asymptotic representation is defined for $n$ co-prime to 6 and sends generators to the following $\C^n$ spanned $e_i$ with $i\in\Z$ and $e_{i+n}=e_i$:

\begin{align*}
\rho_n'(a_1)e_j&=e_{j+1}\\
\rho_n'(a_2)e_j&=\exp\left(\frac{4\pi i}{n}{j\choose 3}\right)e_j\\
\rho_n'(a_3)e_j&=\exp\left(\frac{4\pi i}{n}{j\choose 2}\right)e_j\\
\rho_n'(a_4)e_j&=\exp\left(\frac{2\pi i}{n}j\right)e_j\\
\rho_n'(a_5)&=\Id_{\C^n}.
\end{align*}

And sends the element written uniquely in the form $a_1^{x_1}\cdots a_5^{x_5}\mapsto\rho_n'(a_1)^{x_1}\cdots\rho_n'(a_5)^{x_5}$. Here ${j\choose 3}$ is the polynomial $\frac16j(j-1)(j-2)$. This group can be concretely realized as $\Z^5$ with multiplication given by

$$(x_1,\ldots, x_5)*(y_1,\ldots y_5)=(\eta_1(\mathbf{x,y}),\ldots,\eta_5(\mathbf{x,y}))$$

\begin{align*}
\eta_1(\mathbf{x,y})&=x_1+y_1\\
\eta_2(\mathbf{x,y})&=x_2+y_2\\
\eta_3(\mathbf{x,y})&=x_3+y_3+x_2y_1\\
\eta_4(\mathbf{x,y})&=x_4+y_4+2x_3y_1+2x_2{y_1\choose 2}\\
\eta_5(\mathbf{x,y})&=x_5+y_5+y_1{x_2\choose 2}+x_3y_2+x_2y_1y_2
\end{align*}

by the isomorphism
$a_1^{x_1}\cdots a_5^{x_5}\mapsto(x_1,\ldots,x_5)$. We have verified the fact that these polynomials lead to an associative operation obeying the relations of the group by computer and the fact that this is an isomorphism follows from uniqueness of canonical form. In general these polynomials may be calculated by methods given in [5].

In order to compute a non-torsion cocycle we will develop one as a central extension as in Proposition <ref>. We will do this by computing the dual extension to the relation $[a_4,a_1]=1$ relative to the to the other relations. Thus we get a group $\tilde{\Gamma}$ generated by $\tilde{a}_1,\ldots,\tilde{a}_6$ with the relations

\begin{align*}
\tilde a_2\tilde a_1&=\tilde a_1\tilde a_2a_3\\
\tilde a_3\tilde a_1&=\tilde a_1va_3\tilde a_4^{2}\\
\tilde a_3\tilde a_2&=\tilde a_2\tilde a_3\tilde a_5\\
\tilde a_4\tilde a_1&=\tilde a_1\tilde a_4\tilde a_6\\
\tilde a_i\tilde a_j&=\tilde a_j\tilde a_i&\mbox{for all other pairs $\{i,j\}$.}
\end{align*}

The reader is warned that such an extension cannot be made for any homogeneous relation in any torsion-free finitely generated nilpotent group. Consider the group $\Lambda$ generated by $b_1,\ldots, b_4$ with the relations

\begin{align*}
[b_i,b_j]&=1&\mbox{for all other pairs $\{i,j\}$.}
\end{align*}

In this case the relation $[b_3,b_4]=1$ follows from the other three relations so it is not centrally irredundant relative to the other relations. We have checked by computer that such a problem does not arise from the $\tilde{\Gamma}$ by computing the multiplication polynomials and checking that they do in fact form a group where $\tilde a_6$ has infinite order. 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 <cit.>.

Then $\Tilde{\Gamma}$ can be identified with $\Z^6$ with the multiplication is given by

\begin{align*}
\gamma_i(\mathbf{x,y})&=\eta_i(x_1,\ldots x_5,y_1,\ldots y_5)&\mbox{for }i<6\\
\gamma_6(\mathbf{x,y})&=x_6+y_6+x_4y_1+2x_3{y_1\choose 2}+2x_2{y_1\choose 3}
\end{align*}

From this it follows that a cocycle corresponding to the central extension is given by

$$\sigma(\mathbf{x,y})=x_4y_1+2x_3{y_1\choose 2}+2x_2{y_1\choose 3}.$$

Then for any $n$ co-prime to 6 we may define

with multiplication given by

$$(x_1,\ldots, x_n)*(y_1,\ldots,y_n)=(\bar\eta_1(\mathbf{x,y}),\ldots,\bar\eta_5(\mathbf{x,y}))$$

where $\bar\eta_i$ is $\eta_i$ with each coefficient reduced mod $n$.[Any $\frac12$ coefficient will be the corresponding inverse of 2 mod $n$ which exists since $n$ is co-prime to 6.] Then reducing each coefficient of $\sigma$ mod $n$ we get $\bar{\sigma}$. Note that the fact that $\bar{\sigma}$ is a $\Z/n\Z$-valued cocycle on $Q_n$ implies that $\sigma$ is of $Q_n$-$n$ type. Then we can use the formula from Remark <ref> to define our asymptotic representation. Note that $\ell^2(Q_n)\simeq\ell^2(\Z/n\Z)^{\tensor 5}.$ Treat $\{e_i\}_{i=0,\ldots,n-1}$ as a basis for $\ell^2(\Z/n\Z)$. For ease of notation we will extend $e_i$ to be well-defined for all $i\in\Z$ by the formula $e_{i+n}=e_i$. Thus we get

$$\rho_n(x_1,\ldots,x_5)e_{y_1}\tensor e_{y_2}\tensor\cdots\tensor e_{y_5}=\exp\left(\frac{2\pi i\sigma(\mathbf{x,y})}{n}\right)e_{\eta_1(\mathbf{x,y})}\tensor e_{\eta_2(\mathbf{x,y})}\tensor\cdots\tensor e_{\eta_5(\mathbf{x,y})}.$$

Theorem <ref> guarantees that this formula is well-defined. As in Remark <ref> we can “ignore” the last tensor coordinate since $\sigma(\mathbf{x,y})$ does not depend on $y_5$ and neither does $\eta_i$ 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 $\sigma$ depends only on $\mathbf{x}$ and $y_1$ so we can ignore every tensor coordinate except the first. This gives us the asymptotic representation we started with.

§.§ A Polycyclic Group

Let $\Gamma=\Z^2\rtimes\Z$ where the action of $\Z$ on $\Z^2$ is given by “Arnold's Cat Map:”

$$1\mapsto T=\begin{bmatrix}2&1\\1&1\end{bmatrix}\in\Gl_2(\Z)=\Aut(\Z^2).$$

We will compute a non-torsion cocycle in $H^2(\Gamma;\Z)$. 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 $\Gamma$, compute the multiplication in the central extension. We may consider the following presentation of $\Gamma$. The generators are $a_1,a_2,a_3$
with the relations

\begin{align*}
\end{align*}

Or asymptotic representation is given on $\C^n\tensor\C^n$ with basis $\{e_j\tensor e_k\}_{j,k\in\Z}$ and the convention that $e_{j+n}=e_j$. With this notation the generators are sent to the following operators

\begin{align*}
\rho_n'(a_1)e_j\tensor e_k&=e_{j+1}\tensor e_k\\
\rho_n'(a_2)e_j\tensor e_k&=\exp\left(\frac{2\pi i}{n}j\right)e_j\tensor e_{k+1}\\
\rho_n'(a_3)e_j\tensor e_k&=\exp\left(\frac{2\pi i}{n}\left(jk+j^2+\frac12k^2-j-\frac12k\right)\right)e_{2j+k}\tensor e_{j+k}.
\end{align*}

Each element in the group can be written uniquely as $a_1^{x_1}a_2^{x_2}a_3^{x_3}$ and this element will be sent to the corresponding product of matrices.

We may make an extension by “blowing up” the relation $[a_1,a_2]=1$. Then we may consider $\Tilde{\Gamma}$ to be the group generated by $\tilde{a}_1,\ldots,\tilde{a}_1$ and the relations

\begin{align*}
\tilde a_2\tilde a_1&=\tilde a_4\\
\tilde a_3\tilde a_1&=\tilde a_1^2\tilde a_2\tilde a_3\\
\tilde a_3 \tilde a_2&=\tilde a_1\tilde a_2\tilde a_3\\
\tilde a_4\tilde a_i&=\tilde a_i\tilde a_4&\mbox{for all $i$.}
\end{align*}

Using these relations we may write any element of $\Tilde{\Gamma}$ uniquely in the form $\tilde a_1^{x_1}\cdots\tilde a_4^{x_4}$ with $x_i\in\Z$. As the reader is warned in Remark <ref> 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 $\Gamma$ implicitly by a pair $(v,k)$ with $v=(v^1,v^2)\in\Z^2$ and $k\in\Z$ 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 $\Tilde{\Gamma}$ similarly. To that end we will describe an element of $\Tilde{\Gamma}$ as a triple $(v,k,d)$ with $v=(v^1,v^2)\in\Z^2$ and $k,d\in\Z$. This represents the element $\tilde a_1^{v^1}\tilde a_2^{v^2}\tilde a_3^{k}\tilde a_4^{d}$. To that end we make the following observations:

\begin{align*}
\tilde{a}_2^{v^2}\tilde a_1^{v^1}&=\tilde a_1^{v^1}\tilde a_2^{v^2}\tilde a_3^{v^1v^2}&(7.1)\\
\tilde a_3a_1^{v^1}&=(\tilde a_1^2\tilde a_2)^{v^1}\tilde a_3\\
&=\tilde a_1^{2v^1}\tilde a_2^{v^1}\tilde a_3\tilde a_4^{v^1(v^1-1)}&(7.2)\\
\tilde a_3a_2^{v^2}&=(\tilde a_1\tilde a_2)^{v^2}\tilde a_3\\
&=\tilde a_1^{v^2}\tilde a_2^{v^2}\tilde a_3\tilde a_4^{\frac12v^2(v^2-1)}&(7.3)\\
\tilde a_3\tilde a_1^{v^1}\tilde a_2^{v^2}&=\tilde a_1^{2v^1}\tilde a_2^{v^1}\tilde a_1^{v^2}\tilde a_2^{v^2}\tilde a_3a_4^{v^1(v^1-1)+\frac12v^2(v^2-1)}\\
&=\tilde a_1^{2v^1+v^2}\tilde a_2^{v^1+v^2}\tilde a_3a_4^{v^1(v^1-1)+\frac12v^2(v^2-1)}\tilde a_4^{v^1v^2}.&(7.4)
\end{align*}

These essentially describe the ways we can get “$\tilde a_4$ terms.” We will informally refer to the contributions “(7.1) terms,” “(7.2) terms,” “(7.3) terms,” and “(7.4) terms.” The “(7.4) terms” will refer to the terms in (7.4) that do not appear in (7.2) or (7.3). In order to capture these terms we define the two $\Q$-valued bilinear forms and one linear form as follows:

\begin{align*}
\alpha(v_1\tensor v_2)&=v_1^1v_2^2\\
\beta(v_1\tensor v_2)&=\frac12v_1^2v_2^2+v_1^1v_2^1\\
\gamma(v_1)&=v_1^1+\frac12v_1^1.
\end{align*}

Thus we have shown

$$(v_1,1,0)*(v_2,0,0)=(v_1+Tv_2,1,\alpha(v_2\tensor v_1+v_2\otimes v_2)+\beta(v_2\tensor v_2)-\gamma(v_2)).$$

Note that although $\beta$ and $\gamma$ take rational values in general, $\beta(v_2\tensor v_2)-\gamma(v_2)$ is always an integer. Here the input to $\alpha$ comes from the (7.1) and (7.4) terms while the inputs to $\beta$ and $\gamma$ come from the (7.2) and (7.3) terms. To compute the product in general for positive values of $k_1$ the steps would look like

\begin{align*}
(v_1,k_1,0)*(v_2,k_2,0)=&(v_1,k_1-1,0)*(Tv_2,k_2+1,(\alpha+\beta)(v_2\otimes v_2+v_2\tensor v_2)-\gamma(v_2))\\
=&(v_1,k_1-2,0)*(T^2v_2,k_2+2,\alpha(v_2\tensor v_2+Tv_2\tensor Tv_2)-\gamma(v_2+Tv_2))\\
&*(T^3v_2,k_2+3,\alpha(v_2\tensor v_2+Tv_2\tensor Tv_2+T^2v_2\tensor T^3v_2)\\
\end{align*}

and so on. This motivates the definition

$$S_k=\begin{cases}\sum_{j=0}^{k-1}(T\tensor T)^j&k\ge 0;\\
-\sum_{j=k+1}^{-1}(T\tensor T)^j&k<0.
\end{cases}$$

We cannot use the exponential sum formula to get a closed form for $S_k$ because $T\tensor T-1$ is not invertible. However $T-1$ is invertible, so with the linear terms we may write an inverse formula. We have done enough to motivate our definition of the cocycle. It comes from keeping track of each of the “$\tilde a_3$ terms” when computing multiplication in $\tilde\Gamma$. We define

\begin{align*}
\sigma_1((v_1,k_1),(v_2,k_2))&=\alpha(T^{k_1}v_2\tensor v_1)\\
\sigma_2((v_1,k_1),(v_2,k_2))&=\alpha((S_{k_1}(v_2\tensor v_2))\\
\sigma_3((v_1,k_1),(v_2,k_2))&=\beta(S_{k_1}(v_2\tensor v_2))\\
\sigma_4((v_1,k_1),(v_2,k_2))&=\gamma((T-1)^{-1}(T^{k_1}-1)v_2))
\end{align*}

Then we define our cocycle

Note that $\sigma_4$ is subtracted unlike the others. Here $\sigma_1$ comes from the “(7.1) terms,” $\sigma_2$ comes from the “(7.4) terms,” and $\sigma_3-\sigma_4$ comes from the “(7.2) and (7.3) terms." Before we compute $\partial\sigma$ we observe the following identities about $S_k$

\begin{align*}
S_{k_1+k_2}&=S_{k_1}+(T\tensor T)^{k_1}S_{k_2}\\
(T\tensor T-1)S_k&=(T\tensor T)^k-1
\end{align*}

Now we compute $\partial \sigma$ piece by piece. We will let $g_i\in\Gamma$ be represented as the pair $(v_i,k_i)$. Then we compute

\begin{align*}
\partial\sigma_1(g_1,g_2,g_3)=&\alpha(\\
&-T^{k_1}v_2\tensor v_1\\
&+T^{k_1}(v_2+T^{k_1}v_3)\tensor v_1\\
&+T^{k_2}v_3\tensor v_2)\\
=&\alpha(-T^{k_1+k_2}v_3\tensor T^{k_1}v_2+T^{k_2}v_3\tensor v_2)\\
=&-\alpha(((T\tensor T)^{k_1}-1)(T^{k_2}v_3\tensor v_2))\\
=&-\alpha((T\tensor T-1)S_{k_1}(T^{k_2}v_3\tensor v_2)).
\end{align*}

\begin{align*}
\partial(\sigma_2+\sigma_3)(g_1,g_2,g_3)=&(\alpha+\beta)(\\
&-S_{k_1}(v_2\tensor v_2)\\
&-S_{k_1+k_2}(v_3\tensor v_3)\\
&+S_{k_2}(v_3\tensor v_3))\\
&-S_{k_1}(v_2\tensor v_2)\\
&-S_{k_2}(v_3\tensor v_3)-S_{k_1}(T^{n_2}v_3\tensor T^{n_2}v_3)\\
&+S_{k_2}(v_3\tensor v_3))\\
=&(\alpha+\beta)(S_{k_1}(v_2\tensor T^{k_2}v_3+T^{k_2}v_3\tensor v_2)).
\end{align*}

\begin{align*}
\partial \sigma_4(g_1,g_2,g_3)=&\gamma((T-1)^{-1}(\\
\end{align*}

$$S_{k_1}(T^{k_1}v_3\tensor v_2)=\sum_{i=1}^2u_i\tensor w_i.$$

Note that since $S_{k_1}$ commutes with the map $u\tensor w\mapsto w\tensor u$ by construction so we have that

$$S_{k_1}(v_2\tensor T^{k_2}\tensor v_3)=\sum_{i=1}^2w_i\tensor u_i.$$

Now we have

$$\partial\sigma(g_1,g_2,g_3)=\sum_{i=1}^2\Big(\alpha(-(T\tensor T)(u_i\tensor w_i)+2u_i\tensor w_i+w_i\tensor u_i)+\beta(u_i\tensor w_i+w_i\tensor u_i)\Big).$$

Next we see from the definition of $T$ and $\alpha$ that

$$\alpha((T\tensor T)u_i\tensor w_i)=(2u_i^1+u_i^2)(w_i^2+w_i^1)=2u_i^1w_i^1+2u_i^1w_i^2+u_i^2w_i^1+u_i^2w_i^2.$$

\begin{align*}
\alpha(2u_i\tensor w_i)&=2u_i^1w_i^2\\
\alpha(w_i\tensor u_i)&=u_i^2 w_i^1\\
\beta(u_i\tensor w_i+w_i\tensor u_i)&=2\beta (u_i\tensor w_i)\\
\end{align*}

Thus $\partial\sigma=0$. Secondly we have the 2 chain

Then since $S_0=0$ and $T^0-1=0$ we have that $\sigma_1$ is the only one of the forms to pair nontrivially with $c$. Thus we see

We next investigate finite quotients of $\Gamma$. If $n,m\in\N^+$ so that the order of $T$ reduced mod $n$ in $\Gl_2(\Z/n\Z)$ divides $m$ then a finite quotient of $\Gamma$ can be of the form

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 $\sigma$ is $n$-$Q_n$ compatible. To do this note that the pair $(S_{k+k},T^{k+1})$ can be determined from the pair $(S_k,T^k)$. Both can be done with polynomials that are may be reduced mod $n$. So it follows that

$$(S_k,T^k)\equiv (S_{k+m},T^{k+m})\mod n$$

for some $m$ depending on $n$. It follows that the order of the order of $T$ reduced mod $n$ in $\Gl_2(\Z/n/\Z)$. Thus for odd $n$ we define

Since $n$ is odd we may express the $\frac12$s in the definition of $\sigma$ as the inverse of 2 mod $n$. We may reduce the rest of the operations mod $n$ easily and the operators $T^k$ and $S_k$ are $m$-periodic, so the formula for $\sigma$ determines a cocycle $\sigma'$ which defines a cohomology class in $H^2(Q_n;\Z/n\Z)$. Thus $\sigma$ is of finite type. Our formula for the representation then acts on the space $\C^n\tensor\C^n\tensor\C^m$. We can get a formula from Proposition <ref>, but this formula is messy, since our definition for $\sigma$ is messy. For that reason we will simply check what the generators do:

\begin{align*}
\rho_n(a_1)e_j\tensor e_k\tensor e_\ell&=e_{j+1}\tensor e_k\tensor e_\ell\\
\rho_n(a_2)e_j\tensor e_k\tensor e_\ell&=\exp\left(\frac{2\pi i}{n}j\right)e_j\tensor e_{k+1}\tensor e_\ell\\
\rho_n(a_3)e_j\tensor e_k\tensor e_\ell&=\exp\left(\frac{2\pi i}{n}\left(jk+j^2+\frac12k^2-j-\frac12k\right)\right)e_{2j+k}\tensor e_{j+k}\tensor e_{\ell+1}.
\end{align*}

We can see that this must be an asymptotic representation tensored against a genuine representation in the third tensor coordinate. Thus we can pick a smaller representation by “ignoring” the $e_\ell$ part. This gives us the list of formulas from the start of the subsection.

I would like to thank my advisor Marius Dadarlat for introducing me to the problem and for all his other advice. I would like to thank Ben McReynolds and Mark Pengitore for answering questions I had.

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