In this case we say that $A$ is \CmMark{determinant class}.
\end{dfn*}
\begin{rem*}
If $A$ is right multiplication by a matrix in $M_n(\Z[\Gamma])$ and $\Gamma$ is sofic, then $A$ is determinant class (Elek-Szabo).
\end{rem*}
\begin{dfn*}
Let $M$ be a finite CW-complex.
Assume that $\Gamma=\pi_1M$ is sofic and that $M$ is $l^2$-acyclic.
Then the \CmMark{$l^2$-torsion}$\rho^{(2)}(M)$ is defined as
\begin{align*}
\rho^{(2)}(M) = -\frac{1}{2}\sum_{p \geq 0} (-1)^p p \cdot\ln{\textstyle\det^{(2)}}(\Delta_p^{(2)}\colon l^2 \Gamma\otimes_{\Z\Gamma} C_p(\tilde M) \to l^2 \Gamma\otimes_{\Z\Gamma} C_p(\tilde M)).
\end{align*}
\end{dfn*}
\begin{rem*}
Note that the following remarks are actually propositions.
\begin{itemize}
\item$l^2$-torsion is a homotopy invariant
\item Let $M$ be a closed hyperbolic $3$-manifold.
Then $\rho^{(2)}(M)=\frac{1}{6\pi}\operatorname{vol}(M, g_{\text{hyp}})$.
This implies, in particular, the homotopy invariance of the hyperbolic volume, which also follows non-trivially by Mostow rigidity.
\end{itemize}
\end{rem*}
\subsection{Approximation}
\begin{thm*}[Lück]
Let $M$ be a finite CW-complex with $\Gamma=\pi_1M$ residually finite.
Let $\Gamma=\Gamma_0 > \Gamma_1 > \Gamma_2 > \cdots$ be a \CmMark{residual chain}, i.e. $[\Gamma: \Gamma]=0$, $\Gamma_i \vartriangleleft\Gamma$, $\bigcup\Gamma_i \{1\}$.
The following is a bold conjecture (that probably is only true up to a few more adjectives added, e.g. (arithmetic) hyperbolic manifold).
\begin{conj*}
Let $M^{2n+1}$ be a closed odd-dimeninsional aspherical manifold with residually finite $\Gamma=\pi_1M$ and let $(\Gamma_i)$ be a residual chain in $\Gamma$.
Then $\rho^{(2)}(M)=\lim_{i \to\infty}\frac{\ln |\operatorname{tors} H_n(M_i; \Z)|}{[\Gamma : \Gamma_i]}$.
\end{conj*}
In particular, this would tell us that, if $M$ is a closed hyperbolic 3-manifold, then
Both come from a multiplication with $A \in M_n(\Z[\Gamma])$ (resp. $A_i \in M_n(\Z[\Gamma/\Gamma_i])$), where $A_i$ is the ``reduction'' of $A$ mod $\Gamma_i$.
This follows from spectral calculus, by using the equalities $\tr_{\Gamma}(\pr_{\Ker A})=\mu_A(\{0\})$ and $\lim_{i \to\infty}\tr_{\Gamma/\Gamma_i}(\pr_{\Ker A_i})=\lim_{i \to\infty}\mu_{A_i}(\{0\})$.
To prove the equality on this level it is important to consider the spectral measure in a neighbourhood of $\{0\}$.
Recall that measures $\nu_i$ on the real line \CmMark[weak convergence of measures]{converge weakly}, if $\int_{\R}f\dop\nu_i \to\int_{\R} f\dop\nu$ converges for all $f \in C_c(\R)$.
(In good situations, it is enough to check this for monomials.)
As the right-hand-sides are purely combinatorial, we get $\tr_{\Gamma/\Gamma_i}(A^m)\to\tr_{\Gamma}(A^m)$ and conclude $\mu_{A_i}\to\mu_A$ weakly.
Caveat: For the Borel probability measure on $\R$ we have $\nu_i = i \chi_{(0,\frac{1}{i}]}\dop\lambda\to\nu=\delta_{\{0\}}$ weakly, but $0=\nu(\{0\})\not\to\delta_{\{0\}}(\{0\})=1$.
Without loss of ideas (needed for the general case), we consider
Further, fix $i$, let $n =[\Gamma : \Gamma_i]$ and denote by $0=\lambda_1=\cdots\lambda_m < \lambda_{m+1}\leq\dots\leq\lambda_n$ the eigenvalues of $A_i$.
The characteristic polynomial of $A_i$ is given by $p(z)= z^mq(z)$ for $q \in\Z[t]$ and this yields
\begin{align*}
\lambda_{m+1}\cdots\lambda_n - q(0) \geq 1.
\end{align*}
Small eigenvaolues: $N(\varepsilon)=\#$ eigenvalues of $A_i$ in $(0,\varepsilon)$.
We get $1\leq\lambda_{m+1}\cdots\lambda_n \leq\varepsilon^{N(\varepsilon)}\|A_i\|^n \leq\varepsilon^{N(\varepsilon)}\cdot\text{const}^n$.
Thus taking a logarithm, we obtain $\frac{N(\varepsilon)}{n}\leq\frac{\text{const}}{|\log\varepsilon|}$ (independent of $i$).
Since $\mu_{A_i}((0,\varepsilon))=\frac{N(\varepsilon)}{n}$, we have found an upper bound for it.
Basic measure theory shows that weak convergence yields $\limsup\mu_{A_i}(W)\leq\mu_A(W)$ for a closed set $W$ and $\liminf\mu_{A_i}(U)\geq\mu_A(U)$ for $U$ open.
A flat bundle $(E,\nabla)\to(M,g)$ admits a harmonic metric if and only if $\hol(\nabla)$ is semi-simple.
\end{thm*}
\begin{rem*}
\begin{itemize}
\item$\Gl_n\C/U(n)$ is a non-compacat (continuous) non-positively curved homogeneous riemannian manifold.
\item If $\rho$ is discrete and the image is acting freely on $\Gl_n\C/U(n)$, then the equivariant map $\tilde h$ descends to a map
\begin{align*}
\tilde h \colon M \to\Im(\rho)\backslash(\Gl_n\C/U(n))
\end{align*}
\end{itemize}
\end{rem*}
Now let us fix a hermitian metric $h$ on $E$.
\begin{thm*}[Restatement]
Given a bundle $(E,h)\to(M,g)$ a flat connection $\nabla\in\mathcal F(E)$, there exists an automorphism $\phi\in\mathcal A(E)$ such that $(E,h,\phi^{*}\nabla)$ ($\phi^{*}\nabla=\phi\circ\nabla\circ\phi^{-1}$) is a harmonic flat bundle if and only if $\hol(\nabla)$ is semi-simple.
Moreover, $\phi$ is unique up to $\mathcal A_h(E)=\{\phi\in\mathcal A(E)\mid\phi^{*}h = h\}$.
\end{thm*}
\begin{dfn*}
The space of Harmonic flat bundles is
\begin{align*}
\mathcal{HF}(E, h) := \{\nabla\in\mathcal F(E) \mid (E, \nabla, h) \text{ is a Harmonic flat bundle}\}/\mathcal A_h(E).
\end{align*}
We also call $\mathcal A_h(E)$ the group of \CmMark{gauge transformations} of $(E,h)$.
The space $\mathcal{HF}(E,h)$, while homeomorphic to the space on the right-hand-side, has significantly more structure, i.e. we can make several techniques available to the other side.
Every element $\nabla\in\mathcal{HF}(E,h)$ corresponds to a $\hol(\nabla)$-equivariant map
\begin{align*}
h_{\nabla}\colon\tilde M \to\Gl_n\C/U(n),
\end{align*}
such that $E(h_{\nabla})=\frac{1}{2}\int_F \|\dop h_{\nabla}\|^2\dop V_g \in\R_{\geq}$, where $F$ is a fundamental domain for the $\pi_1M$ action on the universal cover $\tilde M$.
This assignment defines a continuous function
\begin{align*}
E \colon\mathcal{HF}(E,h) \to\R_{\geq 0}
\end{align*}
called the enery function.
Now we will specialize to $M =\Sigma$ a closed, oriented surface of genus $> 1$ equipped with a conformal structure $\sigma$.
In this case, the space $\mathcal HF(E,h)$ is remarkably rich!
Given a connection $\nabla$ on $(E,h)$, there exists a unique decomposition $\nabla=\nabla^h +\psi\in\Omega^1(\End(E))$, where $\nabla^h$ is a unitary connection, i.e. $\dop h(s,s')= h(\nabla^hs,s')+ h(s,\nabla^hs')$, and such that $\psi$ is hermitian, i.e. $h(\psi(s),s')= h(s,\psi(s'))$.
Thus, we have
\begin{align*}
\mathcal{HF}(E,h) = \{ (\nabla^H, \psi) \mid\substack{\nabla^H \text{ is unitary}, \psi\text{ is hermitian},\\ (E,h,\nabla^H + \psi) \text{ is a Harmonic flat bundle}}\}/\mathcal A_h(E).
\end{align*}
In these terms, the harmonic and flat conditions have a nice description.
where if $\psi=\sum\gamma_i \otimes s_i$ for $\gamma_i \in\Omega^1(\Sigma), s_i \in\End(E)$ we put $[\psi,\psi]=\sum\gamma_i \wedge\gamma_i \otimes[s_i, s_i]$, wherein $[\blank,\blank]$ is the commutator.
We observe:
\begin{itemize}
\item If $\nabla^H$ is unitary, then $F(\nabla^H)$ is skew-hermitian,
Recall for the last observation, that we introduced for $\alpha, \beta\in\Omega^1(X,\End(E))$ a pairing $\left<\alpha, \beta\right> =\int\left<\alpha, \beta\right> \dop V_g$.
(Note that, on a surface this expression in 1-forms does only depend on the conformal class of $g$)
Now $\delta^{\nabla^H}$ was the adjoint of $\dop^{\nabla^H}$ with respect to this pairing.
Recall that on a Riemann surface $X$ with $z$ the local complex coordinate, a holomorphic $\kappa$ differential on $X$ is
\begin{itemize}
\item a holomorphic section $\alpha$ of $S^k\T_{\hol}^{*}X$ ($S^k$ denotes the $k$th symmetric power),
\item$\alpha= f(z)\dop z^k$, where $f(z)$ is a holomorphic function.
\end{itemize}
For $\psi\in\Omega^1(X,\End(E))$ we have $\psi^k \in\Gamma(\T^{*}X^{\otimes k}\otimes\End(E))$ and we can take its trace $\tr(\psi^k)\in\Gamma(\T^{*}X^{\otimes k})$, for which we have the following proposition.
\begin{prop*}
If $(\nabla^H, \psi)\in\mathcal{HF}(E,h)$, then
\begin{align*}
\alpha_k := \tr(\psi^k)^{k,0} = f(z) \dop z^k,
\end{align*}
(where $f(z)$ is only $C^{\infty}$ a priori) is actually holomorphic.
Moreover, $\alpha_2$ is the Hopf differential of the corresponding harmonic map.
\end{prop*}
\begin{rem*}
Earlier we saw that the map
\begin{equation*}
\begin{tikzcd}
\{\substack{\text{discrete, faithful}\\\text{repr. }\pi_1\Sigma\to\SL_2\R}\}/\SL_2(\R) \ar{r}{\sim}\ar[hook]{d}&\{\text{hyp. metrics on }\Sigma\}/\Diff_0(\Sigma) \ar{r}{\sim}&\mathrm{QD}(X)\\
\subsection{Another unique feature of $\mathcal{HF}(E,h)$ for a Riemann surface}
$U(1)$-action of $\mathcal{HF}(E,h)\ni(\nabla^H, \psi)$ with $F(\nabla^H)+\frac{1}{2}[\psi,\psi]=0$, $\dop^{\nabla^h}\psi=0$ and $\delta^{\nabla^H}\psi=0$.
Given $X, I$, we get a Hodge $*$-operator
\begin{align*}
* \colon\Omega^1(X, \R) \to\Omega^1(X, \R).
\end{align*}
We write $\psi= sum_{i =1}^k \gamma_i \otimes s_i$ and define $I\psi=\sum_{i =1}^k *\gamma_i \otimes s_i$ and $\theta\cdot\psi= e^{I\theta}\psi=\cos\theta\psi+\sin\theta I\psi$.
\begin{prop*}
We have $(\nabla^H, \psi)\in\mathcal{HF}(E,h)\iff(\nabla^H, \theta\cdot\psi)\in\mathcal{HF}(E,h)$, i.e. this defines a $U(1)$-action on $\mathcal{HF}(E,h)$.
\end{prop*}
The enery $E$ is invariant under this $U(1)$-action $U(1)\curvearrowright\mathcal{HF}(E,h)\xrightarrow{E}\R$ and in fact, we have $\dop E|_{(\nabla^H, \psi)}=0$ if and only if $(\nabla^H, \psi)$ is fixed by the $U(1)$-action.
It was first realized by Atiyah-Bott and used in the exact context by Hitchin, that $E$ can be used as a Morse function to compute the Betti numbers of $\mathcal{HF}(E,h)$.
This has been completely carred out for $\SpecialLinear_2\C$ by Hitchin and still today, this is the only method to understand the Betti numbers of $\Hom_{\text{ss}}(\pi_1\Sigma, \Gl_n\C)/\Gl_n\C$.