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# Steinberg Group as a Lattice in a lie group

Given an integral domain $R$, the Steinberg group $St_n(R)$ is the group given by generators
$e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda)$, $p\neq q$, $1\leq p,q \leq n$

Subject to the relations
\begin{align} e_{ij}(\lambda) e_{ij}(\mu) &= e_{ij}(\lambda+\mu) \ \left[ e_{ij}(\lambda),e_{jk}(\mu) \right] &= e_{ik}(\lambda \mu) && \mbox{for } i \neq k\ \left[ e_{ij}(\lambda),e_{kl}(\mu) \right] &= \mathbf{1} && \mbox{for } i \neq l, j \neq k\ \end{align}

The Steinberg group is the universal central extension of the special linear group over $R$; $Sl_n(R)$.

Is there a description of the Steinberg group $St_n(Z)$, the special linear group over the integers as a lattice in some lie group, and some covering map realizing the universal central extension of $Sln(R)$ ( real coefficients), which restricts to the integral universal central extension of $Sln(Z)$ given by the Steinberg group ?