Steinberg Group as a Lattice in a lie group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:29:07Z http://mathoverflow.net/feeds/question/105184 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105184/steinberg-group-as-a-lattice-in-a-lie-group Steinberg Group as a Lattice in a lie group Nicolas 2012-08-21T18:54:45Z 2012-08-22T00:02:54Z <p>Given an integral domain $R$, the Steinberg group $St_n(R)$ is the group given by generators<br> $e_{pq}(\lambda) := \mathbf{1} + a_{pq}(\lambda)$, $p\neq q$, $1\leq p,q \leq n$</p> <p>Subject to the relations<br> \begin{align} e_{ij}(\lambda) e_{ij}(\mu) &amp;= e_{ij}(\lambda+\mu) \ \left[ e_{ij}(\lambda),e_{jk}(\mu) \right] &amp;= e_{ik}(\lambda \mu) &amp;&amp; \mbox{for } i \neq k\ \left[ e_{ij}(\lambda),e_{kl}(\mu) \right] &amp;= \mathbf{1} &amp;&amp; \mbox{for } i \neq l, j \neq k\ \end{align}</p> <p>The Steinberg group is the universal central extension of the special linear group over $R$; $Sl_n(R)$. </p> <p>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 ? </p> http://mathoverflow.net/questions/105184/steinberg-group-as-a-lattice-in-a-lie-group/105189#105189 Answer by Jim Humphreys for Steinberg Group as a Lattice in a lie group Jim Humphreys 2012-08-21T20:00:55Z 2012-08-22T00:02:54Z <p>EDIT: This started to be an answer, but my recollections were inaccurate. It's essential here to distinguish carefully between what happens when <code>$n=2$</code> (where life is much more complicated) and when <code>$n \geq 3$</code> (where the situation stabilizes). The definition of the Steinberg group differs in these situations.</p> http://mathoverflow.net/questions/105184/steinberg-group-as-a-lattice-in-a-lie-group/105195#105195 Answer by Yves Cornulier for Steinberg Group as a Lattice in a lie group Yves Cornulier 2012-08-21T21:56:38Z 2012-08-21T21:56:38Z <p>For $n\ge 3$ the Steinberg group $\text{St}_n(\mathbf{Z})$ is a lattice in the universal covering of $\text{SL}_n(\mathbf{R})$ (which has fundamental group the cyclic group of order 2). </p> <p>To see that the inverse image of $\text{St}_n(\mathbf{Z})$ in the universal covering of $\text{SL}_n(\mathbf{R})$ is indeed a non trivial central extension, essentially amounts to check that the image of $K_2(\mathbf{Z})$ in the topological $K_2$ or the reals is injective (but I guess that a suitable direct argument, for instance using an element of order 2, can work as well). </p>