<Chapter><Heading>Congruence Subgroups of <M>SL_3(\mathbb Z)</M> (by Giulia Orr&#x00F9;)</Heading>

	Let <M>\Gamma_0(N) = \{\left(\begin{array}{lll}
		a_{11} &amp;a_{12} &amp;a_{13}\\
		a_{21} &amp;a_{22} &amp;a_{23}\\
		a_{31} &amp;a_{32} &amp;a_{33}
		\end{array}\right) \in SL_3(\mathbb Z)\ \colon \ a_{21}\equiv a_{31} \equiv 0 \mod N \}.
</M>
        <Section><Heading>Rational homology in degree 3</Heading>
			The homology <M>H_3(\Gamma_0(N),\mathbb Q)</M> was computed for all <M>10 \le N\le 209</M> and many <M>210\le N\le 250</M> in <Cite Key="vanderkallen"/>. The following command, based on the same method, computes <M>H_3(\Gamma_0(23),\mathbb Q) =\mathbb Q^{4}</M>.
                
<Example>
	<#Include SYSTEM "../lib/Orru/tutorial/tutex/sl3.1.txt">
</Example>

<P/> This command can be used to extend the computations to  the following table of dimensions of <M>H_3(\Gamma_0(N),\mathbb Q)</M> for all <M>10\le N\le 300</M>. The table corrects slips in
<Cite Key="vanderkallen"/> for <M>N=27,92,93,240,250</M>.

<P/>
<M>
	\begin{array}{l|llllllllll}
	x= &amp;0 &amp;1 &amp;2 &amp;3 &amp;4 &amp;5 &amp;6 &amp;7 &amp;8 &amp;9 \\
        N &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\hline\\
	10+x &amp;2 &amp;2 &amp;7 &amp;0 &amp;4 &amp;4 &amp;6 &amp;2 &amp;7&amp;2 \\
20+x &amp;9 &amp;4&amp;8&amp;4&amp;17&amp;4&amp;6&amp;5&amp;13&amp;4\\
30+x &amp;20&amp;4&amp;12&amp;10&amp;10&amp;8&amp;21&amp;4&amp;12&amp;8\\
40+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
50+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
60+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
70+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
80+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
90+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
100+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
110+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
120+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
130+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
140+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
150+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
160+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
170+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
180+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
190+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
200+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
210+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
220+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
230+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
240+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
250+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
260+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
270+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
280+x &amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;\\
290+x &amp;142&amp;78&amp;121&amp;48&amp;239&amp;70&amp;141&amp;107&amp;98&amp;60\\
300+x &amp;325&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;&amp;
	\end{array}
</M>

        </Section>       

	<Section><Heading>Integral homology modulo the primes <M>2</M> and <M>3</M></Heading>
		The group <M>SL_3(\mathbb Z)</M> acts cellularly on a <M>3</M>-dimensional contractible
			regular CW complex <M>K</M>
				with each cell stabilizer group finite of order dividing <M>24</M>. Hence, the finite index subgroup
<M>\Gamma_0(N)</M> also acts on this space.
The quotient <M>Y=K/\Gamma_0(N)</M>
	has integral homology <M>H_n(Y,\mathbb Z)</M> equal to the group homology <M>H_n(\Gamma_0(N),\mathbb Z)</M> modulo any <M>2</M>-torsion and <M>3</M>-torsion. For <M>N=31</M> the
		following example computes <M>H_1(Y,\mathbb Z)=\mathbb Z_{5}</M>, <M>H_2(Y,\mathbb Z)=\mathbb Z_3</M> and <M>H_3(Y,\mathbb Z)= \mathbb Z^4</M>.

<Example>
<#Include SYSTEM "../lib/Orru/tutorial/tutex/sl3.2.txt">
</Example>

	</Section>

	<Section><Heading>Low-dimensional integral homology</Heading>
			The following example computes <M>H_1(\Gamma_0(31),\mathbb Z)= \mathbb Z_{30}</M>,
	<M>H_2(\Gamma_0(31),\mathbb Z)= (\mathbb Z_2)^8 </M>,
	<M>H_3(\Gamma_0(31),\mathbb Z)=(\mathbb Z_6)^2 \oplus (\mathbb Z_{12})^2
	\oplus \mathbb Z^4</M>.

<Example>
<#Include SYSTEM "../lib/Orru/tutorial/tutex/sl3.3.txt">
</Example>
	</Section>
</Chapter>
