Congruence Subgroups of SL_3(\mathbb Z) (by Giulia Orrù)

Let \Gamma_0(N) = \{\left(\begin{array}{lll} a_{11} &a_{12} &a_{13}\\ a_{21} &a_{22} &a_{23}\\ a_{31} &a_{32} &a_{33} \end{array}\right) \in SL_3(\mathbb Z)\ \colon \ a_{21}\equiv a_{31} \equiv 0 \mod N \}.
Rational homology in degree 3
The homology H_3(\Gamma_0(N),\mathbb Q) was computed for all 10 \le N\le 209 and many 210\le N\le 250 in . The following command, based on the same method, computes H_3(\Gamma_0(23),\mathbb Q) =\mathbb Q^{4}.

<#Include SYSTEM "../lib/Orru/tutorial/tutex/sl3.1.txt">

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

\begin{array}{l|llllllllll} x= &0 &1 &2 &3 &4 &5 &6 &7 &8 &9 \\ N &&&&&&&&&&\hline\\ 10+x &2 &2 &7 &0 &4 &4 &6 &2 &7&2 \\ 20+x &9 &4&8&4&17&4&6&5&13&4\\ 30+x &20&4&12&10&10&8&21&4&12&8\\ 40+x &&&&&&&&&&\\ 50+x &&&&&&&&&&\\ 60+x &&&&&&&&&&\\ 70+x &&&&&&&&&&\\ 80+x &&&&&&&&&&\\ 90+x &&&&&&&&&&\\ 100+x &&&&&&&&&&\\ 110+x &&&&&&&&&&\\ 120+x &&&&&&&&&&\\ 130+x &&&&&&&&&&\\ 140+x &&&&&&&&&&\\ 150+x &&&&&&&&&&\\ 160+x &&&&&&&&&&\\ 170+x &&&&&&&&&&\\ 180+x &&&&&&&&&&\\ 190+x &&&&&&&&&&\\ 200+x &&&&&&&&&&\\ 210+x &&&&&&&&&&\\ 220+x &&&&&&&&&&\\ 230+x &&&&&&&&&&\\ 240+x &&&&&&&&&&\\ 250+x &&&&&&&&&&\\ 260+x &&&&&&&&&&\\ 270+x &&&&&&&&&&\\ 280+x &&&&&&&&&&\\ 290+x &142&78&121&48&239&70&141&107&98&60\\ 300+x &325&&&&&&&&& \end{array}
Integral homology modulo the primes 2 and 3
The group SL_3(\mathbb Z) acts cellularly on a 3-dimensional contractible regular CW complex K with each cell stabilizer group finite of order dividing 24. Hence, the finite index subgroup \Gamma_0(N) also acts on this space. The quotient Y=K/\Gamma_0(N) has integral homology H_n(Y,\mathbb Z) equal to the group homology H_n(\Gamma_0(N),\mathbb Z) modulo any 2-torsion and 3-torsion. For N=31 the following example computes H_1(Y,\mathbb Z)=\mathbb Z_{5}, H_2(Y,\mathbb Z)=\mathbb Z_3 and H_3(Y,\mathbb Z)= \mathbb Z^4.

<#Include SYSTEM "../lib/Orru/tutorial/tutex/sl3.2.txt">
Low-dimensional integral homology
The following example computes H_1(\Gamma_0(31),\mathbb Z)= \mathbb Z_{30}, H_2(\Gamma_0(31),\mathbb Z)= (\mathbb Z_2)^8 , H_3(\Gamma_0(31),\mathbb Z)=(\mathbb Z_6)^2 \oplus (\mathbb Z_{12})^2 \oplus \mathbb Z^4.

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