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I’m studying the homology groups of arithmetic groups such as $SL(5,\mathbb{Z})$. I saw in the answer to this post that we can use GAP to compute some of the homology groups for $SL(3,\mathbb{Z})$. Is the same true for $SL(5,\mathbb{Z})$? If so, how many homology groups can it compute?

In general, is there a database of some kind that has stored these types of homology calculations?

I apologize if these questions are naive. I’m very new to GAP and what it’s capabilities are.

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  • $\begingroup$ I imagine what you can compute will largely be dictated by how much system memory GAP can address and how long you can wait. Why don't you try? $\endgroup$
    – Ryan Budney
    Commented Jan 28 at 5:09
  • $\begingroup$ I figured this would be the case. I don’t currently have a device that can support it. Otherwise, I would look into this myself. $\endgroup$
    – Noah B
    Commented Jan 28 at 7:18
  • $\begingroup$ One needs to run "LoadPackage("HAP");;" before Graham's code. Then it will compute the homology of SL(5,Z). $\endgroup$
    – Ryan Budney
    Commented Jan 28 at 18:58

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Graham Ellis would be able to better comment on the correctness of his code for $SL(5,\mathbb Z)$, as he appears to be the author of the HAP package in GAP.

But his code executes quickly and claims to compute the homology of $SL(5,\mathbb Z)$.

$$H_2(SL_5(\mathbb Z), \mathbb Z)$$

LoadPackage("HAP");;
R:=ResolutionArithmeticGroup("SL(5,Z)",4);;
Homology(TensorWithIntegers(R),2);

Returns $[2, 12]$, which I believe means the code is claiming this homology $H_2$ group is isomorphic to $\mathbb Z_2 \oplus \mathbb Z_{12}$. It similarly computes $H_3$ to be $\mathbb Z_2 \oplus \mathbb Z_{24}$, and $H_4$ to $\mathbb Z_{12}$.

I believe the 4 in the call for the resolution is a reference to the length of the resolution, so 4 is longer than required for $H_2$. With the resolution of length 4 call, the code chokes on an $H_5$ computation.

As was pointed out in the comments these answers disagree with what's known in the literature. After digging around in Graham Ellis's documentation a little more I found the comment regarding the ResolutionArithmeticGroup command:

Inputs a positive integer n
 and a string P
 equal to one of the following:

"SL(2,Z)" , "SL(3,Z)" , "PGL(3,Z[i])" ,  PGL(3,Eisenstein_Integers)" , "PSL(4,Z)" , "PSL(4,Z)_b" ,"PSL(4,Z)_c" , "PSL(4,Z)_d" , "Sp(4,Z)"

So it would appear the answer to the question of this thread is no, GAP currently can't compute group homology of $SL_5(\mathbb Z)$.

The documentation is here: https://docs.gap-system.org/pkg/hap/doc/chap0_mj.html

Click on Chapter 5 to find the above.

It's computing something, just not the homology of the group.

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    $\begingroup$ Thanks you for the response! It is strange to me that it computes $H_2(SL(5,\mathbb{Z})$ to be $\mathbb{Z}_2\times \mathbb{Z}_{12}$. The reference given in the answer here has it as $\mathbb{Z}_2$. $\endgroup$
    – Noah B
    Commented Jan 28 at 21:16
  • $\begingroup$ mathoverflow.net/questions/64551/… $\endgroup$
    – Noah B
    Commented Jan 28 at 21:17
  • $\begingroup$ Indeed, note that running your code actually produces an error (until one sets $R$ to be a resolution of some implemented group; and then runs the code again. Here, the package should return an error, but this is not done). $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Jan 29 at 1:05
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    $\begingroup$ HAP can not compute homology of SL(5,Z). The code should say that data for this group is currently unavailable and then list the groups for which it is currently available. It certainly used to do this for such a group, but I must have introduced a bug at some stage. I'll look into it and fix it. GAP needs the finite cell structure of a fundamental domain for a group action plus a list of cell stabilizers. I think that Philippe Elbaz-Vincent and others have this data for SL(5,Z) so it should not be difficult for me to include it. I'll look into this too. Apologies for the confusion caused! $\endgroup$
    – Graham Ellis
    Commented Jan 29 at 8:41
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    $\begingroup$ @Noah B Yes that's the paper I had in mind. I've sent an email asking if I can have a copy of their fundamental domains. $\endgroup$
    – Graham Ellis
    Commented Jan 29 at 16:17

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