10
votes
Accepted
How can I detect the homology image of a unipotent group in the general linear group?
Suppose first that $F$ is a finite field of characteristic $p$. Then $U_n(F)$ is a Sylow $p$-subgroup of $GL_n(F)$, and so using the transfer in group homology one sees that the image of $f_k$ (for $k&...
- 18.2k
7
votes
Accepted
The third homology stability of general linear groups over finite fields
The original paper by Suslin is available here and with some effort you should be able to read it.
If $|\mathbb{F}| = p^r$ then the homology groups $H_*(GL_n(\mathbb{F});\mathbb{F}_\ell)$ with $\...
- 8,167
6
votes
Accepted
Computing homology groups with GAP
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 ...
- 44.3k
6
votes
Small simplicial set models for BG
There is an algebraic result that is relevant, due to Benson and Carlson and stated as Corollary 4.5 in `Complexity and Multiple Complexes' Math. Z. vol 195 (1987) 221--238. Given a finite group $G$, ...
- 3,451
6
votes
Small simplicial set models for BG
Going out on a limb (I may well be messing up badly), I think the answer is yes at least for the CW structure version of the question.
Proof:
Choose a finite presentation of $F$ with generating set $G$...
- 63.9k
4
votes
Small simplicial set models for BG
This question may be somewhat relevant: Small simplicial complexes with torsion in their homology. David Speyer's answer there shows that one can build a simplicial complex $X$ with $H_1(X)=\mathbb{Z}...
- 56.9k
4
votes
Accepted
Homology of symplectic groups in the unstable range
The optimal stabilization range for third homology of the symplectic groups has been determined by Marco Schlichting and Husney Parvez Sarwar in https://arxiv.org/abs/2111.01539.
Their result states ...
- 17.4k
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