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Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:

  1. Is there any special property of $X_G$ due to the group's perfectness?

  2. What can we say about $H_2(X_G;\mathbb{Z})$, is it non-trivial free sometimes?

Any suggestions will be really helpful. Thanks!

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    $\begingroup$ What is the definition of $X_G$? I've never heard of the "representation complex" of a group, and I couldn't find anything with a quick Google search. $\endgroup$
    – Dan Ramras
    May 6, 2022 at 20:40
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    $\begingroup$ That's the presentation complex. $\endgroup$ May 6, 2022 at 21:01
  • $\begingroup$ I do apologize. I have edited. Thanks! $\endgroup$
    – piper1967
    May 6, 2022 at 21:06
  • $\begingroup$ 1. Since $G$ is perfect, $H_1(X_G,\mathbb{Z})=0$ by Hurewicz. 2. $H_2(X_G,\mathbb{Z})$ is always free (since $X_G$ is 2-dimensional) and can certainly be non-trivial. For a silly example, consider the presentation $\langle a\mid a,a\rangle$ of the trivial group, which has $X_G$ homeomorphic to the 2-sphere. $\endgroup$
    – HJRW
    May 7, 2022 at 8:16
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    $\begingroup$ One final comment. Of course, one should choose the most efficient possible presentation for a given group. It seems to be an open question whethe or not every finite simple group has a presentation with $m=n$ -- see this answer and the references therein: math.stackexchange.com/questions/3273061/… . $\endgroup$
    – HJRW
    May 7, 2022 at 12:42

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