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Suppose we have an inclusion of groups $G_1<G_2$. I am curious about what methods there are out there for analyzing the map $H_k(G_1;\mathbb Q)\to H_k(G_2;\mathbb Q)$. In particular, what are tools that might show it is injective? In several of the cases I'm interested in, $G_1\cong\mathbb Z^k$. I.e. I want to show that the top dimensional homology of a certain free abelian subgroup survives to the whole group.

Edit: One example I'm thinking about is abelian subgroups of the mapping class group of a surface generated by disjoint Dehn twists. I know for example, that in a torus with $4k+1$ punctures, that the subgroup generated by $4k+1$ Dehn twists along curves which cut the torus into $4k+1$ punctured annuli has a fundamental class that survives to the whole mapping class group. The method of showing this is rather indirect (constructing an explicit cocycle detecting it in the ribbon graph complex), so a more conceptual explanation would be nice.

The other main class of examples come from subgroups of $\operatorname{Out}(F_n)$, as explained in this paper.

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  • $\begingroup$ Well, the most obvious case in which this happens is when $G_1$ is a retract. If $G_2$ happens to be virtually special and $G_1$ is quasiconvex then $G_1$ is always virtual retract of $G_2$, so one can then try to analyse how 'virtual' the retract really is. $\endgroup$
    – HJRW
    Commented Sep 11, 2015 at 11:49
  • $\begingroup$ @HJRW: A retract seems way too much to hope for in the cases I'm thinking about, but your answer serves as a motivation to learn about virtually special groups. $\endgroup$
    – Jim Conant
    Commented Sep 11, 2015 at 12:14
  • $\begingroup$ It would probably help a lot if you indicated which cases you do have in mind. $\endgroup$
    – HJRW
    Commented Sep 11, 2015 at 12:15
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    $\begingroup$ @HJRW: fair enough. I was hoping to assemble a toolkit of general methods, but I will add some more information about the examples I'm thinking about. $\endgroup$
    – Jim Conant
    Commented Sep 11, 2015 at 12:25
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    $\begingroup$ If your subgroup is normal then there is the Hochschild-Serre spectral sequence. If it's of finite index, there's the transfer. Both of these are explained in Brown's book. $\endgroup$
    – Mark Grant
    Commented Sep 11, 2015 at 12:39

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