Has the homology of the Baumslag Solitar groups B(1,k) been computed?

Question

I'm trying to compute some homology for related groups and was wondering if the homology of these groups have been studied. Using the description of these groups as semidirect products of $\mathbb{Z}[1/k] \ltimes_k \mathbb{Z}$, I thought this would just pop out from the Lyndon-Hothschild-Serre spectral sequence, but I'm finding it a bit more difficult.

Answer 1

By Magnus(-Karrass-Solitar) theorem, one-relator group only has torison in case its relator is a proper power. Then, by Cockroft's theorem (W. H. Cockroft, On two-dimesional alpherical complexes, 1954, rediscovered later by Lyndon and other people, see, for example, "Lyndon Identity theorem"), presentation complex of a torsion-free one-relator group is aspherical, so it is a classifying space for that group.

So... we're lucky, because we can use explicit cellular structure of presentation complex — which happens to be $K(G, 1)$ — to compute (co)homology. Cellular complex for $BS(r, s)$ will be $$\Bbb Z \overset{(0, r-s)}\longrightarrow \Bbb Z^2 \overset{0}\longrightarrow \Bbb Z$$

So, $H_1$ is $\Bbb Z \oplus \Bbb Z/(r-s)$ (which is already clear from presentation), and $H_2$ is zero. Turns out there's not much to study after all.

I want to remark that vanishing of $H_2$ can be obtained without (pretty hard) asphericity result. To make a classifying space out of presentation complex, we only add cells in dimensions 3 and higher, but that can only kill 2-dim homology classes, not add new ones. Now we notice that cellular differential should be nonzero on (the only) 2-cell of presentation complex (because it's a 2-generated group with rank 1 abelianization, and we need something to kill a 1-dim cycle). Same reasoning tells us that $k$-generated $(k-l)$-relator group with abelianisation having rank $l$ will have vanishing $H_2$.

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A different avenue for calculating homology would be using "Mayer-Vietoris" sequence from I. M. Chiswell. Exact sequences associated with a graph of groups — especially if you want to compute homology with nontrivial coefficients.

P. S. I vaguely remember a paper by D. J. S. Robinson, where he computers homology of more general graphs of groups with cyclic edge and vertex groups, but cannot remember the title. It would be interesting to know homology of "cyclic BS-type" groups, like Higman group - i. e. you have generators indexed by $\Bbb Z/n$, and $i$-th generator conjugates the $i+1$-st one into some power. Relation modules of presentations of this type occurred in potential (counter)examples to the relation gap conjecture.