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I believe there is a reasonable notion of $\text{Ext}^1(G,H)$ in the category of groups (where $G$ and $H$ are groups). Is there a decent reference describing this?

My particular situation involves a nilpotent Lie group $G$ with finitely many components. One may form the short-exact sequence $$1\rightarrow G_0\rightarrow G\rightarrow\pi_0(G)\rightarrow 1,$$ an element of $\text{Ext}^1(\pi_0(G),G_0)$. I am curious about conditions under which this sequence splits. I would appreciate any and all references and suggestions.

Thanks!

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  • $\begingroup$ ... But to understand the general case (with your assumptions: $G$ virtually connected nilpotent Lie group), you can probably reduce to the case when $G_0$ is a torus and in general, I expect that if $W$ is the maximal (compact) torus in $G_0$ then the nilpotent extensions of $G_0$ by a finite nilpotent group $F$ should be classified by the same object as the central extensions $W$-by-$F$. $\endgroup$
    – YCor
    Commented Apr 12, 2013 at 22:07
  • $\begingroup$ By Mostow, $G$ has a maximal compact subgroup $K$ and $KG_0=G$. In particular, if you assume in addition that $G_0$ is simply connected, then $G=G_0\rtimes K$ so your extension is split (just assuming $G_0$ nilpotent). In general, since $G$ is nilpotent and $K$ is compact, the action of $K$ on the Lie algebra of $G$ is unipotent and hence trivial, so $[K,G_0]=1$. Note that there are easy non-split extensions, e.g. with $G_0$ the circle and $G/G_0$ noncyclic group of order 4. $\endgroup$
    – YCor
    Commented Apr 13, 2013 at 8:12

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Since $G$ is a Lie group with finitely many components, it has a maximal compact subgroup $K$, unique up to conjugation and $G=KG_0$ (Mostow). Since $G$ is actually nilpotent, $K$ is unique and actually consists of the elements in $G$ contained in a compact subgroup. In particular $K$ is normal in $G$.

From $G$ we can define $K$ as above; conversely if we know $K$ and $G_0$, we know $G$, which is the quotient of $K\times G_0$ by the central subgroup $\{(g,g^{-1}),g\in K_0\}$.

This shows that if $F$ is a finite nilpotent group, the set classifying extensions of $G_0$ by $F$ that are nilpotent is in canonical bijection with the set classifying central extensions of $K_0$ by $F$. The latter is in bijection with $H^2(F,K_0)$, which is a reasonable well-understood object ($K_0$ is a torus).

In particular: 1) If $G_0$ is simply connected we get $G=G_0\times F$ (only the direct extension)

2) Everything boils down to when $G_0$ is a torus. If $S^1$ is the 1-circle, $Q$ is the group of quaternions of order 8 and if we define $G$ as the quotient of $S^1\times Q$ by the diagonal element of order 2, then we get a nontrivial extension (as we see by counting elements of order 2).

NB: you assume explicitly that $G$ is nilpotent. So I don't claim to classify all extensions of $G_0$ by $F$.

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Pages 177-190 of here contain a relatively complete description of group extensions, following Schreier, Baer, Eilenber + MacLane, Hochschild, and Serre.

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