Baer invariant

Definition
Suppose $$G$$ is a group. Suppose $$\mathcal{V}$$ is a subvariety of the variety of groups (note that $$G$$ may or may not be in $$\mathcal{V}$$). The Baer invariant of $$G$$ with respect to $$\mathcal{V}$$, denoted $$\mathcal{V}M(G)$$, is an abelian group defined as follows.

Definition in terms of extensions
Consider possible group extensions of the form:

$$1 \to A \to E \to G \to 1$$

satisfying the condition that the image of $$A$$ in $$E$$ is contained in the $$\mathcal{V}$$-marginal subgroup of $$E$$. Consider the set of defining words for $$\mathcal{V}$$ (note that it suffices to take any generating set of words for the variety). For each word $$w$$ with $$n_w$$ letters, we have a word map $$E^{n_w} \to E$$ (a set map only, not a homomorphism). There is a natural initial object dependent only on $$G$$ and $$\mathcal{V}$$ that has a unique homomorphism to the $$\mathcal{V}$$-verbal subgroup of $$E$$. This natural initial objection, which we will call $$V^\#(G)$$, has a unique homomorphism to the $$\mathcal{V}$$-verbal subgroup $$V(G)$$. The kernel of this homomorphism is the Baer invariant of $$G$$.

Definition in terms of presentation
Suppose $$G$$ is expressed in the form $$F/R$$ where $$F$$ is a free group and $$R$$ is the normal closure of a set of words in $$F$$. Explicitly, any presentation of $$G$$ can be viewed in this manner, where $$F$$ is the free group on symbols corresponding to the generators and $$R$$ is the subgroup obtained as the normal closure of the relation words.

Denote by $$V(F)$$ the verbal subgroup of $$F$$ corresponding to all the words defining the variety $$\mathcal{V}$$, so $$F/V(F)$$ is the largest quotient of $$F$$ that is in $$\mathcal{V}$$. Also, define $$V^*(R,F)$$ as the subgroup generated by all words of the form:

$$v(f_1,f_2,\dots,f_ir,f_{i+1},\dots,f_n)v(f_1,f_2,\dots,f_n)^{-1}$$

where $$v$$ varies over all words defining the variety $$\mathcal{V}$$, $$f_1,f_2,\dots,f_n \in F$$ (and $$n,i$$ are also free to vary) and $$r$$ varies over all of $$R$$.

Then, the Baer invariant of $$G$$ with respect to $$\mathcal{V}$$ is defined as:

$$\mathcal{V}M(G) = \frac{R \cap V(F)}{V^*(R,F)}$$