Marginal subgroup

Marginal subgroup for a single word
Suppose $$w$$ is a word in the letters $$x_1,x_2,\dots,x_n$$ and $$G$$ is a group. The marginal subgroup for $$w$$ in $$G$$ is the set of all $$g \in G$$ such that:

$$\! w(x_1,x_2,\dots,x_n) = w(x_1,x_2,\dots,gx_i,x_{i+1}, \dots x_n) = w(x_1,x_2,\dots,x_ig,x_{i+1},\dots,x_n) \ \forall \ x_1,x_2,\dots,x_n \in G, \forall i \in \{ 1,2,\dots,n \}$$

That this set is a subgroup is readily verified.

Marginal subgroup for a collection of words
The marginal subgroup for a (possibly infinite) collection of words in a group is the intersection of the marginal subgroups for each of the words in that group.

Marginal subgroup for a variety
The marginal subgroup for a subvariety of the variety of groups is defined in the following equivalent ways:


 * 1) It is the marginal subgroup for the collection of all words that become trivial in that variety.
 * 2) It is the marginal subgroup for any collection of words that generates the variety, in the sense that a group is in the variety iff all those words are trivial in it.

Extreme examples

 * The whole group is the marginal subgroup for the word $$w(x) = x$$.
 * The trivial subgroup is the marginal subgroup for the empty word, i.e., the word that evaluates to the identity element for any element.

Power words

 * The marginal subgroup for the power word $$x^2$$ is the set of central elements of order dividing 2.

Facts

 * Marginal subgroup is closed in T0 topological group

Dual property
The notion of marginal subgroup is somewhat dual to the notion of verbal subgroup, which is the subgroup generated by all elements realized using the given word.