Group satisfying no nontrivial identity

Definition in terms of words
A group satisfying no nontrivial identity is a group such that for any word $$w(x_1,x_2,\dots,x_n)$$ with the property that:

$$w(g_1,g_2,\dots,g_n) = e \ \forall g_1,g_2,\dots,g_n \in G$$,

we have that $$w$$ is a trivial word; in other words, if $$F_n$$ is a free group on $$n$$ generators $$a_1, a_2, \dots, a_n$$, $$w(a_1,a_2,\dots,a_n) = e$$.

Definition in terms of verbal subgroups
A group satisfying no nontrivial identity is a group that cannot be expressed as the quotient of a free group by a normal subgroup that contains a nontrivial verbal subgroup.

Opposite properties
Group satisfying a nontrivial identity: This is the precise opposite.

Stronger properties

 * Weaker than::Group having a free non-abelian subgroup