By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{1}=1$, $\underline{x \cdot y}=\underline{x} \cdot \underline{y}$. Notice that the elements of this group have the form $\underline{x_1} \cdot \underline{x_2}^{-1} \cdot \underline{x_3} \cdot \underline{x_4}^{-1} \cdot \dotsc$.
Question 1: Does this group have a name? It is analogous to the Grothendieck group, which is the left adjoint of the forgetful functor from abelian groups to commutative monoids, so perhaps we may call it the non-abelian Grothendieck group?
Question 2: Is there any criterion when an element of this group vanishes?
Best Answer
George Bergman refers to it as the universal enveloping group of the monoid. See Section 3.11, page 81, of his universal algebra notes/forthcoming book: http://math.berkeley.edu/~gbergman/245/
He also references P.M. Cohn's Universal Algebra book, and two papers of Mal'cev in which he establishes conditions for the universal map to be an embedding.
P.S. As far as I know, the construction predates Grothendieck's work; but because of the applications the latter found for the abelian case, his work is better known. In fact, this is not the first time I've seen the general case described as "the nonabelian version of the Grothendieck group"...