Non-abelian Grothendieck group 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?
 A: I'd just like to point out that in the many-object case this has been called the fundamental groupoid of a category by Paré, which would suggest that it could be called the fundamental group of a monoid, a possibility strengthened by the fact that, as Benjamin Steinberg points out in the comments, it's the fundamental group of the classifying space of the monoid. Another possibility might be classifying group, but that's probably ill-advised since it's not clear (to me) what it classifies.
A: 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"...
A: I have seen "(universal) enveloping group of the monoid" used for this construction. Mal'cev has found necessary and sufficient conditions for injectivity of the comparison map.
Anatoliy I.Mal’cev, Über die Einbettung von assoziativen Systemen in Gruppen, Mat. Sb. N.S. 6 (1939) 331-336.
Anatoliy I. Mal’cev, Über die Einbettung von assoziativen Systemen in Gruppen, II, Mat. Sb. N.S. 8 (1940) 251-264.
See also Chapter VII in P. M. Cohn, Universal Algebra, second edition, Reidel, 1981.
