Given a discrete group $\Gamma$ generated by $S$ let $K[S]$ denote the subring of the group-ring $K[\Gamma]$ generated by $S$ (over a commutative ring $K$). The ring $K[S]$ is thus a quotient of the free algebra generated by non-commutative variables indexed by $S$. Are there examples where the free monoid $S^*$ injects into $\Gamma$ but $K[S]$ is not free? (I guess the answer should be yes).

On the other hand, there should also be examples where $K[S]$ is the free algebra of non-commutative polynomials over $S$, although $\Gamma$ is not free (this implies of course that $S^*$ injects into $\Gamma$).

Remark: The subring $K[S]$ is of course isomorphic to $K[\Gamma]$ if $\Gamma$ is a torsion-group.

Given a discrete group $\Gamma$ generated by $S$ let $K[S]$ denote the subring of the group-ring $K[\Gamma]$ generated by $S$ (over a commutative ring $K$). The ring $K[S]$ is thus a quotient of the free algebra generated by non-commutative variables indexed by $S$. Are there examples where the free monoid $S^*$ injects into $\Gamma$ but $K[S]$ is not free? (I guess the answer should be yes).

On the other hand, there should also be examples where $K[S]$ is the free algebra of non-commutative polynomials over $S$, although $\Gamma$ is not free (this implies of course that $S^*$ injects into $\Gamma$).

Remark: The subring $K[S]$ is of course isomorphic to $K[\Gamma]$ if $\Gamma$ is a torsion-group (or more generally, if all elements of $S$ represent torsion elements of $\Gamma$).