Abstract
Let k be a field, let
{\mathfrak{A}_{1}}
be the k-algebra
{k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]}
of Laurent polynomials in
{x_{1},\dots,x_{s}}
, and let
{\mathfrak{A}_{2}}
be the k-algebra
{k[x,y]}
of polynomials in the commutative indeterminates x and y. Let
{\sigma_{1}}
be the monomial k-automorphism of
{\mathfrak{A}_{1}}
given by
{A=(a_{i,j})\in GL_{s}(\mathbb{Z})}
and
{\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}}
,
{1\leq i\leq s}
, and let
{\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])}
. Let
{D_{i}}
,
{1\leq i\leq 2}
, be the ring of fractions of the skew polynomial ring
{\mathfrak{A}_{i}[X;\sigma_{i}]}
, and let
{D_{i}^{\bullet}}
be its multiplicative group. Under a mild restriction for
{D_{1}}
, and in general for
{D_{2}}
, we show that
{D_{i}^{\bullet}}
,
{1\leq i\leq 2}
, contains a free subgroup. If
{i=1}
and
{s=2}
, we show that a noncentral normal subgroup N of
{D_{1}^{\bullet}}
contains a free subgroup.
Random elements of a free profinite group generate a free subgroup
