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Let  k  be a field, let  1{\mathfrak{A}_{1}}    be the  k -algebrak  ⁢   [   x  1   ±  1    ,  …  ,   x  s   ±  1    ]{k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]}    of Laurent polynomials inx  1   ,  …  ,   x  s{x_{1},\dots,x_{s}}   , and let  2{\mathfrak{A}_{2}}    be the  k -algebrak  ⁢   [  x  ,  y  ]{k[x,y]}    of polynomials in the commutative indeterminates  x  and  y . Letσ  1{\sigma_{1}}    be the monomial  k -automorphism of  1{\mathfrak{A}_{1}}    given byA  =   (   a   i  ,  j    )   ∈   G  ⁢   L  s   ⁢   (  ℤ  ){A=(a_{i,j})\in GL_{s}(\mathbb{Z})}    andσ  1   ⁢   (   x  i   )    =    ∏   j  =  1   s    x  j   a   i  ,  j{\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}}   ,1  ≤  i  ≤  s{1\leq i\leq s}   , and letσ  2   ∈    Aut  k   ⁢   (   k  ⁢   [  x  ,  y  ]    ){\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])}   . LetD  i{D_{i}}   ,1  ≤  i  ≤  2{1\leq i\leq 2}   , be the ring of fractions of the skew polynomial ring  i   ⁢   [  X  ;   σ  i   ]{\mathfrak{A}_{i}[X;\sigma_{i}]}   , and letD  i  ∙{D_{i}^{\bullet}}    be its multiplicative group. Under a mild restriction forD  1{D_{1}}   , and in general forD  2{D_{2}}   , we show thatD  i  ∙{D_{i}^{\bullet}}   ,1  ≤  i  ≤  2{1\leq i\leq 2}   , contains a free subgroup. Ifi  =  1{i=1}    ands  =  2{s=2}   , we show that a noncentral normal subgroup  N  ofD  1  ∙{D_{1}^{\bullet}}    contains a free subgroup.

Free subgroups in k ( x 1 ,... , x n )( X ;σ) and k ( x , y )( k ;σ)