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

Abstract

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