A non-abelian free pro-\(p\) group is not linear over a local field
Pink's recent characterization of compact subgroups of \(\text{GL}_n(F)\), where \(F\) is a non-archimedian local field, is applied to prove that a free non-abelian pro-\(p\) group can not be linear over \(F\). This establishes a conjecture of Lubotzky and Shalev. Using a theorem of Zelmanov the authors deduce some group theoretic properties of linear pro-\(p\) groups over local fields.
linear pro-\(p\) groups over local fields
scientific article; zbMATH DE number 1275267