UC San Diego

In 1964, Golod and Shafarevich found a sufficient condition for an algebra presented by generators and relators to be infinite dimensional. This condition gives rise to an analogous condition for a pro-p group to be infinite. Groups and algebras that satisfy this condition are called GS groups and algebras. In 1983, A. Lubotzky exhibited a class of GS groups with the property that all of their finite index subgroups are also GS. The underlying topological structure was essential in Lubotzky's examples. This dissertation is an account of our search for algebraic analogs to these examples and our exploration of conditions under which subalgebras of GS algebras are themselves GS. In Chapter 3 it is proved that finite codimensional subalgebras of finitely presented algebras are finitely presented. However, subalgebras of finite codimension in graded GS algebras are not necessarily GS (Chapter 4). In Chapter 5 we prove that infinitely many Veronese powers of a graded algebra presented by m generators and r relators are GS if r < 1/ 4(m/2-1)². For quadratic algebras, the bound is improved to r < 4/25m². We prove that for a generic quadratic algebra A presented by m generators and r relators, all Veronese powers of A are GS if r less than or equal to 4/ 25m² and all but finitely many Veronese powers of A are not GS if r < 4/25m²