UC San Diego

We consider uniform exponential growth in algebras. We give conditions for the uniform exponential growth of descending-filtered algebras and prove that an N-graded algebra has uniform exponential growth if it has exponential growth. We use this to prove that Golod- Shafarevich algebras and group algebras of Golod- Shafarevich groups have uniform exponential growth. We prove that the twisted Laurent extension of a free commutative polynomial algebra with respect to an endomorphism with some eigenvalue of norm not 1 must have uniform exponential growth. We prove that the group algebra of a (free abelian)-by-cyclic group has polynomially-bounded or uniform exponential group. We prove that the uniform exponential growth of the universal enveloping algebra U of a Lie algebra L implies uniform exponential growth of L, and contrariwise should L be N- graded, and prove the same result for restricted Lie algebras. We use this to give several conditions equivalent to the uniform exponential growth of graded algebra associated to a group algebra filtered by powers of its fundamental ideal