Quantifying local embeddings into finite groups

We study a function LΓ which quantifies the LEF (local embeddability into finite groups) property for finitely generated group Γ. compute this growth in some examples, including certain wreath products. compare with analogous quantitative version of residual finiteness, and exhibit family residually groups nevertheless admit many more local embeddings than they do quotients. Along way, we give new proof that B.H. Neumann's continuous 2-generated contains no presented group, result originally due to Baumslag Miller. versions soficity other metric approximation properties groups. Finally, show there exists “universal” is an upper bound on any given number generators, (for non-cyclic such non-computable.