Linearly Sofic Lie Algebras: Extensions and Embeddings

We prove that any countable sofic (respectively, weakly sofic, hyperlinear, linearly sofic) group can be embedded into a finitely generated sofic (respectively, weakly sofic, hyperlinear, linearly sofic) group. We also prove an analogous result that any countable dimensional linearly sofic Lie algebra can be embedded into a finitely generated linearly sofic Lie algebra. In the course of proving this result, we prove that, over a field of characteristic 0, every extension of a linearly sofic Lie algebra by a Lie algebra with amenable universal enveloping algebra is linearly sofic.