UC San Diego

Metric approximable groups have been studied since the introduction of sofic groups byGromov [ 13]. Since then, further classes of metric approximable groups have been studied, such as hyperlinear [24], linearly sofic [4 ], and weakly sofic groups [11]. Due to their connection with many open problems such as Gottschalk’s Surjunctivity Conjecture [ 12], Connes’ Embedding Problem [7], and Kaplansky’s Direct Finiteness Conjecture [ 18], metric approximable groups have generated much interest. Recently, metric approximability has been extended from groups to associative algebras through linearly sofic associative algebras [ 4]. These algebras were shown to have many similar properties as linearly sofic groups, such as equivalent characterization through metric x ultraproducts and almost representations. Additionally, non-linearly sofic algebras proved easier to find as compared to the case of linearly sofic groups. However, certain properties that hold for groups have not been shown to hold for associative algebras, such as preservation of linear soficity through certain extensions. In this dissertation, we continue the work in [4 ] by extending the definition of linear soficity further to Lie algebras. Lie algebras are a natural object to study in this area, as they have many similarities to both groups and associative algebras. In §3, we define linear soficity for Lie algebras using metric ultraproducts, as well as give an equivalent characterization through the use of almost representations. We also give some examples of linearly sofic Lie algebras. In §4, we show the connection between linear soficity in Lie and associative algebras by showing that, over fields of characteristic 0, a Lie algebra is linearly sofic if and only if its universal enveloping algebra is. In §5, we look at extensions of linearly sofic Lie algebras. We show that any extension of a linearly sofic Lie algebra by a Lie algebra with an amenable universal enveloping algebra is linearly sofic, as in the case of groups. In §6, we use wreath products to show that a countable metric approximable group is embeddable in a finitely generated metric approximable group of the same kind. In addition, we use a similar argument using wreath products to show that a countable dimensional linearly sofic Lie algebra is embeddable in a finitely generated linearly sofic Lie algebra.