Ismael Morales López

A finitely generated group G is termed parafree if it is residually nilpotent and there exists a free group F which has the same isomorphism types of nilpotent quotients as G. These groups were introduced by Baumslag in 1968 and many fundamental and intriguing questions about them still remain open. They have recently received much attention in connection to Remmeslenikov's conjecture on the profinite rigidity of free groups. 

In this thesis we offer a new way of producing parafree groups combining techniques on pro-p groups and L^2-Betti numbers. 

There are two original results. On the one hand, we can completely describe in simple terms which free products of finitely generated groups, with abelian amalgams, are parafree.  Secondly, we can give a description of which abelian HNN extensions of finitely generated groups are parafree. The latter is not as explicit. Nevertheless, if G is a parafree group with two-generated abelianisation, we can describe in very concrete terms which abelian HNN extensions of G are parafree.

The aim of this exposition is to discuss the proofs of the following two results:

• Helfgott's theorem on the rapid growth of generating sets of SL(2, Z/p), which uses techniques from additive combinatorics. 

• The influential work of Bourgain and Gamburd on expanding Cayley graphs of SL(2, Z/p) with respect to the reduction modulo p of a generic subset S of the group SL(2, Z).

These results provided many explicit families of expander graphs. We essentially reproduce their original arguments but we also implement some simplifications and improvements based on subsequent developments due to Babai, Gowers, Nikolov, Pyber and Tao. There is no claim of originality in this exposition, except in a possibly new proof of the quasirandomness of PSL(2, Z/p) that merely uses divisibility and orthogonality relations of characters.