A non-amenable finitely presented group without non-abelian free
Abstract: A finitely generated non-amenable group without free subgroups was first constructed by Olshanskii in 1980. It was a solution of a von Neumann problem. A question of whether there exists a finitely presented example is one of a series of well known problems asking about existence of finitely presented "monsters". Together with A.Yu. Olshanskii, we constructed a finitely presented solution of von Neumann's problem. The solution employs ideas from geometry, group theory and computer science. Other applications of our methods include constructions of finitely presented groups and complexes with weird isoperimetric functions, Higman-type embedding theorems, a characterization of NP-complete groups, etc.