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In this talk I will discuss some new results and methods on free actions of groups. Non-Archimedean infinite words and elimination processes are the main players in the game. These methods were extensively, though sometimes implicitly, used in our joint with Olga Kharlampovich solution of the Tarski's problems. It seems now they can provide an adequate tool to attack some open problems concerning with the algebraic structure of finitely generated groups acting freely on Lambda-trees.
The New York Group Theory Seminar and some of the associated conferences are supported by funds from the National Science Foundation, Dean of Science, Maria Tamargo and Dean of Engineering, Joe Barba.