The proposed research contains two main directions in group theory and geometry: Independence of Group Elements and Geometric Rigidity. The first consists of problems related to the existence of free subgroups, uniform and effective ways of producing such, and analogous questions for finite groups where the analog of independent elements are elements for which the Cayley graph has a large girth, or non-small expanding constant. This line of research began almost a century ago and contains many important works including works of Hausdorff, Banach and Tarski on paradoxical decompositions, works of Margulis, Sullivan and Drinfeld on the Banach-Ruziewicz problem, the classical Tits Alternative, Margulis-Soifer result on maximal subgroups, the recent works of Eskin-Mozes-Oh and Bourgain-Gamburd, etc. Among the famous questions is Milnor's problem on the exponential verses polynomial growth for f.p. groups, originally stated for f.g. groups but reformulated after Grigorchuk's counterexample. Related works of the PI includes a joint work with Breuillard on the topological Tits alternative, where several well known conjectures were solved, e.g. the foliated version of Milnor's problem conjectured by Carriere, and on the uniform Tits alternative which significantly improved Tits' and EMO theorems. A joint work with Glasner on primitive groups where in particular a conjecture of Higman and Neumann was solved. A paper on the deformation varieties where a conjecture of Margulis and Soifer and a conjecture of Goldman were proved. The second involves extensions of Margulis' and Mostow's rigidity theorems to actions of lattices in general topological groups on metric spaces, and extensions of Kazhdan's property (T) for group actions on Banach and metric spaces. This area is very active today. Related work of the PI includes his joint work with Karlsson and Margulis on generalized harmonic maps, and his joint work with Bader, Furman and Monod on actions on Banach spaces.