Research Projects

Quasi isometric rigidity has become a central theme in Geometric group theory, with Gromov's polynomial growth theorem proving that nilpotent groups and lattices in semisimple Lie groups are quasi isometrically rigid. The current research focuses on studying twisted conjugacy problems in groups or classes of groups and their quasi isometric rigidity. Four cases are discussed: twisted conjugacy problem in s-arithmetic groups, quasi-isometric rigidity of groups with the R?-property, z-classes related to twisted conjugacy action of groups on itself, endomorphisms in infinite groups, and topological transitivity property of continuous maps in certain spaces. The proposed research is based on topology and group theory, using techniques from Differential Geometry, Lie groups, and Linear Algebra. The problems include classifying S-arithmetic groups with R?-property, studying the twisted conjugacy problem in other groups, describing a class of groups with geometric R?-property, studying conjugate subgroups to determine R?-property, characterizing transitive maps of dynamical systems, and studying continuous maps of certain dynamical systems to link them with geometric group theory.