The characteristic polynomials of geometric automorphisms of a free group of finite rank at least three form a nowhere dense set in the Zariski topology.

We prove under certain natural conditions the finiteness of the number of isomorphism classes of Zariski dense subgroups in semisimple groups with isomorphic p-adic closures.

We prove the Zariski dense orbit conjecture in positive characteristic for endomorphisms of G a N defined over F p ‾ .

We prove a general Zariski-van Kampen-Lefschetz type theorem for higher homotopy groups of generic and nongeneric pencils on singular open complex spaces.

We define a new topological invariant of complex projective plane curves. As an application, we present examples of arithmetic Zariski pairs.

We prove in a synthetic manner a general Zariski-van Kampen type theorem for higher homotopy groups of pencils on singular complex spaces.