In this talk I'll first recall the notion of a "cellular automaton over a group" (also presenting celebrated Conway's "Game of Life") and then consider the linear version of the so-called "Garden of Eden Theorem" and Gottshalk's "Surjunctivity problem" which arise from symbolic dynamics. After recalling the notion of "amenability" for groups, I'll discuss the Gromov-Weiss notion of a "sofic group". Finally, I'll discuss Kaplansky's conjectures on the non-existence of idempotents and on stable finiteness of group rings and relate these problems to the theory of linear cellular automata.