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.