  In this paper we study obstructions to presentability by products for
finitely generated groups. Along the way we develop both the concept of
acentral subgroups, and the relations between presentability by products on the
one hand, and certain geometric and measure or orbit equivalence invariants of
groups on the other. This leads to many new examples of groups not presentable
by products, including all groups with infinitely many ends, the (outer)
automorphism groups of free groups, Thompson's groups, and even some elementary
amenable groups.
