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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, Thompsons groups, and even some elementary amenable groups.
In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of s
Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably coHopfian.
For oriented connected closed manifolds of the same dimension, there is a transitive relation: $M$ dominates $N$, or $M ge N$, if there exists a continuous map of non-zero degree from $M$ onto $N$. Section 1 is a reminder on the notion of degree (Bro
The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and $operatorname{Out}(mathbb F_n)
Given a group $G$ and a subset $X subset G$, an element $g in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has b