12 research outputs found
Groups not presentable by products
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.Comment: 25 pages, minor changes, to appear in Groups, Geometry, and Dynamic
On stability of non-domination under taking products
We show that non-domination results for targets that are not dominated by
products are stable under Cartesian products.Comment: 6 page
Fundamental classes not representable by products
We prove that rationally essential manifolds with suitably large fundamental
groups do not admit any maps of non-zero degree from products of closed
manifolds of positive dimension. Particular examples include all manifolds of
non-positive sectional curvature of rank one and all irreducible locally
symmetric spaces of non-compact type. For closed manifolds from certain
classes, say non-positively curved ones, or certain surface bundles over
surfaces, we show that they do admit maps of non-zero degree from non-trivial
products if and only if they are virtually diffeomorphic to products.Comment: 22 pages; updated references and corrected a typo; to appear in the
Journal of the London Mathematical Societ
Integral Foliated Simplicial Volume of Aspherical Manifolds
We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth. Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable residually finite fundamental group (being equal to zero in the latter case). However, we show that integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4