Fundamental groups of aspherical manifolds and maps of non-zero degree

Abstract

We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Löh, providing new classes of aspherical manifolds -beyond those non-positively curved ones which were predicted by Gromov -that do not admit maps of non-zero degree from direct products.A sample application is that an aspherical geometric 4-manifold admits a map of non-zero degree from a direct product if and only if it is a virtual product itself. This completes a characterization of the product geometries due to Hillman. Along the way we prove that for certain groups the property IIPP is a criterion for reducibility. This especially implies the vanishing of the simplicial volume of the corresponding aspherical manifolds. It is shown that aspherical manifolds with reducible fundamental groups do always admit maps of non-zero degree from direct products. FUNDAMENTAL GROUPS OF ASPHERICAL MANIFOLDS AND MAPS OF NON-ZERO DEGREE 3 manifolds using maps of non-zero degree [4,38] and the monotonicity of Kodaira dimensions with respect to the existence of maps of non-zero degree [39] will be presented in a subsequent paper [28].

Ordering Thurston’s geometries by maps of nonzero degree

“…For the converse, we have that a closed Sol 4 1 -manifold Z is finitely covered by a non-trivial circle bundle over a closed oriented Sol 3 -manifold and the center of π 1 (Z) is infinite cyclic (Prop. 3.4 [16,Section 6.3.3] for presentations of the fundamental groups of the above manifolds.…”

“…Using the property "not (infinite-index) presentable by products", the following result is an application in [16]: Thus closed 4-manifolds possessing a non-product solvable geometry are not dominated by products. We therefore only need to show the converse, namely, that solvable manifolds do not dominate manifolds modeled on one of the geometries X 3 × R or the reducible H 2 × H 2 geometry.…”

“…We now show that M cannot be dominated by any other (non-hyperbolic) geometric closed aspherical 4-manifold N. Since M is not dominated by products, it suffices to show that M cannot be dominated by a closed manifold N possessing one of the geometries Sol 4 1 , Nil 4 , Sol 4 m =n or Sol 4 0 . For each of those geometries, π 1 (N) has a normal subgroup of infinite index, which is free Abelian of rank one (geometries Sol 4 1 and Nil 4 ) or three (geometries Sol 4 m =n and Sol 4 0 ); see [16,Section 6] for details. If there were a (π 1 -surjective) map of non-zero degree f : N −→ M, then by [13, Theorem IX.6.14] either f would factor through a lower dimensional aspherical manifold or π 1 (M) would be free Abelian of finite rank.…”

Anosov diffeomorphisms of products II. Aspherical manifolds

“…If M carries the geometry Sol 4 1 , then by Theorem 4.5 (see also [15,Prop. 6.15]) a presentation of its fundamental group is given by is a diffeomorphism.…”

“…Using this, one can moreover derive that every closed Sol 4 1 -manifold is a virtually non-trivial circle bundle over a Sol 3 -manifold [15,Prop. 6.15].…”