Fundamental groups of aspherical manifolds that collapse

TL;DR

This paper proves that aspherical manifolds collapsing under certain curvature and diameter bounds have fundamental groups with non-trivial abelian subgroups, ruling out non-elementary hyperbolic groups in large cases.

Contribution

It establishes a link between collapsing geometric sequences of aspherical manifolds and the algebraic structure of their fundamental groups, specifically the existence of abelian normal subgroups.

Findings

Fundamental groups have non-trivial finitely generated abelian normal subgroups.

Collapsed manifolds' fundamental groups cannot be non-elementary hyperbolic.

Results apply to sequences with Ricci curvature bounded below and diameter bounded above.

Abstract

We show that if a sequence $M_n$ of closed aspherical $d$-dimensional Riemannian manifolds with Ricci curvature uniformly bounded below and diameter uniformly bounded above collapses, then for all large enough $n$, the fundamental groups $\pi_1(M_n)$ have non-trivial finitely generated abelian normal subgroups. In particular, the groups $\pi_1(M_n)$ cannot be non-elementary hyperbolic.