Acyclic embeddings of open Riemann surfaces into new examples of elliptic manifolds

TL;DR

This paper explores the properties of elliptic manifolds, providing new examples with free fundamental groups and demonstrating that all open Riemann surfaces can be embedded acyclically into elliptic manifolds, advancing understanding of complex geometry.

Contribution

It introduces new elliptic manifolds with free fundamental groups and shows that all open Riemann surfaces can be embedded acyclically into elliptic manifolds, addressing open questions in the field.

Findings

Quotients of C^n by discrete affine groups are elliptic.

New examples of elliptic manifolds with free fundamental groups.

Open Riemann surfaces can be embedded acyclically into elliptic manifolds.

Abstract

The geometric notion of ellipticity for complex manifolds was introduced by Gromov in his seminal 1989 paper on the Oka principle, and is a sufficient condition for a manifold to be Oka. In the current paper we present contributions to three open questions involving elliptic and Oka manifolds. We show that quotients of C^n by discrete groups of affine transformations are elliptic. Combined with an example of Margulis, this yields new examples of elliptic manifolds with free fundamental groups and vanishing higher homotopy. Finally we show that every open Riemann surface embeds acyclically into an elliptic manifold, giving a partial answer to a question of Larusson.