Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth Have Finitely Generated Fundamental Groups

TL;DR

This paper extends the finite generation of fundamental groups to certain non-smooth metric spaces with non-negative Ricci curvature, using advanced geometric estimates, thus contributing to the understanding of their topological structure.

Contribution

It generalizes Sormani's finite generation result to non-branching RCD(0,N) spaces, supporting Milnor's conjecture in this broader setting.

Findings

Proves finitely generated fundamental groups for these spaces

Utilizes Abresch-Gromoll type excess estimates in non-smooth contexts

Supports Milnor's conjecture for non-compact RCD(0,N) spaces

Abstract

In this paper, we generalize the finite generation result of Sormani to non-branching $RCD(0,N)$ geodesic spaces (and in particular, Alexandrov spaces) with full support measures. This is a special case of the Milnor's Conjecture for complete non-compact $RCD(0,N)$ spaces. One of the key tools we use is the Abresch-Gromoll type excess estimates for non-smooth spaces obtained by Gigli-Mosconii.