Group actions on injective spaces and Helly graphs

TL;DR

This paper reviews group actions on injective metric spaces and Helly graphs, highlighting their properties, constructions, and the interplay between the two theories, with applications to geometric group actions.

Contribution

It provides a comprehensive overview of injective spaces and Helly graphs, including new constructions and insights into group actions on these structures.

Findings

Injective spaces and Helly graphs share fundamental properties.

Various constructions of these spaces support interesting group actions.

Open questions suggest directions for future research.

Abstract

These are lecture notes for a minicourse on group actions on injective spaces and Helly graphs, given at the CRM Montreal in June 2023. We review the basics of injective metric spaces and Helly graphs, emphasizing the parallel between the two theories. We also describe various elementary properties of groups actions on such spaces. We present several constructions of injective metric spaces and Helly graphs with interesting actions of many groups of geometric nature. We also list a few exercises and open questions at the end.