Abstract
We will discuss Helly graphs and injective metric spaces: basic definitions, elementary properties, simple examples. We will focus on nonpositive curvature features, such as local properties, bicombings and classification of isometries. We will present numerous groups acting geometrically by isometries on such spaces, such as hyperbolic groups, braid groups, mapping class groups and lattices in Lie groups.
Previous talk: Thomaes Haettel - Group actions on Helly graphs and injective metric spaces - I-III
Talk
The notes to the talk can be found here.
Teaser:
- Are hyperbolic groups CAT(0)? Open!
- Are braid groups CAT(0)? Open!
- Are CAT(0) groups biautomatic? No!
Well we don’t know the answers to the first two but we know:
- Hyperbolic groups are Helly
- Braids are Helly
- Helly groups are biautomatic.
IV Lattices
We show that balls and intervals “fulfil the Helly property”.
Definition
A lower semi-lattice is flag if