## V. Chepoi

#### 6 - 17 May

### Title: Helly graphs and Helly groups

Helly graphs are the graphs in which the family of balls satisfies the Helly property, i.e., any collection of pairwise intersecting balls has a common vertex. Helly complexes are clique complexes of Helly graphs. Helly groups are groups which act geometrically on Helly complexes.

Helly graphs are discrete analogs of hyperconvex/injective geodesic metric spaces and analogously to such spaces they have a canonical closure property: for any graph G there exists the smallest Helly graph into which G isometrically embeds. Examples of Helly groups are CAT(0) cubical groups, hyperbolic groups, (C4)-(T4) graphical small cancelation groups.

We will present the basic properties, examples, constructions, and characterizations of Helly graphs. In particular, we will present a local-to-global characterization of such graphs. We will also discuss the basic examples Helly groups and show that Helly groups are biautomatic.