Realization and Dismantlability

The Nielsen realization problem asks if a finite group of mapping classes of a surface can be
realized by a group of homeomorphisms. Since its positive solution by Kerckhoff, similar
realization theorems have been proven in other contexts, in particular graphs and handlebodies.
In this talk we will develop a unified combinatorial approach to such questions which gives
simplified proofs for realization theorems for punctured surfaces, graphs and handlebodies.
The key is a non-positive curvature property of graphs called dismantlability, which is enjoyed
by combinatorial complexes (like the arc graph, disk graph or sphere graph) This is joint work
with Damian Osajda and Piotr Przytycki.