Authors: Cesar Ceballos ¹; Jean-Philippe Labbé ¹; Christian Stump ^2,3

• 1 Institute of Mathematics [Berlin]
• 2 Leibniz Universität Hannover=Leibniz University Hannover
• 3 Institut für Algebra, Zahlentheorie und Diskrete Mathematik

We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex.