Carlos Barrera-Rodríguez

Department of Mathematics,
University of California at Davis,
Davis CA 95616, USA.

Title: A collection of multicurve complexes

Abstract: We introduce a new collection of simplicial complexes associated to a connected orientable compact surface $S=S_{g,n}$, called $k$-curve complexes and denoted by $k$-$\cal{C} (S)$. Each complex is realized by: vertices given by multicurves ($(k-1)$-simplices of the original curve complex of $S$) and edges given by a restricted nonfillingness property between vertices. We prove that for each $k$, $1 \leq k \leq 3g+n-5$, the corresponding complex of this collection is connected and we study the coarse geometry of $1$-$\cal{C} (S)$. In particular, we prove that $1$-$\cal{C} (S)$ is hyperbolic. We also show a small application of these complexes to a Heegaard splitting of a manifold and a useful relation with the mapping class group of the surface $S$.