Heegaard splittings and virtually special square complexes

Abstract

We give a new perspective of Heegaard splittings in terms square complexes and Guirardel's notion of a \textit{core} which allows for combinatorial measurement of the obstruction to being a connect sum of Heegaard diagrams. A Heegaard splitting is a decomposition of a closed orientable $3$-manifold into two isomorphic handle bodies that have a shared boundary surface. Usually, a number of curves on the shared boundary surface, called a Heegaard diagram, are used to describe a Heegaard splitting. We define a larger object, the \textit{augmented Heegaard diagram}, by building on methods of Stallings and Guirardel to encode the information of a Heegaard splitting. \textit{Augmented Heegaard diagrams} have several desirable properties: each 2-cell is a square, they have \textit{non-positive combinatorial curvature} and they are \textit{virtually special}. Restricting to manifolds that do not have $S^1 \times S^2$ as a connect summand, augmented Heegaard diagrams are tied to the decomposition of a $3$-manifold via connect sum as described above.