### Heegaard splittings and square complexes

Geometry and Topology Seminar

24th May 2022, 3:15 pm – 4:15 pm

Fry Building, 2.04

A construction of Stallings encodes the information of a Heegaard splitting as a continuous map from the splitting surface to a product of graphs. In this talk we investigate this construction from a more geometric perspective, and find that Heegaard splittings can be encoded as non-positively curved square complexes. Looking at the square complex structure from a geometric group theory view-point, we see that it is intimately connected to another combinatorially tractable square complex: the Guirardel core of a pair of actions on two trees. We then look at the square complexes associated to Heegaard splittings of 3-manifolds and relate some properties of the 3-manifolds to properties of the square complexes.

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