Virtually geometric words and Whitehead's algorithm

Jason Manning, SUNY, Buffalo

Abstract: Whitehead's algorithm takes a collection of (cyclic) words in a free group and reduces them to minimal length. Results of Berge and Zieschang relate the tools used in Whitehead's algorithm to the realizability problem for Heegaard splittings. (This is the question of which finite group presentations are presentations for fundamental groups of 3-manifolds of a special form which comes from a Heegaard diagram.) It turns out these ideas are also applicable to a recent question of Gordon and Wilton about whether every presentation with a single relator is "virtually geometric" - this is related in turn to the question of which hyperbolic groups contain surface subgroups. I'll discuss some of these ideas and answer Gordon and Wilton's question.