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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.