Sang-hyun Kim (김상현), On Gromov Conjecture and Topological Jigsaw Puzzle

Inspired by the famous virtual Haken conjecture in 3–manifold theory, Gromov asked whether every one-ended word-hyperbolic group contains a surface group. One simple, but still captivating case, is when the word-hyperbolic group is given as the double of a free group with a cyclic edge group. In the first part of the talk, I will describe the polygonality of a word in a free group, and a relation between polygonality and Gromov’s question. Polygonality is a combinatorial property, which is very much like solving a “topological jigsaw puzzle”. In the second part, I will describe a reduction to a purely (finite) graph theoretic conjecture using the Whitehead graph. Part I is a joint word with Henry Wilton (Caltech).