Cofinite hopficity and free-by-infinite cyclic groups

Jonathan Hillman

Wednesday 3rd December, 2008 16:00-17:00 Mathematics Building, room 214


[This is joint work with M.Bridson, D.Groves and G.Martin.] A group is cofinitely hopfian if every endomorphism with image of finite index is an automorphism. The need for this concept arose in a topological context (which I shall not explain). It is clearly an extension of a notion introduced long ago by H.Hopf in connection with surface groups. We shall compare this with several related notions, and show that a finitely generated free-by-infinite cyclic group $G=F(r)\rtimes{Z}$ is cofinitely hopfian if and only if it has trivial centre. Using earlier work by Gonzalez-Acuna and Whitten (on cohopficity) it follows that this criterion holds also for knot groups.

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