Maps from Out(F_n) to Out(F_m)

Karen Vogtmann (Cornell University)

Monday 12th October, 2009 16:00-17:00 Mathematics Building, room 204


The group Aut(F_n) of automorphisms of a free group exhibits rigidity phenomena, including the fact that maps from Aut(F_n) to Aut(F_m) have finite image (of order at most two) if m is less than n. On the other hand, for m greater than n there are natural inclusions, given by letting an automorphism act on n generators while fixing the rest. If we consider the outer automorphisms group Out(F_n) instead, the fact that there are basically no maps for m less than n is still true. For m greater than n the situation is not at all clear, since an inner automorphism does not remain inner under the map described above. Bogopolski and Puga constructed embeddings for certain values of m. We give a geometric interpretation for these maps and show that these values of m are the only ones for which this type of construction can work. We then show that there are no imbeddings of Out(F_n) in Out(F_m) if m is between n and 2n. This is joint work with Martin Bridson.

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