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On the geometry of the automorphism group of a free group. (English) Zbl 0836.20045

The groups \(\operatorname{Aut}(F_3)\) and \(\text{Out}(F_3)\) satisfy strictly exponential isoperimetric inequalities; in particular, they are not automatic. For \(n\geq 3\), \(\operatorname{Aut}(F_n)\) and \(\text{Out}(F_n)\) do not admit bounded bicombings of sub-exponential length, hence they cannot act properly and cocompactly by isometries of any simply-connected space of non-positive curvature, and they are not biautomatic.

MSC:

20F28 Automorphism groups of groups
20F65 Geometric group theory
20E05 Free nonabelian groups
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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