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Isometric endomorphisms of free groups. (English) Zbl 1262.20043

Summary: An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries.
Using similar methods, we show that a random fatgraph in a free group is extremal (i.e., is an absolute minimizer for relative Gromov norm) for its boundary; this implies, for instance, that a random element of a free group with commutator length at most \(n\) has commutator length exactly \(n\) and stable commutator length exactly \(n-1/2\). Our methods also let us construct explicit (and computable) quasimorphisms which certify these facts.

MSC:

20F65 Geometric group theory
20E05 Free nonabelian groups
20P05 Probabilistic methods in group theory
20E36 Automorphisms of infinite groups
20F12 Commutator calculus
20F69 Asymptotic properties of groups
57M07 Topological methods in group theory

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