New York Journal of Mathematics
Volume 17 (2011) 713-743


Danny Calegari and Alden Walker

Isometric endomorphisms of free groups

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Published: October 25, 2011
Keywords: Free groups, stable commutator length, Gromov norm, fatgraph, quasimorphism, small cancellation
Subject: 20F65, 20J05, 20E05, 20P05, 57M07

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.


The first author was supported by NSF grant DMS 1005246.

Author information

Danny Calegari:
Department of Mathematics, Caltech, Pasadena CA, 91125

Alden Walker:
Department of Mathematics, Caltech, Pasadena CA, 91125