A Caltech Library Service

Isometric endomorphisms of free groups

Calegari, Danny and Walker, Alden (2011) Isometric endomorphisms of free groups. New York Journal of Mathematics, 17 . pp. 713-743. ISSN 1076-9803.

PDF - Published Version
See Usage Policy.


Use this Persistent URL to link to this item:


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.

Item Type:Article
Related URLs:
URLURL TypeDescription
Additional Information:© 2011 Electronic Journals Project. Received October 2, 2011. Danny Calegari was supported by NSF grant DMS 1005246. We would like to thank Mladen Bestvina and Geo Mess for some useful conversations about this material.
Funding AgencyGrant Number
Subject Keywords:Free groups, stable commutator length, Gromov norm, fatgraph, quasimorphism, small cancellation
Classification Code:2010 MSC: 20F65, 20J05, 20E05, 20P05, 57M07
Record Number:CaltechAUTHORS:20120320-102304082
Persistent URL:
Official Citation:Danny Calegari and Alden Walker Isometric endomorphisms of free groups pp 713-743
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29783
Deposited By: Ruth Sustaita
Deposited On:20 Mar 2012 18:03
Last Modified:18 May 2020 15:54

Repository Staff Only: item control page