 

Danny Calegari and Alden Walker
Isometric endomorphisms of free groups view print


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


Abstract
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 n1/2.
Our methods also let us construct explicit (and computable) quasimorphisms
which certify these facts.


Acknowledgements
The first author was supported by NSF grant DMS 1005246.


Author information
Danny Calegari:
Department of Mathematics, Caltech, Pasadena CA, 91125
dannyc@its.caltech.edu
Alden Walker:
Department of Mathematics, Caltech, Pasadena CA, 91125
awalker@caltech.edu

