The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Paper

Abstract

Given a free group

of rank

with a fixed set of free generators we associate to any homomorphism

from

to a group

with a left-invariant semi-norm a generic stretching factor,

, which is a non-commutative generalization of the translation number. We concentrate on the situation when

corresponds to a free action of

on a simplicial tree

, in particular, when

corresponds to the action of

on its Cayley graph via an automorphism of

. In this case we are able to obtain some detailed ``arithmetic'' information about the possible values of

. We show that

and is a rational number with

for every

. We also prove that the set of all

, where

varies over

, has a gap between 1 and

, and the value 1 is attained only for ``trivial'' reasons. Furthermore, there is an algorithm which, when given

, calculates