The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms
Vadim Kaimanovich, Ilya Kapovich, Paul Schupp
TL;DR
The paper defines and analyzes a noncommutative analogue of translation length, the generic stretching factor λ(φ), for free group homomorphisms into groups with a left-invariant semi-norm. It leverages Kingman’s Subadditive Ergodic Theorem and random-walk models to show λ(φ) exists, is rational with precise arithmetic constraints, and is computable in key cases, particularly for free actions on trees and endomorphisms. A central result is a gap phenomenon in the spectrum of λ over Aut(F_k), with λ=1 only in trivial/simple cases and otherwise λ bounded away from 1 by an explicit amount, alongside rigidity statements tying small distortions to simple automorphisms. The work connects probabilistic methods, regular languages, and geometric group theory (via actions on trees and currents) to provide a unified approach to generic distortion and to pose several natural open problems about rationality, convergence, and uniformity across bases.
Abstract
Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $φ$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $λ(φ)$, which is a non-commutative generalization of the translation number. We concentrate on the situation when $φ:F_k\to Aut(X)$ corresponds to a free action of $F_k$ on a simplicial tree $X$, in particular, when $φ$ corresponds to the action of $F_k$ on its Cayley graph via an automorphism of $F_k$. In this case we are able to obtain some detailed ``arithmetic'' information about the possible values of $λ=λ(φ)$. We show that $λ\ge 1$ and is a rational number with $2kλ\in \mathbb Z[ \frac{1}{2k-1} ]$ for every $φ\in Aut(F_k)$. We also prove that the set of all $λ(φ)$, where $φ$ varies over $Aut(F_k)$, has a gap between 1 and $1+\frac{2k-3}{2k^2-k}$, and the value 1 is attained only for ``trivial'' reasons. Furthermore, there is an algorithm which, when given $φ$, calculates $λ(φ)$.