Patterson–Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for outer space

Ilya Kapovich and Martin Lustig

Vol. 277 (2015), No. 2, 371–398

DOI: 10.2140/pjm.2015.277.371

Abstract

We quantitatively relate the Patterson–Sullivan currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on outer space and to Guirardel’s intersection number. Thus we show that, given $N\ge 2$ and $ϵ>0$, there exists a constant $c=c\left(N,ϵ\right)>0$ such that for any two trees $T,S\in {cv}_N$ of covolume $1$ and injectivity radius $\ge ϵ$, we have

$$|log\langle S,{\mu }_T\rangle -d_L\left(T,S\right)|\le c,$$

where $d_L$ is the asymmetric Lipschitz metric on the Culler–Vogtmann outer space, and where ${\mu }_T$ is the (appropriately normalized) Patterson–Sullivan current corresponding to $T$. As a corollary, we show there exist constants $C_1\ge 1$ and $C_2\ge 1$ (depending on $N,ϵ$) such that for any $T,S$ as above we have

$$\frac{1}{C_1}log{i}_c\left(T,S\right)-C_2\le log\langle S,{\mu }_T\rangle \le C_{1}log{i}_c\left(T,S\right)+C_2,$$

where $i_c$ is the combinatorial version of Guirardel’s intersection number. We apply these results to the properties of generic stretching factors of free group automorphisms. In particular, we show that for any $N\ge 2$, there exists a constant $0<{\rho }_N<1$ such that for every automorphism $\varphi$ of $F_N=F\left(A\right)$, we have

$$0<{\rho }_N\le \frac{{\lambda }_A\left(\varphi \right)}{{\Lambda }_A\left(\varphi \right)}\le 1.$$

Here ${\lambda }_A$ is the generic stretching factor of $\varphi$ with respect to the free basis $A$ of $F_N$ and ${\Lambda }_A\left(\varphi \right)$ is the extremal stretching factor of $\varphi$ with respect to $A$.

Keywords

Culler–Vogtmann's outer space, Patterson–Sullivan measures, geodesic currents

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 57M07, 37B99, 37D40

Milestones

Received: 26 August 2014

Revised: 12 February 2015

Accepted: 12 February 2015

Published: 15 September 2015