Dynamics on freebycyclic groups
Abstract
Given a freebycyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible traintrack map $f$, we construct a $K(G,1)$ $2$complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudoAnosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $\epsilon \in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $\Theta_u \subset X$ such that: (1) the inclusion $\Theta_u \to X$ is $\pi_1$injective and $\pi_1(\Theta_u) = \mathrm{ker}(u)$, (2) $u(\epsilon) = \chi(\Theta_u)$, (3) $\Theta_u \subset X$ is a section of the semiflow and the first return map to $\Theta_u$ is an expanding irreducible train track map representing $\varphi_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{\varphi_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $\varphi_u$ is precisely $\mathfrak H(u)$, (5) if $\varphi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $\varphi_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.
 Publication:

arXiv eprints
 Pub Date:
 January 2013
 arXiv:
 arXiv:1301.7739
 Bibcode:
 2013arXiv1301.7739D
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Dynamical Systems;
 Mathematics  Group Theory;
 20F65 (Primary);
 57M;
 37B;
 37E (Secondary)
 EPrint:
 v7: Minor organizational and stylistic changes incorporating referee's suggestions. Notably, section 6.3 in v6 has been moved to section 4.5 in v7. 67 pages, 13 figures. Final version