Notes of Christopher Leininger’s Cambridge lecture 23-06-2017

Free-by-cyclic groups and trees

Joint work with S. Dowdall and I. Kapovich.

The Bieri-Neumann-Strebel invariant is an open subset ${\sigma G}$ of ${H^1(G)=Hom(G,{\mathbb R})}$, it is the set of ${u}$ such that ${\omega_u^{-1}({\mathbb R}_+)\rightarrow \hat X}$ is surjective on ${\pi_1}$. Here, ${\hat X}$ is the torsion free abelian cover of ${X=BG}$ and ${\omega_u}$ is an equivariant map ${\hat X\rightarrow{\mathbb R}}$ representing ${u}$.

If ${G}$ is free-by-cyclic, one can refine

$\displaystyle \begin{array}{rcl} \Sigma_{\mathbb Z} G=\{u\in\Sigma G\,;\,u(G)={\mathbb Z}\}. \end{array}$

Geoghegan-Mihalik-Sapir-Wise show that for every ${u\in \Sigma_{\mathbb Z} G}$, ${ker(u)}$ is locally free and there exists an outer automorphism ${\phi_u}$ and a finitely generated subgroup ${Q_u such that ${G=Q_u *_{\phi_u}}$. In particular, if ${u\in \Sigma_{\mathbb Z} G\cap(-\Sigma_{\mathbb Z} G)}$, then one can take ${Q_u=ker(u)}$.

From now on, we assume that ${\phi}$ is atoroidal and fully irreducible. Then ${G}$ is hyperbolic, and there exists an expanding irreducible train track representative (Bestvina-Handel). Let ${X=X_f}$ be the mapping torus. It carries the suspension of ${\phi}$, which is a one-sided flow (action of semi-group ${({\mathbb R}_+,+)}$). The representative ${\omega_u}$ of integral cohomology class ${u}$ factors to a map ${X\rightarrow S^1}$. Let ${S\subset H^1(G)}$ be the subset of cohomology classes ${u}$ such that the representative can be chosen to be increasing along the flow. Then

Theorem 1

1. ${S}$ is a component of ${\Sigma G}$. It is a rational polyhedral cone.
2. For ${u\in S_{\mathbb Z}}$, inverse images of points are cross-sections ${\Gamma_u}$ of the flow. The first return map ${f_u}$ is an expanding irreducible train track representative of ${\phi_u:Q_u\rightarrow Q_u}$, with ${\lambda(f_u)=\lambda(\phi_u)}$.

Stretch factors ${\lambda(f_u)}$ form a nice function on ${S}$.

Theorem 2 (Algom-Kfir-Hironaka-Rafi) There exists an ${{\mathbb R}}$-analytic, convex function ${h:S\rightarrow{\mathbb R}}$ such that for all ${u\in S}$ such that for al ${u\in S}$ and ${t>0}$,

1. ${\lim_{u\rightarrow\partial S}h(u)=+\infty}$.
2. ${h(tu)=\frac{1}{t}h(u)}$.
3. If ${u\in S_Z}$, then ${h(u)=\log(\lambda(f_u))=\log(\lambda(\phi_u))}$.