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<ker(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))}.


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