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Relative Free Splitting Complexes III: Stable Translation Lengths and Filling Paths

Michael Handel, Lee Mosher

Abstract

This is the last of a three part work about relative free splitting complexes $\mathcal{FS}(Γ,\mathscr{A})$ and their actions by relative outer automorphism groups $\text{Out}(Γ;\mathscr{A})$. We obtain quantitative relations between the stable translation length $τ_φ$ and the relative train track dynamics of~$φ\in \Out(Γ;\A)$. First, if $φ$ has an orbit with diameter bounded below by a certain constant $Ω(Γ;\mathscr{A}) \ge 1$ then $φ$ has a filling attracting lamination. Also, there is a positive lower bound $τ_φ\ge A(Γ;\mathscr{A}) > 0$ amongst all $φ$ which have a filling attracting lamination. Both proofs rely on a study of \emph{filling paths} in a free splitting. These results are all new even for $\text{Out}(F_n)$.

Relative Free Splitting Complexes III: Stable Translation Lengths and Filling Paths

Abstract

This is the last of a three part work about relative free splitting complexes and their actions by relative outer automorphism groups . We obtain quantitative relations between the stable translation length and the relative train track dynamics of~. First, if has an orbit with diameter bounded below by a certain constant then has a filling attracting lamination. Also, there is a positive lower bound amongst all which have a filling attracting lamination. Both proofs rely on a study of \emph{filling paths} in a free splitting. These results are all new even for .

Paper Structure

This paper contains 59 sections, 29 theorems, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Strong Two Over All Theorem (Section \ref{['SectionSTOAT']}).
  3. The implication (4)$\implies$(1) of Theorem B (Section \ref{['SectionNoFillingLam']}).
  4. The lower bound of Theorem A (Section \ref{['SectionLowerBound']}).
  5. Filling paths and the Strong Two Over All Theorem
  6. Question:
  7. An example.
  8. Statement of the Strong Two Over All Theorem
  9. Terminology review.
  10. Filling paths
  11. The statement: noniterated and iterated forms
  12. A filling criterion for paths, expressed using protoforests
  13. Remarks.
  14. A continued example.
  15. A new example.
  16. ...and 44 more sections

Key Result

Theorem 2.4

For any group $\Gamma$ and any free factor system $\mathscr A$ there exists a constant $\Theta = \Theta(\Gamma;\mathscr A)$ such that for any foldable map $f \colon S \to T$ of free splittings of $\Gamma$ rel $\mathscr A$, if $d(S,T) \ge \Theta$ then there exist two natural edges $E_i \subset S$$(i=

Figures (3)

  • Figure 1: Letting $\mu=3\kappa+\omega$, assuming $<1$ free splitting unit between $T_{I-\mu p}$ and $T_I$, and setting $I=I_0$, $I_1=I-\kappa p$, and $I_2 = I - \mu p = I_1 - (2\kappa+\omega)p$, there is a collapse expand diagram as depicted with a pullback sequence $\beta_i \subset R_i$ of constant component complexity.
  • Figure 2: The Main Projection Diagram. This is a projection diagram from $R$ to $T_{I_2} \mapsto\cdots\mapsto T_{J_0}$ of maximal depth $\Delta$ satisfying $I_0 \le \Delta \le J_0$ (by the Restricted Projection Property). The portion of this diagram between column $I_2$ and column $I_0$ is a collapse expand diagram to which Lemma \ref{['LemmaUniformLiftingCrossing']} may be applied. The diagram also depicts the map $g_{I_0}$, which is one of the foldable maps $g_i \colon R_i \to R$ obtained by composing arrows in the $R$-row.
  • Figure 3: Associated to the monotonically nested tile sequence $\eta_0 \subset \ldots \subset \eta_m \subset \ldots$ is this matrix of monotonic relation sequences, such that in each column of relations one equation holds if and only if all equations hold (see Lemma \ref{['LemmaNestingOfFillingSupport']}). In the final row, the upper bound $\mathop{\mathrm{KR}}\nolimits(\Gamma;\mathscr A) = \left| \mathscr A \right| + \mathop{\mathrm{corank}}\nolimits(\mathscr A)$ on relative Kurosh ranks $\mathop{\mathrm{KR}}\nolimits(\eta_i)=\mathop{\mathrm{KR}}\nolimits(F_{\text{\!\tiny min}}(\eta_i))$ is found in HandelMosher:RelComplexHypII.

Theorems & Definitions (69)

  • proof : Proof of Theorem B
  • Definition 2.1: Lifting paths through collapse maps
  • Definition 2.2: Interior Crossings, and Filling Paths
  • Definition 2.3: Tiles
  • Theorem 2.4: Strong Two Over All Theorem (noniterated form)
  • Theorem 2.5: Strong Two Over All Theorem (iterated form)
  • proof : Proof of the iterated form, assuming the noniterated form
  • Definition 2.6: Protoforests in a free splitting
  • Definition 2.7: The overlap protoforest of a path in a free splitting
  • Definition 2.8: The filling support of a path in a free splitting
  • ...and 59 more